{"id":75,"date":"2016-04-21T22:43:46","date_gmt":"2016-04-21T22:43:46","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstats1xmaster\/?post_type=chapter&#038;p=75"},"modified":"2021-07-06T22:22:27","modified_gmt":"2021-07-06T22:22:27","slug":"histograms-frequency-polygons-and-time-series-graphs","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/frontrange-introstats1\/chapter\/histograms-frequency-polygons-and-time-series-graphs\/","title":{"raw":"Histograms, Frequency Polygons, and Time Series Graphs","rendered":"Histograms, Frequency Polygons, and Time Series Graphs"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul id=\"list123523\">\r\n \t<li>Display data graphically and interpret graphs: histograms, frequency polygons and time series graphs.<\/li>\r\n<\/ul>\r\n<\/div>\r\nFor most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets. A rule of thumb is to use a histogram when the data set consists of [latex]100[\/latex] values or more.\r\n\r\nA\u00a0<strong>histogram<\/strong> consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either <strong>frequency<\/strong> or <strong>relative frequency<\/strong> (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data.\r\n\r\nThe relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:\r\n<ul>\r\n \t<li>[latex]f[\/latex] = frequency<\/li>\r\n \t<li>[latex]n[\/latex] = total number of data values (or the sum of the individual frequencies), and<\/li>\r\n \t<li>[latex]RF[\/latex] = relative frequency,<\/li>\r\n<\/ul>\r\nthen [latex]\\displaystyle{R}{F}=\\frac{{f}}{{n}}[\/latex]\r\n\r\nFor example, if three students in Mr. Ahab's English class of [latex]40[\/latex] students received from [latex]90[\/latex]% to [latex]100[\/latex]%, then, [latex]\\displaystyle{f}={3},{n}={40}[\/latex], and [latex]{R}{F}=\\frac{{f}}{{n}}=\\frac{{3}}{{40}}={0.075}[\/latex]. [latex]7.5[\/latex]% of the students received [latex]90\u2013100[\/latex]%. [latex]90\u2013100[\/latex]% are quantitative measures.\r\n\r\n<strong>To construct a histogram<\/strong>, first decide how many <strong>bars<\/strong> or <strong>intervals<\/strong>, also called <strong>classes<\/strong>, represent the data. Many histograms consist of five to [latex]15[\/latex] bars or classes for clarity. The number of bars needs to be chosen. Choose a starting point for the first interval to be less than the smallest data value. A <strong>convenient starting point<\/strong> is a lower value carried out to one more decimal place than the value with the most decimal places. For example, if the value with the most decimal places is [latex]6.1[\/latex] and this is the smallest value, a convenient starting point is [latex]6.05[\/latex] ([latex]6.1 \u2013 0.05 = 6.05[\/latex]). We say that [latex]6.05[\/latex] has more precision. If the value with the most decimal places is [latex]2.23[\/latex] and the lowest value is [latex]1.5[\/latex], a convenient starting point is [latex]1.495[\/latex] ([latex]1.5 \u2013 0.005 = 1.495[\/latex]). If the value with the most decimal places is [latex]3.234[\/latex] and the lowest value is [latex]1.0[\/latex], a convenient starting point is [latex]0.9995[\/latex] ([latex]1.0 \u2013 0.0005 = 0.9995[\/latex]). If all the data happen to be integers and the smallest value is two, then a convenient starting point is [latex]1.5[\/latex] ([latex]2 \u2013 0.5 = 1.5[\/latex]). Also, when the starting point and other boundaries are carried to one additional decimal place, no data value will fall on a boundary. The next two examples go into detail about how to construct a histogram using continuous data and how to create a histogram using discrete data.\r\n\r\nWatch the following video for an example of how to draw a histogram.\r\n\r\nhttps:\/\/youtu.be\/4eLJGG2Ad30\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe following data are the heights (in inches to the nearest half inch) of [latex]100[\/latex] male semiprofessional soccer players. The heights are <strong>continuous<\/strong> data, since height is measured.\r\n\r\n[latex]60[\/latex]; [latex]60.5[\/latex]; [latex]61[\/latex]; [latex]61[\/latex]; [latex]61.5[\/latex]\r\n\r\n[latex]63.5[\/latex]; [latex]63.5[\/latex]; [latex]63.5[\/latex]\r\n\r\n[latex]64[\/latex]; [latex]64[\/latex]; [latex]64[\/latex]; [latex]64[\/latex]; [latex]64[\/latex]; [latex]64[\/latex]; [latex]64[\/latex]; [latex]64.5[\/latex]; [latex]64.5[\/latex]; [latex]64.5[\/latex]; [latex]64.5[\/latex]; [latex]64.5[\/latex]; [latex]64.5[\/latex]; [latex]64.5[\/latex]; [latex]64.566[\/latex]; [latex]66[\/latex]; [latex]66[\/latex]; [latex]66[\/latex]; [latex]66[\/latex]; [latex]66[\/latex]; [latex]66[\/latex]; [latex]66[\/latex]; [latex]66[\/latex]; [latex]66[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67.5[\/latex]; [latex]67.5[\/latex]; [latex]67.5[\/latex]; [latex]67.5[\/latex]; [latex]67.5[\/latex]; [latex]67.5[\/latex]; [latex]67.5[\/latex]\r\n\r\n[latex]68[\/latex]; [latex]68[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]69.5[\/latex]; [latex]69.5[\/latex]; [latex]69.5[\/latex]; [latex]69.5[\/latex]; [latex]69.5[\/latex]\r\n\r\n[latex]70[\/latex]; [latex]70[\/latex]; [latex]70[\/latex]; [latex]70[\/latex]; [latex]70[\/latex]; [latex]70[\/latex]; [latex]70.5[\/latex]; [latex]70.5[\/latex]; [latex]70.5[\/latex]; [latex]71[\/latex]; [latex]71[\/latex]; [latex]71[\/latex]\r\n\r\n[latex]72[\/latex]; [latex]72[\/latex]; [latex]72[\/latex]; [latex]72.5[\/latex]; [latex]72.5[\/latex]; [latex]73[\/latex]; [latex]73.5[\/latex]; [latex]74[\/latex]\r\n\r\nThe smallest data value is [latex]60[\/latex]. Since the data with the most decimal places has one decimal (for instance, [latex]61.5[\/latex]), we want our starting point to have two decimal places. Since the numbers [latex]0.5[\/latex], [latex]0.05[\/latex], [latex]0.005[\/latex], etc. are convenient numbers, use [latex]0.05[\/latex] and subtract it from [latex]60[\/latex], the smallest value, for the convenient starting point.\r\n\r\n[latex]60 \u2013 0.05 = 59.95[\/latex] which is more precise than, say, [latex]61.5[\/latex] by one decimal place. The starting point is, then, [latex]59.95[\/latex].\r\n\r\nThe largest value is [latex]74[\/latex], so [latex]74 + 0.05 = 74.05[\/latex] is the ending value.\r\n\r\nNext, calculate the width of each bar or class interval. To calculate this width, subtract the starting point from the ending value and divide by the number of bars (you must choose the number of bars you desire). Suppose you choose eight bars.\r\n\r\n<center>[latex]\\displaystyle\\frac{{{74.05}-{59.95}}}{{8}}={1.76}[\/latex]<\/center>\r\n<div class=\"textbox shaded\">\r\n<h3>Note<\/h3>\r\nWe will round up to two and make each bar or class interval two units wide. Rounding up to two is one way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using [latex]1.76[\/latex] as the width would also work. A guideline that is followed by some for the width of a bar or class interval is to take the square root of the number of data values and then round to the nearest whole number, if necessary. For example, if there are [latex]150[\/latex] values of data, take the square root of [latex]150[\/latex] and round to [latex]12[\/latex] bars or intervals.\r\n\r\n<\/div>\r\nThe boundaries are:\r\n<ul>\r\n \t<li>[latex]59.95[\/latex]<\/li>\r\n \t<li>[latex]59.95 + 2 = 61.95[\/latex]<\/li>\r\n \t<li>[latex]61.95 + 2 = 63.95[\/latex]<\/li>\r\n \t<li>[latex]63.95 + 2 = 65.95[\/latex]<\/li>\r\n \t<li>[latex]65.95 + 2 = 67.95[\/latex]<\/li>\r\n \t<li>[latex]67.95 + 2 = 69.95[\/latex]<\/li>\r\n \t<li>[latex]69.95 + 2 = 71.95[\/latex]<\/li>\r\n \t<li>[latex]71.95 + 2 = 73.95[\/latex]<\/li>\r\n \t<li>[latex]73.95 + 2 = 75.95[\/latex]<\/li>\r\n<\/ul>\r\nThe heights [latex]60[\/latex] through [latex]61.5[\/latex] inches are in the interval [latex]59.95\u201361.95[\/latex]. The heights that are [latex]63.5[\/latex] are in the interval [latex]61.95\u201363.95[\/latex]. The heights that are [latex]64[\/latex] through [latex]64.5[\/latex] are in the interval [latex]63.95\u201365.95[\/latex]. The heights [latex]66[\/latex] through [latex]67.5[\/latex] are in the interval [latex]65.95\u201367.95[\/latex]. The heights [latex]68[\/latex] through [latex]69.5[\/latex] are in the interval [latex]67.95\u201369.95[\/latex]. The heights [latex]70[\/latex] through [latex]71[\/latex] are in the interval [latex]69.95\u201371.95[\/latex]. The heights [latex]72[\/latex] through [latex]73.5[\/latex] are in the interval [latex]71.95\u201373.95[\/latex]. The height [latex]74[\/latex] is in the interval [latex]73.95\u201375.95[\/latex].\r\n\r\nWe can create a frequency distribution to organize this data using our class boundaries.\r\n\r\n<img class=\"alignnone size-full wp-image-2208\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5658\/2016\/04\/30172121\/Screenshot-2021-06-30-112046.png\" alt=\"freq dist soccer\" width=\"200\" height=\"186\" \/>\r\n\r\nThe following histogram displays the heights on the [latex]x[\/latex]-axis and frequency on the [latex]y[\/latex]-axis.\r\n\r\n<img class=\"size-full wp-image-2209 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5658\/2016\/04\/30172138\/Screenshot-2021-06-30-112105.png\" alt=\"histogram soccer\" width=\"618\" height=\"285\" \/>\r\n\r\nThe following histogram displays the heights on the [latex]x[\/latex]-axis and relative frequency on the [latex]y[\/latex]-axis.\r\n\r\n<center><img src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/i4ge-qvk23y6i#fixme#fixme#fixme\" alt=\"Histogram consists of 8 bars with the y-axis in increments of 0.05 from 0-0.4 and the x-axis in intervals of 2 from 59.95-75.95.\" \/><\/center><\/div>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>USING EXCEL<\/strong>\r\n\r\nTo construct a <strong>histogram from a grouped frequency distribution in Excel<\/strong>:\r\n<ol>\r\n \t<li>Select the cells containing the class boundaries and the frequency and select Insert.<\/li>\r\n \t<li>Choose 2D Column Graph which will generate a graph with the classes along the horizontal axis and frequency along the vertical axis.<\/li>\r\n \t<li>Visually, histograms differ from column\/bar graphs in that the bars touch. Edit the column graph by reducing the Gap Width to zero.<\/li>\r\n \t<li>Add in Axis Titles and an appropriate Chart Title.<\/li>\r\n<\/ol>\r\n<em>*Note, there is an Insert Histogram option, but that requires the Data Analysis Toolpak to create a valid histogram. If you already have the data grouped into a frequency distribution, this is the easiest method to create a valid histogram that matches the frequency distribution.<\/em>\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nThe following data are the shoe sizes of [latex]50[\/latex] male students. The sizes are continuous data since shoe size is measured. Construct a histogram and calculate the width of each bar or class interval. Suppose you choose six bars.\r\n\r\n[latex]9[\/latex]; [latex]9[\/latex]; [latex]9.5[\/latex]; [latex]9.5[\/latex]; [latex]10[\/latex]; [latex]10[\/latex]; [latex]10[\/latex]; [latex]10[\/latex]; [latex]10[\/latex]; [latex]10[\/latex]; [latex]10.5[\/latex]; [latex]10.5[\/latex]; [latex]10.5[\/latex]; [latex]10.5[\/latex]; [latex]10.5[\/latex]; [latex]10.5[\/latex]; [latex]10.5[\/latex]; [latex]10.5[\/latex]\r\n\r\n[latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11.5[\/latex]; [latex]11.5[\/latex]; [latex]11.5[\/latex]; [latex]11.5[\/latex]; [latex]11.5[\/latex]; [latex]11.5[\/latex]; [latex]11.5[\/latex]\r\n\r\n[latex]12[\/latex]; [latex]12[\/latex]; [latex]12[\/latex]; [latex]12[\/latex]; [latex]12[\/latex]; [latex]12[\/latex]; [latex]12[\/latex]; [latex]12.5[\/latex]; [latex]12.5[\/latex]; [latex]12.5[\/latex]; [latex]12.5[\/latex]; [latex]14[\/latex]\r\n\r\n[reveal-answer q=\"283391\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283391\"]\r\n\r\nSmallest value: [latex]9[\/latex]\r\n\r\nLargest value: [latex]14[\/latex]\r\n\r\nConvenient starting value: [latex]9 \u2013 0.05 = 8.95[\/latex]\r\n\r\nConvenient ending value: [latex]14 + 0.05 = 14.05[\/latex]\r\n\r\n[latex]\\displaystyle\\frac{{{14.05}-{8.95}}}{{6}}={0.85}[\/latex]\r\n\r\nThe calculations suggest using [latex]0.85[\/latex] as the width of each bar or class interval. You can also use an interval with a width equal to one.\u00a0 Starting with 8.95 and using a class width of [latex]1[\/latex], we get the following frequency distribution:\r\n\r\n<img class=\"alignnone size-full wp-image-2211\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5658\/2016\/04\/30173229\/Screenshot-2021-06-30-113217.png\" alt=\"freq dist\" width=\"183\" height=\"139\" \/>\r\n\r\nWhich gives us the following histogram with shoes sizes\u00a0on the [latex]x[\/latex]-axis and frequency on the [latex]y[\/latex]-axis.\r\n\r\n<img class=\"size-full wp-image-2212 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5658\/2016\/04\/30173420\/Screenshot-2021-06-30-113407.png\" alt=\"shoe size histogram\" width=\"469\" height=\"282\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nThe following data are the number of books bought by 50 part-time college students at ABC College. The number of books is <strong>discrete data<\/strong>, since books are counted.\r\n\r\n[latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]\r\n\r\n[latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]\r\n\r\n[latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]\r\n\r\n[latex]4[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]\r\n\r\n[latex]5[\/latex]; [latex]5[\/latex]; [latex]5[\/latex]; [latex]5[\/latex]; [latex]5[\/latex]\r\n\r\n[latex]6[\/latex]; [latex]6[\/latex]\r\n\r\nEleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books.\r\n\r\nBecause the data are integers, subtract [latex]0.5[\/latex] from [latex]1[\/latex], the smallest data value and add [latex]0.5[\/latex] to [latex]6[\/latex], the largest data value. Then the starting point is [latex]0.5[\/latex] and the ending value is [latex]6.5[\/latex].\r\n\r\nNext, calculate the width of each bar or class interval. If the data are discrete and there are not too many different values, a width that places the data values in the middle of the bar or class interval is the most convenient. Since the data consist of the numbers [latex]1[\/latex], [latex]2[\/latex], [latex]3[\/latex], [latex]4[\/latex], [latex]5[\/latex], [latex]6[\/latex], and the starting point is [latex]0.5[\/latex], a width of one places the [latex]1[\/latex] in the middle of the interval from [latex]0.5[\/latex] to [latex]1.5[\/latex], the [latex]2[\/latex] in the middle of the interval from [latex]1.5[\/latex] to [latex]2.5[\/latex], the [latex]3[\/latex] in the middle of the interval from [latex]2.5[\/latex] to [latex]3.5[\/latex], the [latex]4[\/latex] in the middle of the interval from _______ to _______, the [latex]5[\/latex] in the middle of the interval from _______ to _______, and the _______ in the middle of the interval from _______ to _______ .\r\n\r\n[reveal-answer q=\"283392\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283392\"]\r\n<ul>\r\n \t<li>3.5 to 4.5<\/li>\r\n \t<li>4.5 to 5.5<\/li>\r\n \t<li>6<\/li>\r\n \t<li>5.5 to 6.5<\/li>\r\n<\/ul>\r\nCalculate the number of bars as follows:\r\n\r\n<center>[latex]\\displaystyle\\frac{{{6.5}-{0.5}}}{{\\text{number of bars}}}={1}[\/latex]<\/center>where [latex]1[\/latex] is the width of a bar. Therefore, bars = [latex]6[\/latex].\r\n\r\nThe following histogram displays the number of books on the [latex]x[\/latex]-axis and the frequency on the [latex]y[\/latex]-axis.\r\n\r\n<img src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/rspb-i3l23y6i#fixme#fixme#fixme\" alt=\"Histogram consists of 6 bars with the y-axis in increments of 2 from 0-16 and the x-axis in intervals of 1 from 0.5-6.5.\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nThe following data are the number of sports played by 50 student athletes. The number of sports is discrete data since sports are counted.\r\n\r\n[latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]\r\n\r\n[latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]\r\n\r\n[latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]\r\n\r\n[latex]20[\/latex] student athletes play one sport. [latex]22[\/latex] student athletes play two sports. Eight student athletes play three sports.\r\n\r\n<em>Fill in the blanks for the following sentence.<\/em> Since the data consist of the numbers [latex]1[\/latex], [latex]2[\/latex], [latex]3[\/latex], and the starting point is [latex]0.5[\/latex], a width of one places the [latex]1[\/latex] in the middle of the interval [latex]0.5[\/latex] to _____, the [latex]2[\/latex] in the middle of the interval from _____ to _____, and the [latex]3[\/latex] in the middle of the interval from _____ to _____.\r\n\r\n[reveal-answer q=\"283393\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283393\"]\r\n[latex]1.5[\/latex]\r\n\r\n[latex]1.5[\/latex] to [latex]2.5[\/latex]\r\n\r\n[latex]2.5[\/latex] to [latex]3.5[\/latex]\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUsing this data set, construct a histogram.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th colspan=\"5\">Number of Hours My Classmates Spent Playing Video Games on Weekends<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]9.95[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]2.25[\/latex]<\/td>\r\n<td>[latex]16.75[\/latex]<\/td>\r\n<td>[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]19.5[\/latex]<\/td>\r\n<td>[latex]22.5[\/latex]<\/td>\r\n<td>[latex]7.5[\/latex]<\/td>\r\n<td>[latex]15[\/latex]<\/td>\r\n<td>[latex]12.75[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5.5[\/latex]<\/td>\r\n<td>[latex]11[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]20.75[\/latex]<\/td>\r\n<td>[latex]17.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]23[\/latex]<\/td>\r\n<td>[latex]21.9[\/latex]<\/td>\r\n<td>[latex]24[\/latex]<\/td>\r\n<td>[latex]23.75[\/latex]<\/td>\r\n<td>[latex]18[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]20[\/latex]<\/td>\r\n<td>[latex]15[\/latex]<\/td>\r\n<td>[latex]22.9[\/latex]<\/td>\r\n<td>[latex]18.8[\/latex]<\/td>\r\n<td>[latex]20.5[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"283394\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283394\"]\r\n\r\n<center><img src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/0r7f-h9l23y6i#fixme#fixme#fixme\" alt=\"This is a histogram that matches the supplied data. The x-axis consists of 5 bars in intervals of 5 from 0 to 25. The y-axis is marked in increments of 1 from 0 to 10. The x-axis shows the number of hours spent playing video games on the weekends, and the y-axis shows the number of students.\" \/><\/center>Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary, but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nThe following data represent the number of employees at various restaurants in New York City. Using this data, create a histogram.\r\n[latex]22[\/latex]; [latex]35[\/latex]; [latex]15[\/latex]; [latex]26[\/latex]; [latex]40[\/latex]; [latex]28[\/latex]; [latex]18[\/latex]; [latex]20[\/latex]; [latex]25[\/latex]; [latex]34[\/latex]; [latex]39[\/latex]; [latex]42[\/latex]; [latex]24[\/latex]; [latex]22[\/latex]; [latex]19[\/latex]; [latex]27[\/latex]; [latex]22[\/latex]; [latex]34[\/latex]; [latex]40[\/latex]; [latex]20[\/latex]; [latex]38[\/latex]; and [latex]28[\/latex]\r\nUse [latex]10\u201319[\/latex] as the first interval.\r\n\r\n[reveal-answer q=\"982488\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"982488\"]\r\n\r\n<img class=\"size-full wp-image-2166 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5658\/2016\/04\/28175529\/Capture.jpg\" alt=\"\" width=\"143\" height=\"102\" \/>\r\n\r\n<img class=\"size-full wp-image-2167 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5658\/2016\/04\/28175604\/histogram.jpg\" alt=\"histogram\" width=\"468\" height=\"282\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox tryit\"><header>\r\n<h3 class=\"title\" data-type=\"title\">COLLABORATIVE EXERCISE<\/h3>\r\nCount the money (bills and change) in your pocket or purse. Your instructor will record the amounts. As a class, construct a histogram displaying the data. Discuss how many intervals you think is appropriate. You may want to experiment with the number of intervals.\r\n\r\n<\/header><\/div>\r\n<h1 data-type=\"title\">Frequency Polygons<\/h1>\r\nFrequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons.\r\n\r\nTo construct a frequency polygon, first examine the data and decide on the number of intervals, or class intervals, to use on the [latex]x[\/latex]-axis and [latex]y[\/latex]-axis. After choosing the appropriate ranges, begin plotting the data points. After all the points are plotted, draw line segments to connect them.\r\n<div id=\"fs-idm37683552\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><header>\r\n<div class=\"title\" data-label-parent=\"\" data-type=\"title\">\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\n<div id=\"example4\" class=\"example\" data-type=\"example\">\r\n\r\nA frequency polygon was constructed from the frequency table below.\r\n<table id=\"fs-idp57619648\" summary=\"\">\r\n<thead>\r\n<tr>\r\n<th colspan=\"4\">Frequency Distribution for Calculus Final Test Scores<\/th>\r\n<\/tr>\r\n<tr>\r\n<th>Lower Bound<\/th>\r\n<th>Upper Bound<\/th>\r\n<th>Frequency<\/th>\r\n<th>Cumulative Frequency<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]49.5[\/latex]<\/td>\r\n<td>[latex]59.5[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]59.5[\/latex]<\/td>\r\n<td>[latex]69.5[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]15[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]69.5[\/latex]<\/td>\r\n<td>[latex]79.5[\/latex]<\/td>\r\n<td>[latex]30[\/latex]<\/td>\r\n<td>[latex]45[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]79.5[\/latex]<\/td>\r\n<td>[latex]89.5[\/latex]<\/td>\r\n<td>[latex]40[\/latex]<\/td>\r\n<td>[latex]85[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]89.5[\/latex]<\/td>\r\n<td>[latex]99.5[\/latex]<\/td>\r\n<td>[latex]15[\/latex]<\/td>\r\n<td>[latex]100[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n<figure id=\"eip-idm19499056\"><span id=\"fs-idm4822176\" data-type=\"media\" data-display=\"block\" data-alt=\"A frequency polygon was constructed from the frequency table below.\"> <img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214233\/CNX_Stats_C02_M05a_001.jpg\" alt=\"A frequency polygon was constructed from the frequency table below.\" width=\"350\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\r\n&nbsp;\r\n\r\nThe values 54.5, 64.5, 74.5, 84.5 and 94.5 are the <strong>midpoints<\/strong> between the lower and upper bounds for each class.\r\n\r\nThe first label on the [latex]x[\/latex]-axis is [latex]44.5[\/latex]. This represents an interval extending from [latex]39.5[\/latex] to [latex]49.5[\/latex]. Since the lowest test score is [latex]54.5[\/latex], this interval is used only to allow the graph to touch the [latex]x[\/latex]-axis. The point labeled [latex]54.5[\/latex] represents the next interval, or the first \u201creal\u201d interval from the table, and contains five scores. This reasoning is followed for each of the remaining intervals with the point [latex]104.5[\/latex] representing the interval from [latex]99.5[\/latex] to [latex]109.5[\/latex]. Again, this interval contains no data and is only used so that the graph will touch the [latex]x[\/latex]-axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.\r\n\r\n<\/div>\r\n<div id=\"fs-idm37683552\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<div id=\"fs-idm37683552\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><header>\r\n<div class=\"title\" data-label-parent=\"\" data-type=\"title\">\r\n<h3>Try It<\/h3>\r\n<\/div>\r\n<\/header>\r\n<div id=\"eip-idp12732512\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"eip-idp12732768\" class=\"problem\" data-type=\"problem\">\r\n\r\nConstruct a frequency polygon of U.S. Presidents\u2019 ages at inauguration shown in the table.\r\n<table id=\"fs-idp36852784\" summary=\"\">\r\n<thead>\r\n<tr>\r\n<th>Age at Inauguration<\/th>\r\n<th>Frequency<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]41.5\u201346.5[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]46.5\u201351.5[\/latex]<\/td>\r\n<td>[latex]11[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]51.5\u201356.5[\/latex]<\/td>\r\n<td>[latex]14[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]56.5\u201361.5[\/latex]<\/td>\r\n<td>[latex]9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]61.5\u201366.5[\/latex]<\/td>\r\n<td>[latex]4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]66.5\u201371.5[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"523753\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"523753\"]\r\n\r\n<img class=\"size-full wp-image-2161 alignleft\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5658\/2016\/04\/28174520\/frequency-polygon-table.jpg\" alt=\"frequency polygon table\" width=\"346\" height=\"203\" \/>\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n&nbsp;\r\n\r\n<img class=\"alignnone size-full wp-image-2170\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5658\/2016\/04\/28175804\/ages-frequency-polygon1.jpg\" alt=\"\" width=\"479\" height=\"285\" \/>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-idm37683552\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\r\n<div id=\"eip-idp12732512\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"eip-idp12732768\" class=\"problem\" data-type=\"problem\">\r\n\r\n<span style=\"font-size: 1rem; text-align: initial;\">[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nFrequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets.\r\n\r\n<\/div>\r\n<\/header><\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\n<div id=\"fs-idp21707856\" class=\"example\" data-type=\"example\">\r\n\r\nWe will construct an overlay frequency polygon comparing the scores\u00a0with the students\u2019 final numeric grade.\r\n<table id=\"fs-idm10950720\" summary=\"\">\r\n<thead>\r\n<tr>\r\n<th colspan=\"4\">Frequency Distribution for Calculus Final Test Scores<\/th>\r\n<\/tr>\r\n<tr>\r\n<th>Lower Bound<\/th>\r\n<th>Upper Bound<\/th>\r\n<th>Frequency<\/th>\r\n<th>Cumulative Frequency<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]49.5[\/latex]<\/td>\r\n<td>[latex]59.5[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]59.5[\/latex]<\/td>\r\n<td>[latex]69.5[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]15[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]69.5[\/latex]<\/td>\r\n<td>[latex]79.5[\/latex]<\/td>\r\n<td>[latex]30[\/latex]<\/td>\r\n<td>[latex]45[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]79.5[\/latex]<\/td>\r\n<td>[latex]89.5[\/latex]<\/td>\r\n<td>[latex]40[\/latex]<\/td>\r\n<td>[latex]85[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]89.5[\/latex]<\/td>\r\n<td>[latex]99.5[\/latex]<\/td>\r\n<td>[latex]15[\/latex]<\/td>\r\n<td>[latex]100[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"fs-idp39914624\" summary=\"\">\r\n<thead>\r\n<tr>\r\n<th colspan=\"4\">Frequency Distribution for Calculus Final Grades<\/th>\r\n<\/tr>\r\n<tr>\r\n<th>Lower Bound<\/th>\r\n<th>Upper Bound<\/th>\r\n<th>Frequency<\/th>\r\n<th>Cumulative Frequency<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]49.5[\/latex]<\/td>\r\n<td>[latex]59.5[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]59.5[\/latex]<\/td>\r\n<td>[latex]69.5[\/latex]<\/td>\r\n<td>[latex]10[\/latex]<\/td>\r\n<td>[latex]20[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]69.5[\/latex]<\/td>\r\n<td>[latex]79.5[\/latex]<\/td>\r\n<td>[latex]30[\/latex]<\/td>\r\n<td>[latex]50[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]79.5[\/latex]<\/td>\r\n<td>[latex]89.5[\/latex]<\/td>\r\n<td>[latex]45[\/latex]<\/td>\r\n<td>[latex]95[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]89.5[\/latex]<\/td>\r\n<td>[latex]99.5[\/latex]<\/td>\r\n<td>[latex]5[\/latex]<\/td>\r\n<td>[latex]100[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<figure id=\"eip-id1165746871888\"><span id=\"fs-idm24364960\" data-type=\"media\" data-display=\"block\" data-alt=\"This is an overlay frequency polygon that matches the supplied data. The x-axis shows the grades, and the y-axis shows the frequency.\"> <img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214235\/CNX_Stats_C02_M05a_002N.jpg\" alt=\"This is an overlay frequency polygon that matches the supplied data. The x-axis shows the grades, and the y-axis shows the frequency.\" width=\"350\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\r\n<\/div>\r\n<\/div>\r\nSuppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with this data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.\r\n\r\nOne feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don\u2018t have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph.\r\n<h1 data-type=\"title\">Constructing a Time Series Graph<\/h1>\r\nTo construct a time series graph, we must look at both pieces of our <strong>paired data set<\/strong>. We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\n<div id=\"fs-idp7658080\" class=\"example\" data-type=\"example\">\r\n<div id=\"fs-idp4161680\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-idm51003184\" class=\"problem\" data-type=\"problem\">\r\n\r\nThe following data shows the Annual Consumer Price Index, each month, for ten years. Construct a time series graph for the Annual Consumer Price Index data only.\r\n<table id=\"fs-idp49086512\" summary=\"\"><colgroup> <col data-align=\"center\" \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr valign=\"middle\">\r\n<th>Year<\/th>\r\n<th>Jan<\/th>\r\n<th>Feb<\/th>\r\n<th>Mar<\/th>\r\n<th>Apr<\/th>\r\n<th>May<\/th>\r\n<th>Jun<\/th>\r\n<th>Jul<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2003<\/strong><\/td>\r\n<td>[latex]181.7[\/latex]<\/td>\r\n<td>[latex]183.1[\/latex]<\/td>\r\n<td>[latex]184.2[\/latex]<\/td>\r\n<td>[latex]183.8[\/latex]<\/td>\r\n<td>[latex]183.5[\/latex]<\/td>\r\n<td>[latex]183.7[\/latex]<\/td>\r\n<td>[latex]183.9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2004<\/strong><\/td>\r\n<td>[latex]185.2[\/latex]<\/td>\r\n<td>[latex]186.2[\/latex]<\/td>\r\n<td>[latex]187.4[\/latex]<\/td>\r\n<td>[latex]188.0[\/latex]<\/td>\r\n<td>[latex]189.1[\/latex]<\/td>\r\n<td>[latex]189.7[\/latex]<\/td>\r\n<td>[latex]189.4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2005<\/strong><\/td>\r\n<td>[latex]190.7[\/latex]<\/td>\r\n<td>[latex]191.8[\/latex]<\/td>\r\n<td>[latex]193.3[\/latex]<\/td>\r\n<td>[latex]194.6[\/latex]<\/td>\r\n<td>[latex]194.4[\/latex]<\/td>\r\n<td>[latex]194.5[\/latex]<\/td>\r\n<td>[latex]195.4[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2006<\/strong><\/td>\r\n<td>[latex]198.3[\/latex]<\/td>\r\n<td>[latex]198.7[\/latex]<\/td>\r\n<td>[latex]199.8[\/latex]<\/td>\r\n<td>[latex]201.5[\/latex]<\/td>\r\n<td>[latex]202.5[\/latex]<\/td>\r\n<td>[latex]202.9[\/latex]<\/td>\r\n<td>[latex]203.5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2007<\/strong><\/td>\r\n<td>[latex]202.416[\/latex]<\/td>\r\n<td>[latex]203.499[\/latex]<\/td>\r\n<td>[latex]205.352[\/latex]<\/td>\r\n<td>[latex]206.686[\/latex]<\/td>\r\n<td>[latex]207.949[\/latex]<\/td>\r\n<td>[latex]208.352[\/latex]<\/td>\r\n<td>[latex]208.299[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2008<\/strong><\/td>\r\n<td>[latex]211.080[\/latex]<\/td>\r\n<td>[latex]211.693[\/latex]<\/td>\r\n<td>[latex]213.528[\/latex]<\/td>\r\n<td>[latex]214.823[\/latex]<\/td>\r\n<td>[latex]216.632[\/latex]<\/td>\r\n<td>[latex]218.815[\/latex]<\/td>\r\n<td>[latex]219.964[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2009<\/strong><\/td>\r\n<td>[latex]211.143[\/latex]<\/td>\r\n<td>[latex]212.193[\/latex]<\/td>\r\n<td>[latex]212.709[\/latex]<\/td>\r\n<td>[latex]213.240[\/latex]<\/td>\r\n<td>[latex]213.856[\/latex]<\/td>\r\n<td>[latex]215.693[\/latex]<\/td>\r\n<td>[latex]215.351[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2010<\/strong><\/td>\r\n<td>[latex]216.687[\/latex]<\/td>\r\n<td>[latex]216.741[\/latex]<\/td>\r\n<td>[latex]217.631[\/latex]<\/td>\r\n<td>[latex]218.009[\/latex]<\/td>\r\n<td>[latex]218.178[\/latex]<\/td>\r\n<td>[latex]217.965[\/latex]<\/td>\r\n<td>[latex]218.011[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2011<\/strong><\/td>\r\n<td>[latex]220.223[\/latex]<\/td>\r\n<td>[latex]221.309[\/latex]<\/td>\r\n<td>[latex]223.467[\/latex]<\/td>\r\n<td>[latex]224.906[\/latex]<\/td>\r\n<td>[latex]225.964[\/latex]<\/td>\r\n<td>[latex]225.722[\/latex]<\/td>\r\n<td>[latex]225.922[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2012<\/strong><\/td>\r\n<td>[latex]226.665[\/latex]<\/td>\r\n<td>[latex]227.663[\/latex]<\/td>\r\n<td>[latex]229.392[\/latex]<\/td>\r\n<td>[latex]230.085[\/latex]<\/td>\r\n<td>[latex]229.815[\/latex]<\/td>\r\n<td>[latex]229.478[\/latex]<\/td>\r\n<td>[latex]229.104[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"fs-idm48042992\" summary=\"\"><colgroup> <col data-align=\"center\" \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr>\r\n<th>Year<\/th>\r\n<th>Aug<\/th>\r\n<th>Sep<\/th>\r\n<th>Oct<\/th>\r\n<th>Nov<\/th>\r\n<th>Dec<\/th>\r\n<th>Annual<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2003<\/strong><\/td>\r\n<td>[latex]184.6[\/latex]<\/td>\r\n<td>[latex]185.2[\/latex]<\/td>\r\n<td>[latex]185.0[\/latex]<\/td>\r\n<td>[latex]184.5[\/latex]<\/td>\r\n<td>[latex]184.3[\/latex]<\/td>\r\n<td>[latex]184.0[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2004<\/strong><\/td>\r\n<td>[latex]189.5[\/latex]<\/td>\r\n<td>[latex]189.9[\/latex]<\/td>\r\n<td>[latex]190.9[\/latex]<\/td>\r\n<td>[latex]191.0[\/latex]<\/td>\r\n<td>[latex]190.3[\/latex]<\/td>\r\n<td>[latex]188.9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2005<\/strong><\/td>\r\n<td>[latex]196.4[\/latex]<\/td>\r\n<td>[latex]198.8[\/latex]<\/td>\r\n<td>[latex]199.2[\/latex]<\/td>\r\n<td>[latex]197.6[\/latex]<\/td>\r\n<td>[latex]196.8[\/latex]<\/td>\r\n<td>[latex]195.3[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2006<\/strong><\/td>\r\n<td>[latex]203.9[\/latex]<\/td>\r\n<td>[latex]202.9[\/latex]<\/td>\r\n<td>[latex]201.8[\/latex]<\/td>\r\n<td>[latex]201.5[\/latex]<\/td>\r\n<td>[latex]201.8[\/latex]<\/td>\r\n<td>[latex]201.6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2007<\/strong><\/td>\r\n<td>[latex]207.917[\/latex]<\/td>\r\n<td>[latex]208.490[\/latex]<\/td>\r\n<td>[latex]208.936[\/latex]<\/td>\r\n<td>[latex]210.177[\/latex]<\/td>\r\n<td>[latex]210.036[\/latex]<\/td>\r\n<td>[latex]207.342[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2008<\/strong><\/td>\r\n<td>[latex]219.086[\/latex]<\/td>\r\n<td>[latex]218.783[\/latex]<\/td>\r\n<td>[latex]216.573[\/latex]<\/td>\r\n<td>[latex]212.425[\/latex]<\/td>\r\n<td>[latex]210.228[\/latex]<\/td>\r\n<td>[latex]215.303[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2009<\/strong><\/td>\r\n<td>[latex]215.834[\/latex]<\/td>\r\n<td>[latex]215.969[\/latex]<\/td>\r\n<td>[latex]216.177[\/latex]<\/td>\r\n<td>[latex]216.330[\/latex]<\/td>\r\n<td>[latex]215.949[\/latex]<\/td>\r\n<td>[latex]214.537[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2010<\/strong><\/td>\r\n<td>[latex]218.312[\/latex]<\/td>\r\n<td>[latex]218.439[\/latex]<\/td>\r\n<td>[latex]218.711[\/latex]<\/td>\r\n<td>[latex]218.803[\/latex]<\/td>\r\n<td>[latex]219.179[\/latex]<\/td>\r\n<td>[latex]218.056[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2011<\/strong><\/td>\r\n<td>[latex]226.545[\/latex]<\/td>\r\n<td>[latex]226.889[\/latex]<\/td>\r\n<td>[latex]226.421[\/latex]<\/td>\r\n<td>[latex]226.230[\/latex]<\/td>\r\n<td>[latex]225.672[\/latex]<\/td>\r\n<td>[latex]224.939[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><strong data-effect=\"bold\">2012<\/strong><\/td>\r\n<td>[latex]230.379[\/latex]<\/td>\r\n<td>[latex]231.407[\/latex]<\/td>\r\n<td>[latex]231.317[\/latex]<\/td>\r\n<td>[latex]230.221[\/latex]<\/td>\r\n<td>[latex]229.601[\/latex]<\/td>\r\n<td>[latex]229.594[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"283400\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"283400\"]\r\n\r\n<center><img class=\"alignnone wp-image-1923\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21113533\/618b7e70623b342d3cd528c45b9972a816b698d2-300x116.jpg\" alt=\"This is the time series graph that matches the supplied data. The x-axis shows years from 2003 to 2012, and the y-axis shows the annual CPI\" width=\"503\" height=\"195\" \/><\/center>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-idm11029344\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><header>\r\n<div class=\"title\" data-label-parent=\"\" data-type=\"title\">\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n<header>\r\n<div class=\"title\" data-label-parent=\"\" data-type=\"title\"><\/div>\r\n<\/header>\r\n<div id=\"fs-idm10087584\" class=\"exercise\" data-type=\"exercise\">\r\n<div id=\"fs-idp2285280\" class=\"problem\" data-type=\"problem\">\r\n\r\nThe following table is a portion of a data set from www.worldbank.org. Use the table to construct a time series graph for CO<sub>2<\/sub> emissions for the United States.\u00a0 Units are kilotons (kt).\r\n<table id=\"fs-idp17543168\" summary=\"\">\r\n<thead>\r\n<tr>\r\n<th colspan=\"4\">CO2 Emissions<\/th>\r\n<\/tr>\r\n<tr>\r\n<th><\/th>\r\n<th>Ukraine<\/th>\r\n<th>United Kingdom<\/th>\r\n<th>United States<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>2003<\/td>\r\n<td>[latex]352,259[\/latex]<\/td>\r\n<td>[latex]540,640[\/latex]<\/td>\r\n<td>[latex]5,681,664[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2004<\/td>\r\n<td>[latex]343,121[\/latex]<\/td>\r\n<td>[latex]540,409[\/latex]<\/td>\r\n<td>[latex]5,790,761[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2005<\/td>\r\n<td>[latex]339,029[\/latex]<\/td>\r\n<td>[latex]541,990[\/latex]<\/td>\r\n<td>[latex]5,826,394[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2006<\/td>\r\n<td>[latex]327,797[\/latex]<\/td>\r\n<td>[latex]542,045[\/latex]<\/td>\r\n<td>[latex]5,737,615[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2007<\/td>\r\n<td>[latex]328,357[\/latex]<\/td>\r\n<td>[latex]528,631[\/latex]<\/td>\r\n<td>[latex]5,828,697[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2008<\/td>\r\n<td>[latex]323,657[\/latex]<\/td>\r\n<td>[latex]522,247[\/latex]<\/td>\r\n<td>[latex]5,656,839[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>2009<\/td>\r\n<td>[latex]272,176[\/latex]<\/td>\r\n<td>[latex]474,579[\/latex]<\/td>\r\n<td>[latex]5,299,563[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[reveal-answer q=\"866433\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"866433\"]\r\n\r\n<img class=\"alignnone size-full wp-image-2159 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5658\/2016\/04\/28173608\/CO2-time-series.jpg\" alt=\"CO2 time series\" width=\"686\" height=\"372\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/header><\/div>\r\n<h2 data-type=\"title\">Uses of a Time Series Graph<\/h2>\r\nTime series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot.\r\n<h2>Concept Review<\/h2>\r\nA <strong>histogram<\/strong> is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets.\r\n\r\nA <strong>frequency polygon<\/strong> can also be used when graphing large data sets with data points that repeat. The data usually goes on [latex]y[\/latex]-axis with the frequency being graphed on the [latex]x[\/latex]-axis.\r\n\r\n<strong>Time series graphs<\/strong> can be helpful when looking at large amounts of data for one variable over a period of time.\r\n<h1 data-type=\"title\">References<\/h1>\r\nData on annual homicides in Detroit, 1961\u201373, from Gunst &amp; Mason\u2019s book \u2018Regression Analysis and its Application\u2019, Marcel Dekker\r\n\r\n\u201cTimeline: Guide to the U.S. Presidents: Information on every president\u2019s birthplace, political party, term of office, and more.\u201d Scholastic, 2013. Available online at http:\/\/www.scholastic.com\/teachers\/article\/timeline-guide-us-presidents (accessed April 3, 2013).\r\n\r\n\u201cPresidents.\u201d Fact Monster. Pearson Education, 2007. Available online at http:\/\/www.factmonster.com\/ipka\/A0194030.html (accessed April 3, 2013).\r\n\r\n\u201cFood Security Statistics.\u201d Food and Agriculture Organization of the United Nations. Available online at http:\/\/www.fao.org\/economic\/ess\/ess-fs\/en\/ (accessed April 3, 2013).\r\n\r\n\u201cConsumer Price Index.\u201d United States Department of Labor: Bureau of Labor Statistics. Available online at http:\/\/data.bls.gov\/pdq\/SurveyOutputServlet (accessed April 3, 2013).\r\n\r\n\u201cCO2 emissions (kt).\u201d The World Bank, 2013. Available online at http:\/\/databank.worldbank.org\/data\/home.aspx (accessed April 3, 2013).\r\n\r\n\u201cBirths Time Series Data.\u201d General Register Office For Scotland, 2013. Available online at http:\/\/www.gro-scotland.gov.uk\/statistics\/theme\/vital-events\/births\/time-series.html (accessed April 3, 2013).\r\n\r\n\u201cDemographics: Children under the age of 5 years underweight.\u201d Indexmundi. Available online at http:\/\/www.indexmundi.com\/g\/r.aspx?t=50&amp;v=2224&amp;aml=en (accessed April 3, 2013).\r\n\r\nGunst, Richard, Robert Mason. <em data-effect=\"italics\">Regression Analysis and Its Application: A Data-Oriented Approach<\/em>. CRC Press: 1980.\r\n\r\n\u201cOverweight and Obesity: Adult Obesity Facts.\u201d Centers for Disease Control and Prevention. Available online at http:\/\/www.cdc.gov\/obesity\/data\/adult.html (accessed September 13, 2013).","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul id=\"list123523\">\n<li>Display data graphically and interpret graphs: histograms, frequency polygons and time series graphs.<\/li>\n<\/ul>\n<\/div>\n<p>For most of the work you do in this book, you will use a histogram to display the data. One advantage of a histogram is that it can readily display large data sets. A rule of thumb is to use a histogram when the data set consists of [latex]100[\/latex] values or more.<\/p>\n<p>A\u00a0<strong>histogram<\/strong> consists of contiguous (adjoining) boxes. It has both a horizontal axis and a vertical axis. The horizontal axis is labeled with what the data represents (for instance, distance from your home to school). The vertical axis is labeled either <strong>frequency<\/strong> or <strong>relative frequency<\/strong> (or percent frequency or probability). The graph will have the same shape with either label. The histogram (like the stemplot) can give you the shape of the data, the center, and the spread of the data.<\/p>\n<p>The relative frequency is equal to the frequency for an observed value of the data divided by the total number of data values in the sample. (Remember, frequency is defined as the number of times an answer occurs.) If:<\/p>\n<ul>\n<li>[latex]f[\/latex] = frequency<\/li>\n<li>[latex]n[\/latex] = total number of data values (or the sum of the individual frequencies), and<\/li>\n<li>[latex]RF[\/latex] = relative frequency,<\/li>\n<\/ul>\n<p>then [latex]\\displaystyle{R}{F}=\\frac{{f}}{{n}}[\/latex]<\/p>\n<p>For example, if three students in Mr. Ahab&#8217;s English class of [latex]40[\/latex] students received from [latex]90[\/latex]% to [latex]100[\/latex]%, then, [latex]\\displaystyle{f}={3},{n}={40}[\/latex], and [latex]{R}{F}=\\frac{{f}}{{n}}=\\frac{{3}}{{40}}={0.075}[\/latex]. [latex]7.5[\/latex]% of the students received [latex]90\u2013100[\/latex]%. [latex]90\u2013100[\/latex]% are quantitative measures.<\/p>\n<p><strong>To construct a histogram<\/strong>, first decide how many <strong>bars<\/strong> or <strong>intervals<\/strong>, also called <strong>classes<\/strong>, represent the data. Many histograms consist of five to [latex]15[\/latex] bars or classes for clarity. The number of bars needs to be chosen. Choose a starting point for the first interval to be less than the smallest data value. A <strong>convenient starting point<\/strong> is a lower value carried out to one more decimal place than the value with the most decimal places. For example, if the value with the most decimal places is [latex]6.1[\/latex] and this is the smallest value, a convenient starting point is [latex]6.05[\/latex] ([latex]6.1 \u2013 0.05 = 6.05[\/latex]). We say that [latex]6.05[\/latex] has more precision. If the value with the most decimal places is [latex]2.23[\/latex] and the lowest value is [latex]1.5[\/latex], a convenient starting point is [latex]1.495[\/latex] ([latex]1.5 \u2013 0.005 = 1.495[\/latex]). If the value with the most decimal places is [latex]3.234[\/latex] and the lowest value is [latex]1.0[\/latex], a convenient starting point is [latex]0.9995[\/latex] ([latex]1.0 \u2013 0.0005 = 0.9995[\/latex]). If all the data happen to be integers and the smallest value is two, then a convenient starting point is [latex]1.5[\/latex] ([latex]2 \u2013 0.5 = 1.5[\/latex]). Also, when the starting point and other boundaries are carried to one additional decimal place, no data value will fall on a boundary. The next two examples go into detail about how to construct a histogram using continuous data and how to create a histogram using discrete data.<\/p>\n<p>Watch the following video for an example of how to draw a histogram.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Histograms | Applying mathematical reasoning | Pre-Algebra | Khan Academy\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/4eLJGG2Ad30?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The following data are the heights (in inches to the nearest half inch) of [latex]100[\/latex] male semiprofessional soccer players. The heights are <strong>continuous<\/strong> data, since height is measured.<\/p>\n<p>[latex]60[\/latex]; [latex]60.5[\/latex]; [latex]61[\/latex]; [latex]61[\/latex]; [latex]61.5[\/latex]<\/p>\n<p>[latex]63.5[\/latex]; [latex]63.5[\/latex]; [latex]63.5[\/latex]<\/p>\n<p>[latex]64[\/latex]; [latex]64[\/latex]; [latex]64[\/latex]; [latex]64[\/latex]; [latex]64[\/latex]; [latex]64[\/latex]; [latex]64[\/latex]; [latex]64.5[\/latex]; [latex]64.5[\/latex]; [latex]64.5[\/latex]; [latex]64.5[\/latex]; [latex]64.5[\/latex]; [latex]64.5[\/latex]; [latex]64.5[\/latex]; [latex]64.566[\/latex]; [latex]66[\/latex]; [latex]66[\/latex]; [latex]66[\/latex]; [latex]66[\/latex]; [latex]66[\/latex]; [latex]66[\/latex]; [latex]66[\/latex]; [latex]66[\/latex]; [latex]66[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]66.5[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67[\/latex]; [latex]67.5[\/latex]; [latex]67.5[\/latex]; [latex]67.5[\/latex]; [latex]67.5[\/latex]; [latex]67.5[\/latex]; [latex]67.5[\/latex]; [latex]67.5[\/latex]<\/p>\n<p>[latex]68[\/latex]; [latex]68[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]69[\/latex]; [latex]69.5[\/latex]; [latex]69.5[\/latex]; [latex]69.5[\/latex]; [latex]69.5[\/latex]; [latex]69.5[\/latex]<\/p>\n<p>[latex]70[\/latex]; [latex]70[\/latex]; [latex]70[\/latex]; [latex]70[\/latex]; [latex]70[\/latex]; [latex]70[\/latex]; [latex]70.5[\/latex]; [latex]70.5[\/latex]; [latex]70.5[\/latex]; [latex]71[\/latex]; [latex]71[\/latex]; [latex]71[\/latex]<\/p>\n<p>[latex]72[\/latex]; [latex]72[\/latex]; [latex]72[\/latex]; [latex]72.5[\/latex]; [latex]72.5[\/latex]; [latex]73[\/latex]; [latex]73.5[\/latex]; [latex]74[\/latex]<\/p>\n<p>The smallest data value is [latex]60[\/latex]. Since the data with the most decimal places has one decimal (for instance, [latex]61.5[\/latex]), we want our starting point to have two decimal places. Since the numbers [latex]0.5[\/latex], [latex]0.05[\/latex], [latex]0.005[\/latex], etc. are convenient numbers, use [latex]0.05[\/latex] and subtract it from [latex]60[\/latex], the smallest value, for the convenient starting point.<\/p>\n<p>[latex]60 \u2013 0.05 = 59.95[\/latex] which is more precise than, say, [latex]61.5[\/latex] by one decimal place. The starting point is, then, [latex]59.95[\/latex].<\/p>\n<p>The largest value is [latex]74[\/latex], so [latex]74 + 0.05 = 74.05[\/latex] is the ending value.<\/p>\n<p>Next, calculate the width of each bar or class interval. To calculate this width, subtract the starting point from the ending value and divide by the number of bars (you must choose the number of bars you desire). Suppose you choose eight bars.<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle\\frac{{{74.05}-{59.95}}}{{8}}={1.76}[\/latex]<\/div>\n<div class=\"textbox shaded\">\n<h3>Note<\/h3>\n<p>We will round up to two and make each bar or class interval two units wide. Rounding up to two is one way to prevent a value from falling on a boundary. Rounding to the next number is often necessary even if it goes against the standard rules of rounding. For this example, using [latex]1.76[\/latex] as the width would also work. A guideline that is followed by some for the width of a bar or class interval is to take the square root of the number of data values and then round to the nearest whole number, if necessary. For example, if there are [latex]150[\/latex] values of data, take the square root of [latex]150[\/latex] and round to [latex]12[\/latex] bars or intervals.<\/p>\n<\/div>\n<p>The boundaries are:<\/p>\n<ul>\n<li>[latex]59.95[\/latex]<\/li>\n<li>[latex]59.95 + 2 = 61.95[\/latex]<\/li>\n<li>[latex]61.95 + 2 = 63.95[\/latex]<\/li>\n<li>[latex]63.95 + 2 = 65.95[\/latex]<\/li>\n<li>[latex]65.95 + 2 = 67.95[\/latex]<\/li>\n<li>[latex]67.95 + 2 = 69.95[\/latex]<\/li>\n<li>[latex]69.95 + 2 = 71.95[\/latex]<\/li>\n<li>[latex]71.95 + 2 = 73.95[\/latex]<\/li>\n<li>[latex]73.95 + 2 = 75.95[\/latex]<\/li>\n<\/ul>\n<p>The heights [latex]60[\/latex] through [latex]61.5[\/latex] inches are in the interval [latex]59.95\u201361.95[\/latex]. The heights that are [latex]63.5[\/latex] are in the interval [latex]61.95\u201363.95[\/latex]. The heights that are [latex]64[\/latex] through [latex]64.5[\/latex] are in the interval [latex]63.95\u201365.95[\/latex]. The heights [latex]66[\/latex] through [latex]67.5[\/latex] are in the interval [latex]65.95\u201367.95[\/latex]. The heights [latex]68[\/latex] through [latex]69.5[\/latex] are in the interval [latex]67.95\u201369.95[\/latex]. The heights [latex]70[\/latex] through [latex]71[\/latex] are in the interval [latex]69.95\u201371.95[\/latex]. The heights [latex]72[\/latex] through [latex]73.5[\/latex] are in the interval [latex]71.95\u201373.95[\/latex]. The height [latex]74[\/latex] is in the interval [latex]73.95\u201375.95[\/latex].<\/p>\n<p>We can create a frequency distribution to organize this data using our class boundaries.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-2208\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5658\/2016\/04\/30172121\/Screenshot-2021-06-30-112046.png\" alt=\"freq dist soccer\" width=\"200\" height=\"186\" \/><\/p>\n<p>The following histogram displays the heights on the [latex]x[\/latex]-axis and frequency on the [latex]y[\/latex]-axis.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2209 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5658\/2016\/04\/30172138\/Screenshot-2021-06-30-112105.png\" alt=\"histogram soccer\" width=\"618\" height=\"285\" \/><\/p>\n<p>The following histogram displays the heights on the [latex]x[\/latex]-axis and relative frequency on the [latex]y[\/latex]-axis.<\/p>\n<div style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/i4ge-qvk23y6i#fixme#fixme#fixme\" alt=\"Histogram consists of 8 bars with the y-axis in increments of 0.05 from 0-0.4 and the x-axis in intervals of 2 from 59.95-75.95.\" \/><\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<p><strong>USING EXCEL<\/strong><\/p>\n<p>To construct a <strong>histogram from a grouped frequency distribution in Excel<\/strong>:<\/p>\n<ol>\n<li>Select the cells containing the class boundaries and the frequency and select Insert.<\/li>\n<li>Choose 2D Column Graph which will generate a graph with the classes along the horizontal axis and frequency along the vertical axis.<\/li>\n<li>Visually, histograms differ from column\/bar graphs in that the bars touch. Edit the column graph by reducing the Gap Width to zero.<\/li>\n<li>Add in Axis Titles and an appropriate Chart Title.<\/li>\n<\/ol>\n<p><em>*Note, there is an Insert Histogram option, but that requires the Data Analysis Toolpak to create a valid histogram. If you already have the data grouped into a frequency distribution, this is the easiest method to create a valid histogram that matches the frequency distribution.<\/em><\/p>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>The following data are the shoe sizes of [latex]50[\/latex] male students. The sizes are continuous data since shoe size is measured. Construct a histogram and calculate the width of each bar or class interval. Suppose you choose six bars.<\/p>\n<p>[latex]9[\/latex]; [latex]9[\/latex]; [latex]9.5[\/latex]; [latex]9.5[\/latex]; [latex]10[\/latex]; [latex]10[\/latex]; [latex]10[\/latex]; [latex]10[\/latex]; [latex]10[\/latex]; [latex]10[\/latex]; [latex]10.5[\/latex]; [latex]10.5[\/latex]; [latex]10.5[\/latex]; [latex]10.5[\/latex]; [latex]10.5[\/latex]; [latex]10.5[\/latex]; [latex]10.5[\/latex]; [latex]10.5[\/latex]<\/p>\n<p>[latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11[\/latex]; [latex]11.5[\/latex]; [latex]11.5[\/latex]; [latex]11.5[\/latex]; [latex]11.5[\/latex]; [latex]11.5[\/latex]; [latex]11.5[\/latex]; [latex]11.5[\/latex]<\/p>\n<p>[latex]12[\/latex]; [latex]12[\/latex]; [latex]12[\/latex]; [latex]12[\/latex]; [latex]12[\/latex]; [latex]12[\/latex]; [latex]12[\/latex]; [latex]12.5[\/latex]; [latex]12.5[\/latex]; [latex]12.5[\/latex]; [latex]12.5[\/latex]; [latex]14[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283391\">Show Solution<\/span><\/p>\n<div id=\"q283391\" class=\"hidden-answer\" style=\"display: none\">\n<p>Smallest value: [latex]9[\/latex]<\/p>\n<p>Largest value: [latex]14[\/latex]<\/p>\n<p>Convenient starting value: [latex]9 \u2013 0.05 = 8.95[\/latex]<\/p>\n<p>Convenient ending value: [latex]14 + 0.05 = 14.05[\/latex]<\/p>\n<p>[latex]\\displaystyle\\frac{{{14.05}-{8.95}}}{{6}}={0.85}[\/latex]<\/p>\n<p>The calculations suggest using [latex]0.85[\/latex] as the width of each bar or class interval. You can also use an interval with a width equal to one.\u00a0 Starting with 8.95 and using a class width of [latex]1[\/latex], we get the following frequency distribution:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-2211\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5658\/2016\/04\/30173229\/Screenshot-2021-06-30-113217.png\" alt=\"freq dist\" width=\"183\" height=\"139\" \/><\/p>\n<p>Which gives us the following histogram with shoes sizes\u00a0on the [latex]x[\/latex]-axis and frequency on the [latex]y[\/latex]-axis.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2212 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5658\/2016\/04\/30173420\/Screenshot-2021-06-30-113407.png\" alt=\"shoe size histogram\" width=\"469\" height=\"282\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>The following data are the number of books bought by 50 part-time college students at ABC College. The number of books is <strong>discrete data<\/strong>, since books are counted.<\/p>\n<p>[latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]<\/p>\n<p>[latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]<\/p>\n<p>[latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]<\/p>\n<p>[latex]4[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]; [latex]4[\/latex]<\/p>\n<p>[latex]5[\/latex]; [latex]5[\/latex]; [latex]5[\/latex]; [latex]5[\/latex]; [latex]5[\/latex]<\/p>\n<p>[latex]6[\/latex]; [latex]6[\/latex]<\/p>\n<p>Eleven students buy one book. Ten students buy two books. Sixteen students buy three books. Six students buy four books. Five students buy five books. Two students buy six books.<\/p>\n<p>Because the data are integers, subtract [latex]0.5[\/latex] from [latex]1[\/latex], the smallest data value and add [latex]0.5[\/latex] to [latex]6[\/latex], the largest data value. Then the starting point is [latex]0.5[\/latex] and the ending value is [latex]6.5[\/latex].<\/p>\n<p>Next, calculate the width of each bar or class interval. If the data are discrete and there are not too many different values, a width that places the data values in the middle of the bar or class interval is the most convenient. Since the data consist of the numbers [latex]1[\/latex], [latex]2[\/latex], [latex]3[\/latex], [latex]4[\/latex], [latex]5[\/latex], [latex]6[\/latex], and the starting point is [latex]0.5[\/latex], a width of one places the [latex]1[\/latex] in the middle of the interval from [latex]0.5[\/latex] to [latex]1.5[\/latex], the [latex]2[\/latex] in the middle of the interval from [latex]1.5[\/latex] to [latex]2.5[\/latex], the [latex]3[\/latex] in the middle of the interval from [latex]2.5[\/latex] to [latex]3.5[\/latex], the [latex]4[\/latex] in the middle of the interval from _______ to _______, the [latex]5[\/latex] in the middle of the interval from _______ to _______, and the _______ in the middle of the interval from _______ to _______ .<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283392\">Show Solution<\/span><\/p>\n<div id=\"q283392\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li>3.5 to 4.5<\/li>\n<li>4.5 to 5.5<\/li>\n<li>6<\/li>\n<li>5.5 to 6.5<\/li>\n<\/ul>\n<p>Calculate the number of bars as follows:<\/p>\n<div style=\"text-align: center;\">[latex]\\displaystyle\\frac{{{6.5}-{0.5}}}{{\\text{number of bars}}}={1}[\/latex]<\/div>\n<p>where [latex]1[\/latex] is the width of a bar. Therefore, bars = [latex]6[\/latex].<\/p>\n<p>The following histogram displays the number of books on the [latex]x[\/latex]-axis and the frequency on the [latex]y[\/latex]-axis.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/rspb-i3l23y6i#fixme#fixme#fixme\" alt=\"Histogram consists of 6 bars with the y-axis in increments of 2 from 0-16 and the x-axis in intervals of 1 from 0.5-6.5.\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>The following data are the number of sports played by 50 student athletes. The number of sports is discrete data since sports are counted.<\/p>\n<p>[latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]; [latex]1[\/latex]<\/p>\n<p>[latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]; [latex]2[\/latex]<\/p>\n<p>[latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]; [latex]3[\/latex]<\/p>\n<p>[latex]20[\/latex] student athletes play one sport. [latex]22[\/latex] student athletes play two sports. Eight student athletes play three sports.<\/p>\n<p><em>Fill in the blanks for the following sentence.<\/em> Since the data consist of the numbers [latex]1[\/latex], [latex]2[\/latex], [latex]3[\/latex], and the starting point is [latex]0.5[\/latex], a width of one places the [latex]1[\/latex] in the middle of the interval [latex]0.5[\/latex] to _____, the [latex]2[\/latex] in the middle of the interval from _____ to _____, and the [latex]3[\/latex] in the middle of the interval from _____ to _____.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283393\">Show Solution<\/span><\/p>\n<div id=\"q283393\" class=\"hidden-answer\" style=\"display: none\">\n[latex]1.5[\/latex]<\/p>\n<p>[latex]1.5[\/latex] to [latex]2.5[\/latex]<\/p>\n<p>[latex]2.5[\/latex] to [latex]3.5[\/latex]\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Using this data set, construct a histogram.<\/p>\n<table>\n<thead>\n<tr>\n<th colspan=\"5\">Number of Hours My Classmates Spent Playing Video Games on Weekends<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]9.95[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]2.25[\/latex]<\/td>\n<td>[latex]16.75[\/latex]<\/td>\n<td>[latex]0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]19.5[\/latex]<\/td>\n<td>[latex]22.5[\/latex]<\/td>\n<td>[latex]7.5[\/latex]<\/td>\n<td>[latex]15[\/latex]<\/td>\n<td>[latex]12.75[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5.5[\/latex]<\/td>\n<td>[latex]11[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]20.75[\/latex]<\/td>\n<td>[latex]17.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]23[\/latex]<\/td>\n<td>[latex]21.9[\/latex]<\/td>\n<td>[latex]24[\/latex]<\/td>\n<td>[latex]23.75[\/latex]<\/td>\n<td>[latex]18[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]20[\/latex]<\/td>\n<td>[latex]15[\/latex]<\/td>\n<td>[latex]22.9[\/latex]<\/td>\n<td>[latex]18.8[\/latex]<\/td>\n<td>[latex]20.5[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283394\">Show Solution<\/span><\/p>\n<div id=\"q283394\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\"><img decoding=\"async\" src=\"https:\/\/textimgs.s3.amazonaws.com\/DE\/stats\/0r7f-h9l23y6i#fixme#fixme#fixme\" alt=\"This is a histogram that matches the supplied data. The x-axis consists of 5 bars in intervals of 5 from 0 to 25. The y-axis is marked in increments of 1 from 0 to 10. The x-axis shows the number of hours spent playing video games on the weekends, and the y-axis shows the number of students.\" \/><\/div>\n<p>Some values in this data set fall on boundaries for the class intervals. A value is counted in a class interval if it falls on the left boundary, but not if it falls on the right boundary. Different researchers may set up histograms for the same data in different ways. There is more than one correct way to set up a histogram.\n<\/p><\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>The following data represent the number of employees at various restaurants in New York City. Using this data, create a histogram.<br \/>\n[latex]22[\/latex]; [latex]35[\/latex]; [latex]15[\/latex]; [latex]26[\/latex]; [latex]40[\/latex]; [latex]28[\/latex]; [latex]18[\/latex]; [latex]20[\/latex]; [latex]25[\/latex]; [latex]34[\/latex]; [latex]39[\/latex]; [latex]42[\/latex]; [latex]24[\/latex]; [latex]22[\/latex]; [latex]19[\/latex]; [latex]27[\/latex]; [latex]22[\/latex]; [latex]34[\/latex]; [latex]40[\/latex]; [latex]20[\/latex]; [latex]38[\/latex]; and [latex]28[\/latex]<br \/>\nUse [latex]10\u201319[\/latex] as the first interval.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q982488\">Show Answer<\/span><\/p>\n<div id=\"q982488\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2166 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5658\/2016\/04\/28175529\/Capture.jpg\" alt=\"\" width=\"143\" height=\"102\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2167 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5658\/2016\/04\/28175604\/histogram.jpg\" alt=\"histogram\" width=\"468\" height=\"282\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox tryit\">\n<header>\n<h3 class=\"title\" data-type=\"title\">COLLABORATIVE EXERCISE<\/h3>\n<p>Count the money (bills and change) in your pocket or purse. Your instructor will record the amounts. As a class, construct a histogram displaying the data. Discuss how many intervals you think is appropriate. You may want to experiment with the number of intervals.<\/p>\n<\/header>\n<\/div>\n<h1 data-type=\"title\">Frequency Polygons<\/h1>\n<p>Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons.<\/p>\n<p>To construct a frequency polygon, first examine the data and decide on the number of intervals, or class intervals, to use on the [latex]x[\/latex]-axis and [latex]y[\/latex]-axis. After choosing the appropriate ranges, begin plotting the data points. After all the points are plotted, draw line segments to connect them.<\/p>\n<div id=\"fs-idm37683552\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<header>\n<div class=\"title\" data-label-parent=\"\" data-type=\"title\">\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<div id=\"example4\" class=\"example\" data-type=\"example\">\n<p>A frequency polygon was constructed from the frequency table below.<\/p>\n<table id=\"fs-idp57619648\" summary=\"\">\n<thead>\n<tr>\n<th colspan=\"4\">Frequency Distribution for Calculus Final Test Scores<\/th>\n<\/tr>\n<tr>\n<th>Lower Bound<\/th>\n<th>Upper Bound<\/th>\n<th>Frequency<\/th>\n<th>Cumulative Frequency<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]49.5[\/latex]<\/td>\n<td>[latex]59.5[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]59.5[\/latex]<\/td>\n<td>[latex]69.5[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]15[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]69.5[\/latex]<\/td>\n<td>[latex]79.5[\/latex]<\/td>\n<td>[latex]30[\/latex]<\/td>\n<td>[latex]45[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]79.5[\/latex]<\/td>\n<td>[latex]89.5[\/latex]<\/td>\n<td>[latex]40[\/latex]<\/td>\n<td>[latex]85[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]89.5[\/latex]<\/td>\n<td>[latex]99.5[\/latex]<\/td>\n<td>[latex]15[\/latex]<\/td>\n<td>[latex]100[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<figure id=\"eip-idm19499056\"><span id=\"fs-idm4822176\" data-type=\"media\" data-display=\"block\" data-alt=\"A frequency polygon was constructed from the frequency table below.\"> <img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214233\/CNX_Stats_C02_M05a_001.jpg\" alt=\"A frequency polygon was constructed from the frequency table below.\" width=\"350\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\n<p>&nbsp;<\/p>\n<p>The values 54.5, 64.5, 74.5, 84.5 and 94.5 are the <strong>midpoints<\/strong> between the lower and upper bounds for each class.<\/p>\n<p>The first label on the [latex]x[\/latex]-axis is [latex]44.5[\/latex]. This represents an interval extending from [latex]39.5[\/latex] to [latex]49.5[\/latex]. Since the lowest test score is [latex]54.5[\/latex], this interval is used only to allow the graph to touch the [latex]x[\/latex]-axis. The point labeled [latex]54.5[\/latex] represents the next interval, or the first \u201creal\u201d interval from the table, and contains five scores. This reasoning is followed for each of the remaining intervals with the point [latex]104.5[\/latex] representing the interval from [latex]99.5[\/latex] to [latex]109.5[\/latex]. Again, this interval contains no data and is only used so that the graph will touch the [latex]x[\/latex]-axis. Looking at the graph, we say that this distribution is skewed because one side of the graph does not mirror the other side.<\/p>\n<\/div>\n<div id=\"fs-idm37683552\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<div id=\"fs-idm37683552\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\"><\/div>\n<\/div>\n<\/div>\n<\/header>\n<header>\n<div class=\"title\" data-label-parent=\"\" data-type=\"title\">\n<h3>Try It<\/h3>\n<\/div>\n<\/header>\n<div id=\"eip-idp12732512\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"eip-idp12732768\" class=\"problem\" data-type=\"problem\">\n<p>Construct a frequency polygon of U.S. Presidents\u2019 ages at inauguration shown in the table.<\/p>\n<table id=\"fs-idp36852784\" summary=\"\">\n<thead>\n<tr>\n<th>Age at Inauguration<\/th>\n<th>Frequency<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]41.5\u201346.5[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]46.5\u201351.5[\/latex]<\/td>\n<td>[latex]11[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]51.5\u201356.5[\/latex]<\/td>\n<td>[latex]14[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]56.5\u201361.5[\/latex]<\/td>\n<td>[latex]9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]61.5\u201366.5[\/latex]<\/td>\n<td>[latex]4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]66.5\u201371.5[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q523753\">Show Answer<\/span><\/p>\n<div id=\"q523753\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2161 alignleft\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5658\/2016\/04\/28174520\/frequency-polygon-table.jpg\" alt=\"frequency polygon table\" width=\"346\" height=\"203\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-2170\" style=\"font-size: 1rem; text-align: initial;\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5658\/2016\/04\/28175804\/ages-frequency-polygon1.jpg\" alt=\"\" width=\"479\" height=\"285\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-idm37683552\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<div id=\"eip-idp12732512\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"eip-idp12732768\" class=\"problem\" data-type=\"problem\">\n<p><span style=\"font-size: 1rem; text-align: initial;\"><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Frequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<div id=\"fs-idp21707856\" class=\"example\" data-type=\"example\">\n<p>We will construct an overlay frequency polygon comparing the scores\u00a0with the students\u2019 final numeric grade.<\/p>\n<table id=\"fs-idm10950720\" summary=\"\">\n<thead>\n<tr>\n<th colspan=\"4\">Frequency Distribution for Calculus Final Test Scores<\/th>\n<\/tr>\n<tr>\n<th>Lower Bound<\/th>\n<th>Upper Bound<\/th>\n<th>Frequency<\/th>\n<th>Cumulative Frequency<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]49.5[\/latex]<\/td>\n<td>[latex]59.5[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]59.5[\/latex]<\/td>\n<td>[latex]69.5[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]15[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]69.5[\/latex]<\/td>\n<td>[latex]79.5[\/latex]<\/td>\n<td>[latex]30[\/latex]<\/td>\n<td>[latex]45[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]79.5[\/latex]<\/td>\n<td>[latex]89.5[\/latex]<\/td>\n<td>[latex]40[\/latex]<\/td>\n<td>[latex]85[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]89.5[\/latex]<\/td>\n<td>[latex]99.5[\/latex]<\/td>\n<td>[latex]15[\/latex]<\/td>\n<td>[latex]100[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"fs-idp39914624\" summary=\"\">\n<thead>\n<tr>\n<th colspan=\"4\">Frequency Distribution for Calculus Final Grades<\/th>\n<\/tr>\n<tr>\n<th>Lower Bound<\/th>\n<th>Upper Bound<\/th>\n<th>Frequency<\/th>\n<th>Cumulative Frequency<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]49.5[\/latex]<\/td>\n<td>[latex]59.5[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]59.5[\/latex]<\/td>\n<td>[latex]69.5[\/latex]<\/td>\n<td>[latex]10[\/latex]<\/td>\n<td>[latex]20[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]69.5[\/latex]<\/td>\n<td>[latex]79.5[\/latex]<\/td>\n<td>[latex]30[\/latex]<\/td>\n<td>[latex]50[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]79.5[\/latex]<\/td>\n<td>[latex]89.5[\/latex]<\/td>\n<td>[latex]45[\/latex]<\/td>\n<td>[latex]95[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]89.5[\/latex]<\/td>\n<td>[latex]99.5[\/latex]<\/td>\n<td>[latex]5[\/latex]<\/td>\n<td>[latex]100[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<figure id=\"eip-id1165746871888\"><span id=\"fs-idm24364960\" data-type=\"media\" data-display=\"block\" data-alt=\"This is an overlay frequency polygon that matches the supplied data. The x-axis shows the grades, and the y-axis shows the frequency.\"> <img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21214235\/CNX_Stats_C02_M05a_002N.jpg\" alt=\"This is an overlay frequency polygon that matches the supplied data. The x-axis shows the grades, and the y-axis shows the frequency.\" width=\"350\" data-media-type=\"image\/jpg\" \/><\/span><\/figure>\n<\/div>\n<\/div>\n<p>Suppose that we want to study the temperature range of a region for an entire month. Every day at noon we note the temperature and write this down in a log. A variety of statistical studies could be done with this data. We could find the mean or the median temperature for the month. We could construct a histogram displaying the number of days that temperatures reach a certain range of values. However, all of these methods ignore a portion of the data that we have collected.<\/p>\n<p>One feature of the data that we may want to consider is that of time. Since each date is paired with the temperature reading for the day, we don\u2018t have to think of the data as being random. We can instead use the times given to impose a chronological order on the data. A graph that recognizes this ordering and displays the changing temperature as the month progresses is called a time series graph.<\/p>\n<h1 data-type=\"title\">Constructing a Time Series Graph<\/h1>\n<p>To construct a time series graph, we must look at both pieces of our <strong>paired data set<\/strong>. We start with a standard Cartesian coordinate system. The horizontal axis is used to plot the date or time increments, and the vertical axis is used to plot the values of the variable that we are measuring. By doing this, we make each point on the graph correspond to a date and a measured quantity. The points on the graph are typically connected by straight lines in the order in which they occur.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<div id=\"fs-idp7658080\" class=\"example\" data-type=\"example\">\n<div id=\"fs-idp4161680\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-idm51003184\" class=\"problem\" data-type=\"problem\">\n<p>The following data shows the Annual Consumer Price Index, each month, for ten years. Construct a time series graph for the Annual Consumer Price Index data only.<\/p>\n<table id=\"fs-idp49086512\" summary=\"\">\n<colgroup>\n<col data-align=\"center\" \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr valign=\"middle\">\n<th>Year<\/th>\n<th>Jan<\/th>\n<th>Feb<\/th>\n<th>Mar<\/th>\n<th>Apr<\/th>\n<th>May<\/th>\n<th>Jun<\/th>\n<th>Jul<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong data-effect=\"bold\">2003<\/strong><\/td>\n<td>[latex]181.7[\/latex]<\/td>\n<td>[latex]183.1[\/latex]<\/td>\n<td>[latex]184.2[\/latex]<\/td>\n<td>[latex]183.8[\/latex]<\/td>\n<td>[latex]183.5[\/latex]<\/td>\n<td>[latex]183.7[\/latex]<\/td>\n<td>[latex]183.9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">2004<\/strong><\/td>\n<td>[latex]185.2[\/latex]<\/td>\n<td>[latex]186.2[\/latex]<\/td>\n<td>[latex]187.4[\/latex]<\/td>\n<td>[latex]188.0[\/latex]<\/td>\n<td>[latex]189.1[\/latex]<\/td>\n<td>[latex]189.7[\/latex]<\/td>\n<td>[latex]189.4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">2005<\/strong><\/td>\n<td>[latex]190.7[\/latex]<\/td>\n<td>[latex]191.8[\/latex]<\/td>\n<td>[latex]193.3[\/latex]<\/td>\n<td>[latex]194.6[\/latex]<\/td>\n<td>[latex]194.4[\/latex]<\/td>\n<td>[latex]194.5[\/latex]<\/td>\n<td>[latex]195.4[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">2006<\/strong><\/td>\n<td>[latex]198.3[\/latex]<\/td>\n<td>[latex]198.7[\/latex]<\/td>\n<td>[latex]199.8[\/latex]<\/td>\n<td>[latex]201.5[\/latex]<\/td>\n<td>[latex]202.5[\/latex]<\/td>\n<td>[latex]202.9[\/latex]<\/td>\n<td>[latex]203.5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">2007<\/strong><\/td>\n<td>[latex]202.416[\/latex]<\/td>\n<td>[latex]203.499[\/latex]<\/td>\n<td>[latex]205.352[\/latex]<\/td>\n<td>[latex]206.686[\/latex]<\/td>\n<td>[latex]207.949[\/latex]<\/td>\n<td>[latex]208.352[\/latex]<\/td>\n<td>[latex]208.299[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">2008<\/strong><\/td>\n<td>[latex]211.080[\/latex]<\/td>\n<td>[latex]211.693[\/latex]<\/td>\n<td>[latex]213.528[\/latex]<\/td>\n<td>[latex]214.823[\/latex]<\/td>\n<td>[latex]216.632[\/latex]<\/td>\n<td>[latex]218.815[\/latex]<\/td>\n<td>[latex]219.964[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">2009<\/strong><\/td>\n<td>[latex]211.143[\/latex]<\/td>\n<td>[latex]212.193[\/latex]<\/td>\n<td>[latex]212.709[\/latex]<\/td>\n<td>[latex]213.240[\/latex]<\/td>\n<td>[latex]213.856[\/latex]<\/td>\n<td>[latex]215.693[\/latex]<\/td>\n<td>[latex]215.351[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">2010<\/strong><\/td>\n<td>[latex]216.687[\/latex]<\/td>\n<td>[latex]216.741[\/latex]<\/td>\n<td>[latex]217.631[\/latex]<\/td>\n<td>[latex]218.009[\/latex]<\/td>\n<td>[latex]218.178[\/latex]<\/td>\n<td>[latex]217.965[\/latex]<\/td>\n<td>[latex]218.011[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">2011<\/strong><\/td>\n<td>[latex]220.223[\/latex]<\/td>\n<td>[latex]221.309[\/latex]<\/td>\n<td>[latex]223.467[\/latex]<\/td>\n<td>[latex]224.906[\/latex]<\/td>\n<td>[latex]225.964[\/latex]<\/td>\n<td>[latex]225.722[\/latex]<\/td>\n<td>[latex]225.922[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">2012<\/strong><\/td>\n<td>[latex]226.665[\/latex]<\/td>\n<td>[latex]227.663[\/latex]<\/td>\n<td>[latex]229.392[\/latex]<\/td>\n<td>[latex]230.085[\/latex]<\/td>\n<td>[latex]229.815[\/latex]<\/td>\n<td>[latex]229.478[\/latex]<\/td>\n<td>[latex]229.104[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"fs-idm48042992\" summary=\"\">\n<colgroup>\n<col data-align=\"center\" \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr>\n<th>Year<\/th>\n<th>Aug<\/th>\n<th>Sep<\/th>\n<th>Oct<\/th>\n<th>Nov<\/th>\n<th>Dec<\/th>\n<th>Annual<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td><strong data-effect=\"bold\">2003<\/strong><\/td>\n<td>[latex]184.6[\/latex]<\/td>\n<td>[latex]185.2[\/latex]<\/td>\n<td>[latex]185.0[\/latex]<\/td>\n<td>[latex]184.5[\/latex]<\/td>\n<td>[latex]184.3[\/latex]<\/td>\n<td>[latex]184.0[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">2004<\/strong><\/td>\n<td>[latex]189.5[\/latex]<\/td>\n<td>[latex]189.9[\/latex]<\/td>\n<td>[latex]190.9[\/latex]<\/td>\n<td>[latex]191.0[\/latex]<\/td>\n<td>[latex]190.3[\/latex]<\/td>\n<td>[latex]188.9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">2005<\/strong><\/td>\n<td>[latex]196.4[\/latex]<\/td>\n<td>[latex]198.8[\/latex]<\/td>\n<td>[latex]199.2[\/latex]<\/td>\n<td>[latex]197.6[\/latex]<\/td>\n<td>[latex]196.8[\/latex]<\/td>\n<td>[latex]195.3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">2006<\/strong><\/td>\n<td>[latex]203.9[\/latex]<\/td>\n<td>[latex]202.9[\/latex]<\/td>\n<td>[latex]201.8[\/latex]<\/td>\n<td>[latex]201.5[\/latex]<\/td>\n<td>[latex]201.8[\/latex]<\/td>\n<td>[latex]201.6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">2007<\/strong><\/td>\n<td>[latex]207.917[\/latex]<\/td>\n<td>[latex]208.490[\/latex]<\/td>\n<td>[latex]208.936[\/latex]<\/td>\n<td>[latex]210.177[\/latex]<\/td>\n<td>[latex]210.036[\/latex]<\/td>\n<td>[latex]207.342[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">2008<\/strong><\/td>\n<td>[latex]219.086[\/latex]<\/td>\n<td>[latex]218.783[\/latex]<\/td>\n<td>[latex]216.573[\/latex]<\/td>\n<td>[latex]212.425[\/latex]<\/td>\n<td>[latex]210.228[\/latex]<\/td>\n<td>[latex]215.303[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">2009<\/strong><\/td>\n<td>[latex]215.834[\/latex]<\/td>\n<td>[latex]215.969[\/latex]<\/td>\n<td>[latex]216.177[\/latex]<\/td>\n<td>[latex]216.330[\/latex]<\/td>\n<td>[latex]215.949[\/latex]<\/td>\n<td>[latex]214.537[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">2010<\/strong><\/td>\n<td>[latex]218.312[\/latex]<\/td>\n<td>[latex]218.439[\/latex]<\/td>\n<td>[latex]218.711[\/latex]<\/td>\n<td>[latex]218.803[\/latex]<\/td>\n<td>[latex]219.179[\/latex]<\/td>\n<td>[latex]218.056[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">2011<\/strong><\/td>\n<td>[latex]226.545[\/latex]<\/td>\n<td>[latex]226.889[\/latex]<\/td>\n<td>[latex]226.421[\/latex]<\/td>\n<td>[latex]226.230[\/latex]<\/td>\n<td>[latex]225.672[\/latex]<\/td>\n<td>[latex]224.939[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><strong data-effect=\"bold\">2012<\/strong><\/td>\n<td>[latex]230.379[\/latex]<\/td>\n<td>[latex]231.407[\/latex]<\/td>\n<td>[latex]231.317[\/latex]<\/td>\n<td>[latex]230.221[\/latex]<\/td>\n<td>[latex]229.601[\/latex]<\/td>\n<td>[latex]229.594[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q283400\">Show Solution<\/span><\/p>\n<div id=\"q283400\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-1923\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/132\/2016\/04\/21113533\/618b7e70623b342d3cd528c45b9972a816b698d2-300x116.jpg\" alt=\"This is the time series graph that matches the supplied data. The x-axis shows years from 2003 to 2012, and the y-axis shows the annual CPI\" width=\"503\" height=\"195\" \/><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-idm11029344\" class=\"note statistics try ui-has-child-title\" data-type=\"note\" data-has-label=\"true\" data-label=\"\">\n<header>\n<div class=\"title\" data-label-parent=\"\" data-type=\"title\">\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<\/div>\n<\/div>\n<\/header>\n<header>\n<div class=\"title\" data-label-parent=\"\" data-type=\"title\"><\/div>\n<\/header>\n<div id=\"fs-idm10087584\" class=\"exercise\" data-type=\"exercise\">\n<div id=\"fs-idp2285280\" class=\"problem\" data-type=\"problem\">\n<p>The following table is a portion of a data set from www.worldbank.org. Use the table to construct a time series graph for CO<sub>2<\/sub> emissions for the United States.\u00a0 Units are kilotons (kt).<\/p>\n<table id=\"fs-idp17543168\" summary=\"\">\n<thead>\n<tr>\n<th colspan=\"4\">CO2 Emissions<\/th>\n<\/tr>\n<tr>\n<th><\/th>\n<th>Ukraine<\/th>\n<th>United Kingdom<\/th>\n<th>United States<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>2003<\/td>\n<td>[latex]352,259[\/latex]<\/td>\n<td>[latex]540,640[\/latex]<\/td>\n<td>[latex]5,681,664[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>2004<\/td>\n<td>[latex]343,121[\/latex]<\/td>\n<td>[latex]540,409[\/latex]<\/td>\n<td>[latex]5,790,761[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>2005<\/td>\n<td>[latex]339,029[\/latex]<\/td>\n<td>[latex]541,990[\/latex]<\/td>\n<td>[latex]5,826,394[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>2006<\/td>\n<td>[latex]327,797[\/latex]<\/td>\n<td>[latex]542,045[\/latex]<\/td>\n<td>[latex]5,737,615[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>2007<\/td>\n<td>[latex]328,357[\/latex]<\/td>\n<td>[latex]528,631[\/latex]<\/td>\n<td>[latex]5,828,697[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>2008<\/td>\n<td>[latex]323,657[\/latex]<\/td>\n<td>[latex]522,247[\/latex]<\/td>\n<td>[latex]5,656,839[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>2009<\/td>\n<td>[latex]272,176[\/latex]<\/td>\n<td>[latex]474,579[\/latex]<\/td>\n<td>[latex]5,299,563[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q866433\">Show Answer<\/span><\/p>\n<div id=\"q866433\" class=\"hidden-answer\" style=\"display: none\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-2159 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5658\/2016\/04\/28173608\/CO2-time-series.jpg\" alt=\"CO2 time series\" width=\"686\" height=\"372\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<h2 data-type=\"title\">Uses of a Time Series Graph<\/h2>\n<p>Time series graphs are important tools in various applications of statistics. When recording values of the same variable over an extended period of time, sometimes it is difficult to discern any trend or pattern. However, once the same data points are displayed graphically, some features jump out. Time series graphs make trends easy to spot.<\/p>\n<h2>Concept Review<\/h2>\n<p>A <strong>histogram<\/strong> is a graphic version of a frequency distribution. The graph consists of bars of equal width drawn adjacent to each other. The horizontal scale represents classes of quantitative data values and the vertical scale represents frequencies. The heights of the bars correspond to frequency values. Histograms are typically used for large, continuous, quantitative data sets.<\/p>\n<p>A <strong>frequency polygon<\/strong> can also be used when graphing large data sets with data points that repeat. The data usually goes on [latex]y[\/latex]-axis with the frequency being graphed on the [latex]x[\/latex]-axis.<\/p>\n<p><strong>Time series graphs<\/strong> can be helpful when looking at large amounts of data for one variable over a period of time.<\/p>\n<h1 data-type=\"title\">References<\/h1>\n<p>Data on annual homicides in Detroit, 1961\u201373, from Gunst &amp; Mason\u2019s book \u2018Regression Analysis and its Application\u2019, Marcel Dekker<\/p>\n<p>\u201cTimeline: Guide to the U.S. Presidents: Information on every president\u2019s birthplace, political party, term of office, and more.\u201d Scholastic, 2013. Available online at http:\/\/www.scholastic.com\/teachers\/article\/timeline-guide-us-presidents (accessed April 3, 2013).<\/p>\n<p>\u201cPresidents.\u201d Fact Monster. Pearson Education, 2007. Available online at http:\/\/www.factmonster.com\/ipka\/A0194030.html (accessed April 3, 2013).<\/p>\n<p>\u201cFood Security Statistics.\u201d Food and Agriculture Organization of the United Nations. Available online at http:\/\/www.fao.org\/economic\/ess\/ess-fs\/en\/ (accessed April 3, 2013).<\/p>\n<p>\u201cConsumer Price Index.\u201d United States Department of Labor: Bureau of Labor Statistics. Available online at http:\/\/data.bls.gov\/pdq\/SurveyOutputServlet (accessed April 3, 2013).<\/p>\n<p>\u201cCO2 emissions (kt).\u201d The World Bank, 2013. Available online at http:\/\/databank.worldbank.org\/data\/home.aspx (accessed April 3, 2013).<\/p>\n<p>\u201cBirths Time Series Data.\u201d General Register Office For Scotland, 2013. Available online at http:\/\/www.gro-scotland.gov.uk\/statistics\/theme\/vital-events\/births\/time-series.html (accessed April 3, 2013).<\/p>\n<p>\u201cDemographics: Children under the age of 5 years underweight.\u201d Indexmundi. Available online at http:\/\/www.indexmundi.com\/g\/r.aspx?t=50&amp;v=2224&amp;aml=en (accessed April 3, 2013).<\/p>\n<p>Gunst, Richard, Robert Mason. <em data-effect=\"italics\">Regression Analysis and Its Application: A Data-Oriented Approach<\/em>. CRC Press: 1980.<\/p>\n<p>\u201cOverweight and Obesity: Adult Obesity Facts.\u201d Centers for Disease Control and Prevention. Available online at http:\/\/www.cdc.gov\/obesity\/data\/adult.html (accessed September 13, 2013).<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-75\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>OpenStax, Statistics, Histograms, Frequency Polygons, and Time Series Graphs. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.34:11\/Introductory_Statistics\">http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.34:11\/Introductory_Statistics<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Introductory Statistics . <strong>Authored by<\/strong>: Barbara Illowski, Susan Dean. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44\">http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">All rights reserved content<\/div><ul class=\"citation-list\"><li>Histograms. <strong>Authored by<\/strong>: Khan Academy. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/4eLJGG2Ad30\">https:\/\/youtu.be\/4eLJGG2Ad30<\/a>. <strong>License<\/strong>: <em>All Rights Reserved<\/em>. <strong>License Terms<\/strong>: Standard YouTube License<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":6,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"OpenStax, Statistics, Histograms, Frequency Polygons, and Time Series Graphs\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.34:11\/Introductory_Statistics\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Histograms\",\"author\":\"Khan Academy\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/4eLJGG2Ad30\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"cc\",\"description\":\"Introductory Statistics \",\"author\":\"Barbara Illowski, Susan Dean\",\"organization\":\"Open Stax\",\"url\":\"http:\/\/cnx.org\/contents\/30189442-6998-4686-ac05-ed152b91b9de@17.44\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at 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