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.

A histogram 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 frequency or relative frequency (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.

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:

• [latex]f[/latex] = frequency
• [latex]n[/latex] = total number of data values (or the sum of the individual frequencies), and
• [latex]RF[/latex] = relative frequency,

then [latex]\displaystyle{R}{F}=\frac{{f}}{{n}}[/latex]

For 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]90–100[/latex]%. [latex]90–100[/latex]% are quantitative measures.

To construct a histogram, first decide how many bars or intervals, also called classes, 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 convenient starting point 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 – 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 – 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 – 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 – 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.

Watch the following video for an example of how to draw a histogram.

Example

The following data are the heights (in inches to the nearest half inch) of [latex]100[/latex] male semiprofessional soccer players. The heights are continuous data, since height is measured.

[latex]60[/latex]; [latex]60.5[/latex]; [latex]61[/latex]; [latex]61[/latex]; [latex]61.5[/latex]

[latex]63.5[/latex]; [latex]63.5[/latex]; [latex]63.5[/latex]

[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]

[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]

[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]

[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]

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.

[latex]60 – 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].

The largest value is [latex]74[/latex], so [latex]74 + 0.05 = 74.05[/latex] is the ending value.

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.

[latex]\displaystyle\frac{{{74.05}-{59.95}}}{{8}}={1.76}[/latex]

Note

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.

The boundaries are:

• [latex]59.95[/latex]
• [latex]59.95 + 2 = 61.95[/latex]
• [latex]61.95 + 2 = 63.95[/latex]
• [latex]63.95 + 2 = 65.95[/latex]
• [latex]65.95 + 2 = 67.95[/latex]
• [latex]67.95 + 2 = 69.95[/latex]
• [latex]69.95 + 2 = 71.95[/latex]
• [latex]71.95 + 2 = 73.95[/latex]
• [latex]73.95 + 2 = 75.95[/latex]

The heights [latex]60[/latex] through [latex]61.5[/latex] inches are in the interval [latex]59.95–61.95[/latex]. The heights that are [latex]63.5[/latex] are in the interval [latex]61.95–63.95[/latex]. The heights that are [latex]64[/latex] through [latex]64.5[/latex] are in the interval [latex]63.95–65.95[/latex]. The heights [latex]66[/latex] through [latex]67.5[/latex] are in the interval [latex]65.95–67.95[/latex]. The heights [latex]68[/latex] through [latex]69.5[/latex] are in the interval [latex]67.95–69.95[/latex]. The heights [latex]70[/latex] through [latex]71[/latex] are in the interval [latex]69.95–71.95[/latex]. The heights [latex]72[/latex] through [latex]73.5[/latex] are in the interval [latex]71.95–73.95[/latex]. The height [latex]74[/latex] is in the interval [latex]73.95–75.95[/latex].

The following histogram displays the heights on the [latex]x[/latex]-axis and relative frequency on the [latex]y[/latex]-axis.

USING THE TI-83, 83+, 84, 84+ CALCULATOR

Create the histogram for Example 2.

• Press Y=. Press CLEAR to delete any equations.
• Press STAT 1:EDIT. If L1 has data in it, arrow up into the name L1, press CLEAR and then arrow down. If necessary, do the same for L2.
• Into L1, enter [latex]1[/latex], [latex]2[/latex], [latex]3[/latex], [latex]4[/latex], [latex]5[/latex], [latex]6[/latex].
• Into L2, enter [latex]11[/latex], [latex]10[/latex], [latex]16[/latex], [latex]6[/latex], [latex]5[/latex], [latex]2[/latex].
• Press WINDOW. Set Xmin = [latex].5[/latex], Xscl = [latex](6.5 – .5)/6[/latex], Ymin = [latex]–1[/latex], Ymax = [latex]20[/latex], Yscl = [latex]1[/latex], Xres = [latex]1[/latex].
• Press 2nd Y=. Start by pressing 4:Plotsoff ENTER.
• Press 2nd Y=. Press 1:Plot1. Press ENTER. Arrow down to TYPE. Arrow to the 3rd picture (histogram). Press ENTER.
• Arrow down to Xlist: Enter L1 (2nd 1). Arrow down to Freq. Enter L2 (2nd 2).
• Press GRAPH.
• Use the TRACE key and the arrow keys to examine the histogram.

Example

Using this data set, construct a histogram.

Number of Hours My Classmates Spent Playing Video Games on Weekends
[latex]9.95[/latex]	[latex]10[/latex]	[latex]2.25[/latex]	[latex]16.75[/latex]	[latex]0[/latex]
[latex]19.5[/latex]	[latex]22.5[/latex]	[latex]7.5[/latex]	[latex]15[/latex]	[latex]12.75[/latex]
[latex]5.5[/latex]	[latex]11[/latex]	[latex]10[/latex]	[latex]20.75[/latex]	[latex]17.5[/latex]
[latex]23[/latex]	[latex]21.9[/latex]	[latex]24[/latex]	[latex]23.75[/latex]	[latex]18[/latex]
[latex]20[/latex]	[latex]15[/latex]	[latex]22.9[/latex]	[latex]18.8[/latex]	[latex]20.5[/latex]

Show Solution

Solution:

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.

Frequency Polygons

Frequency polygons are analogous to line graphs, and just as line graphs make continuous data visually easy to interpret, so too do frequency polygons.

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.

example

A frequency polygon was constructed from the frequency table below.

Frequency Distribution for Calculus Final Test Scores
Lower Bound	Upper Bound	Frequency	Cumulative Frequency
[latex]49.5[/latex]	[latex]59.5[/latex]	[latex]5[/latex]	[latex]5[/latex]
[latex]59.5[/latex]	[latex]69.5[/latex]	[latex]10[/latex]	[latex]15[/latex]
[latex]69.5[/latex]	[latex]79.5[/latex]	[latex]30[/latex]	[latex]45[/latex]
[latex]79.5[/latex]	[latex]89.5[/latex]	[latex]40[/latex]	[latex]85[/latex]
[latex]89.5[/latex]	[latex]99.5[/latex]	[latex]15[/latex]	[latex]100[/latex]

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 “real” 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.

Try It

Construct a frequency polygon of U.S. Presidents’ ages at inauguration shown in the table.

Age at Inauguration	Frequency
[latex]41.5–46.5[/latex]	[latex]4[/latex]
[latex]46.5–51.5[/latex]	[latex]11[/latex]
[latex]51.5–56.5[/latex]	[latex]14[/latex]
[latex]56.5–61.5[/latex]	[latex]9[/latex]
[latex]61.5–66.5[/latex]	[latex]4[/latex]
[latex]66.5–71.5[/latex]	[latex]2[/latex]

Frequency polygons are useful for comparing distributions. This is achieved by overlaying the frequency polygons drawn for different data sets.

example

We will construct an overlay frequency polygon comparing the scores with the students’ final numeric grade.

Frequency Distribution for Calculus Final Test Scores
Lower Bound	Upper Bound	Frequency	Cumulative Frequency
[latex]49.5[/latex]	[latex]59.5[/latex]	[latex]5[/latex]	[latex]5[/latex]
[latex]59.5[/latex]	[latex]69.5[/latex]	[latex]10[/latex]	[latex]15[/latex]
[latex]69.5[/latex]	[latex]79.5[/latex]	[latex]30[/latex]	[latex]45[/latex]
[latex]79.5[/latex]	[latex]89.5[/latex]	[latex]40[/latex]	[latex]85[/latex]
[latex]89.5[/latex]	[latex]99.5[/latex]	[latex]15[/latex]	[latex]100[/latex]

Frequency Distribution for Calculus Final Grades
Lower Bound	Upper Bound	Frequency	Cumulative Frequency
[latex]49.5[/latex]	[latex]59.5[/latex]	[latex]10[/latex]	[latex]10[/latex]
[latex]59.5[/latex]	[latex]69.5[/latex]	[latex]10[/latex]	[latex]20[/latex]
[latex]69.5[/latex]	[latex]79.5[/latex]	[latex]30[/latex]	[latex]50[/latex]
[latex]79.5[/latex]	[latex]89.5[/latex]	[latex]45[/latex]	[latex]95[/latex]
[latex]89.5[/latex]	[latex]99.5[/latex]	[latex]5[/latex]	[latex]100[/latex]

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.

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‘t 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.

Constructing a Time Series Graph

To construct a time series graph, we must look at both pieces of our paired data set. 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.

Example

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.

Year	Jan	Feb	Mar	Apr	May	Jun	Jul
2003	[latex]181.7[/latex]	[latex]183.1[/latex]	[latex]184.2[/latex]	[latex]183.8[/latex]	[latex]183.5[/latex]	[latex]183.7[/latex]	[latex]183.9[/latex]
2004	[latex]185.2[/latex]	[latex]186.2[/latex]	[latex]187.4[/latex]	[latex]188.0[/latex]	[latex]189.1[/latex]	[latex]189.7[/latex]	[latex]189.4[/latex]
2005	[latex]190.7[/latex]	[latex]191.8[/latex]	[latex]193.3[/latex]	[latex]194.6[/latex]	[latex]194.4[/latex]	[latex]194.5[/latex]	[latex]195.4[/latex]
2006	[latex]198.3[/latex]	[latex]198.7[/latex]	[latex]199.8[/latex]	[latex]201.5[/latex]	[latex]202.5[/latex]	[latex]202.9[/latex]	[latex]203.5[/latex]
2007	[latex]202.416[/latex]	[latex]203.499[/latex]	[latex]205.352[/latex]	[latex]206.686[/latex]	[latex]207.949[/latex]	[latex]208.352[/latex]	[latex]208.299[/latex]
2008	[latex]211.080[/latex]	[latex]211.693[/latex]	[latex]213.528[/latex]	[latex]214.823[/latex]	[latex]216.632[/latex]	[latex]218.815[/latex]	[latex]219.964[/latex]
2009	[latex]211.143[/latex]	[latex]212.193[/latex]	[latex]212.709[/latex]	[latex]213.240[/latex]	[latex]213.856[/latex]	[latex]215.693[/latex]	[latex]215.351[/latex]
2010	[latex]216.687[/latex]	[latex]216.741[/latex]	[latex]217.631[/latex]	[latex]218.009[/latex]	[latex]218.178[/latex]	[latex]217.965[/latex]	[latex]218.011[/latex]
2011	[latex]220.223[/latex]	[latex]221.309[/latex]	[latex]223.467[/latex]	[latex]224.906[/latex]	[latex]225.964[/latex]	[latex]225.722[/latex]	[latex]225.922[/latex]
2012	[latex]226.665[/latex]	[latex]227.663[/latex]	[latex]229.392[/latex]	[latex]230.085[/latex]	[latex]229.815[/latex]	[latex]229.478[/latex]	[latex]229.104[/latex]

Year	Aug	Sep	Oct	Nov	Dec	Annual
2003	[latex]184.6[/latex]	[latex]185.2[/latex]	[latex]185.0[/latex]	[latex]184.5[/latex]	[latex]184.3[/latex]	[latex]184.0[/latex]
2004	[latex]189.5[/latex]	[latex]189.9[/latex]	[latex]190.9[/latex]	[latex]191.0[/latex]	[latex]190.3[/latex]	[latex]188.9[/latex]
2005	[latex]196.4[/latex]	[latex]198.8[/latex]	[latex]199.2[/latex]	[latex]197.6[/latex]	[latex]196.8[/latex]	[latex]195.3[/latex]
2006	[latex]203.9[/latex]	[latex]202.9[/latex]	[latex]201.8[/latex]	[latex]201.5[/latex]	[latex]201.8[/latex]	[latex]201.6[/latex]
2007	[latex]207.917[/latex]	[latex]208.490[/latex]	[latex]208.936[/latex]	[latex]210.177[/latex]	[latex]210.036[/latex]	[latex]207.342[/latex]
2008	[latex]219.086[/latex]	[latex]218.783[/latex]	[latex]216.573[/latex]	[latex]212.425[/latex]	[latex]210.228[/latex]	[latex]215.303[/latex]
2009	[latex]215.834[/latex]	[latex]215.969[/latex]	[latex]216.177[/latex]	[latex]216.330[/latex]	[latex]215.949[/latex]	[latex]214.537[/latex]
2010	[latex]218.312[/latex]	[latex]218.439[/latex]	[latex]218.711[/latex]	[latex]218.803[/latex]	[latex]219.179[/latex]	[latex]218.056[/latex]
2011	[latex]226.545[/latex]	[latex]226.889[/latex]	[latex]226.421[/latex]	[latex]226.230[/latex]	[latex]225.672[/latex]	[latex]224.939[/latex]
2012	[latex]230.379[/latex]	[latex]231.407[/latex]	[latex]231.317[/latex]	[latex]230.221[/latex]	[latex]229.601[/latex]	[latex]229.594[/latex]

Show Solution

Uses of a Time Series Graph

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.

Concept Review

A histogram 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. A frequency polygon 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. Time series graphs can be helpful when looking at large amounts of data for one variable over a period of time.

References

Data on annual homicides in Detroit, 1961–73, from Gunst & Mason’s book ‘Regression Analysis and its Application’, Marcel Dekker

“Timeline: Guide to the U.S. Presidents: Information on every president’s birthplace, political party, term of office, and more.” Scholastic, 2013. Available online at http://www.scholastic.com/teachers/article/timeline-guide-us-presidents (accessed April 3, 2013).

“Presidents.” Fact Monster. Pearson Education, 2007. Available online at http://www.factmonster.com/ipka/A0194030.html (accessed April 3, 2013).

“Food Security Statistics.” Food and Agriculture Organization of the United Nations. Available online at http://www.fao.org/economic/ess/ess-fs/en/ (accessed April 3, 2013).

“Consumer Price Index.” United States Department of Labor: Bureau of Labor Statistics. Available online at http://data.bls.gov/pdq/SurveyOutputServlet (accessed April 3, 2013).

“CO2 emissions (kt).” The World Bank, 2013. Available online at http://databank.worldbank.org/data/home.aspx (accessed April 3, 2013).

“Births Time Series Data.” 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).

“Demographics: Children under the age of 5 years underweight.” Indexmundi. Available online at http://www.indexmundi.com/g/r.aspx?t=50&v=2224&aml=en (accessed April 3, 2013).

Gunst, Richard, Robert Mason. Regression Analysis and Its Application: A Data-Oriented Approach. CRC Press: 1980.

“Overweight and Obesity: Adult Obesity Facts.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/obesity/data/adult.html (accessed September 13, 2013).