{"id":1148,"date":"2021-10-14T21:24:45","date_gmt":"2021-10-14T21:24:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/?post_type=chapter&p=1148"},"modified":"2021-12-01T21:02:51","modified_gmt":"2021-12-01T21:02:51","slug":"5-2-2-reading-charts-and-graphs-continuous-data","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/uvu-introductoryalgebra\/chapter\/5-2-2-reading-charts-and-graphs-continuous-data\/","title":{"raw":"5.2.2: Reading Charts and Graphs\u2013Continuous Data","rendered":"5.2.2: Reading Charts and Graphs\u2013Continuous Data"},"content":{"raw":"
\r\n
\r\n
\r\n
\r\n

Learning Objectives<\/h3>\r\n
    \r\n \t
  • Determine whether a variable is independent or dependent<\/li>\r\n \t
  • Read a histogram<\/li>\r\n \t
  • Determine the spread, shape, center, and outliers in a histogram<\/li>\r\n \t
  • Use relative frequency to calculate the frequency of an interval<\/li>\r\n \t
  • Read a line graph<\/li>\r\n \t
  • Determine the basic shape of a trend from a line graph<\/li>\r\n<\/ul>\r\n<\/div>\r\n
    \r\n

    KEY words<\/h3>\r\n
      \r\n \t
    • Outliers<\/strong>:\u00a0data points that fall outside the overall pattern of the distribution.<\/li>\r\n \t
    • Histogram:<\/strong> Most often used with grouped continuous data<\/li>\r\n \t
    • Line Graph:<\/strong> Most often used to show trends<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

      Continuous Data Charts<\/h2>\r\n

      Histograms<\/h3>\r\nOne advantage of a histogram is that it can readily display large data sets when the independent variable is discrete or grouped continuous. When the data is continuous the\u00a0histogram divides the variable values into equal-sized intervals. The height of the bars shows the frequency (the number of individuals in each interval) or the relative frequency (percent of the total number of individuals).<\/span>\r\n
      \r\n

      Example<\/h3>\r\n

      A Histogram of Hip Measurements<\/h2>\r\nHere we have a histogram that shows hip girth measurements for 507 adults who exercise regularly. (Hip girth<\/em> is the measurement around the hips.)\r\n\r\nEach interval is 5 cm wide and includes the lower measurement but not the higher measurement. The higher measurement belongs to the next interval on the right. For example, the interval 90-95 includes measurements of 90 cm but not 95 cm. The height of each bar indicates the number of individuals with hip measurements in an interval. We can see that 48 adults have hip measurements between 85 and 90 cm, and 97 adults have hip measurements between 100 and 105 cm.\r\n\r\n\"Histogram\r\n\r\nIn the histogram, the count is the number of individuals in each interval. The count is also called the frequency<\/strong><\/em>. From these counts, we can determine a percentage of individuals within a given interval of variable values. This percentage of the frequency compared to the total of all frequencies is called a relative frequency<\/strong><\/em>.\r\n\r\nThe following questions require us to calculate relative frequencies:\r\n\r\n1. Approximately what percentage of the sample has hip measurements between 85 and 90 cm?\r\n\r\nAnswer: <\/strong>Of the 507 adults in the data set, 48 have hip measurements between 85 and 90 cm.\r\n\r\n48 out of 507 is 48 \u00f7 507 \u2248 0.095 = 9.5%\r\n\r\nSo approximately 9.5% of the adults in this sample have hip girths between 85 and 90 cm.\r\n\r\n \r\n\r\n2. A pants manufacturer plans to produce three sizes of sweatpants. Size Large will fit hip girths of 100 cm to 130 cm. What percentage of the sample will wear size Large sweatpants?\r\n\r\nAnswer: <\/strong>Of the 507 adults in the data set, 158 adults (97 + 42 + 15 + 3 + 1) = 158 have hip measurements of 100 cm or more.\r\n\r\n158 out of 507 is 158 \u00f7 507 \u2248 0.312 = 31.2%\r\n\r\nSo 31.2% of the adults in this sample will wear size Large sweatpants.\r\n\r\n \r\n\r\n<\/div>\r\n \r\n\r\nThe histogram in figure 5 illustrates the heights (which were rounded to the nearest half-inch) of [latex]100[\/latex] male, semiprofessional soccer players. The heights are continuous<\/strong> data, since height is measured.\u00a0Height is displayed on the horizontal axis and relative frequency (%) on the vertical axis.\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"490\"]\"Histogram Figure 1. An example of a histogram[\/caption]\r\n\r\nEach bar represents the percent of players whose height falls in an interval of width [latex]2\"[\/latex]. For example, [latex]0.05(=5\\text{%})[\/latex] of the players are between [latex]59.95\"[\/latex] and [latex]61.95\"[\/latex] tall. Since there are 100 players represented, that means that [latex]5[\/latex] players ([latex]5\\text{% of }100[\/latex]) are between\u00a0[latex]59.95\"[\/latex] and [latex]61.95\"[\/latex]. Since the heights were rounded to the nearest half-inch, this means that there were [latex]5[\/latex] players whose heights rounded to [latex]60\"[\/latex], [latex]60.5\"[\/latex], [latex]61\"[\/latex] or [latex]61.5\"[\/latex].\r\n\r\nIt is easy to identify the mode<\/em><\/strong> as it is found at the tallest bar. Most of the players are between [latex]65.95\"[\/latex] and [latex]67.95\"[\/latex] tall. Indeed [latex]40\\text{%}[\/latex] of the players fall in that height interval, so, since there are [latex]100[\/latex] players, [latex]40[\/latex] of the players are between [latex]65.95\"[\/latex] and [latex]67.95\"[\/latex] tall.\r\n
      \r\n

      Try It<\/h3>\r\nHere is a histogram of the distribution of grades on a quiz.\r\n\r\n\"Histogram\r\n\r\nhttps:\/\/assessments.lumenlearning.com\/assessments\/3425https:\/\/assessments.lumenlearning.com\/assessments\/3426\r\n\r\n \r\n\r\n<\/div>\r\n \r\n\r\nIn the next example, we use a histogram to describe the shape, center, and spread of the distribution of a quantitative variable.\r\n
      \r\n

      EXAMPLE<\/h3>\r\n

      Oscar for Best Actress<\/h2>\r\nThe histogram shows the ages of the actresses who won an Oscar for Best Actress from 1970 to 2001.\r\n
      \"Histogram<\/div>\r\n \r\n\r\nTotal Count:\u00a0<\/strong>There are a total of 32 years from 1970 to 2001, so there are a total of 32 Oscars that were awarded.\r\n\r\nShape:<\/strong>\u00a0Most of the data lives towards the left of the graph. This means that most of the Oscar winners for Best Actress are young. More precisely, we see that 91% (29 of 32) of the winners were under 50 years of age, and 56% (18 of 32) of the winners are under the age of 35.\r\n\r\nCenter:<\/strong> The distribution of ages appears to be centered between 30 and 35 years; 28% (9 of 32) of the winners are in this age range.\r\n\r\nSpread:<\/strong> The data ranges from about 20 to about 80, so the overall range is approximately 60. There is a lot of variability in the ages of actresses who have won the Oscar for Best Actress.\r\n\r\nOutliers:<\/strong> Winners older than 60 are unusual. There are three outliers: one in each of the following intervals: 60\u201365, 70\u201375, 75\u201380.\r\n\r\nNow we summarize all of these observations in a paragraph:\r\n\r\nBetween the years of 1970 and 2001, the Oscar for Best Actress was awarded most often to young actresses: 56% (18 of 32) of the winners were under the age of 35, with 28% (9 of 32) of the winners between 30 and 35 years of age. Winners ranged in age from about 20 to about 80, but only 3 of the 32 were over 60.\r\n\r\nHere is a paragraph that uses more formal vocabulary to summarize the distribution of ages:\r\n\r\nBetween the years of 1970 and 2001, the Oscar for Best Actress was awarded most often to young actresses. The distribution of ages is skewed to the right: 56% (18 of 32) of the winners were under the age of 35, with the center of the distribution between 30 and 35 years of age. With winners ranging in age from about 20 to about 80, the overall range of the distribution is about 60. But much of this variability is due to three outliers who were older than 60 when they won the award.\r\n\r\n<\/div>\r\n
      \r\n

      Example<\/h3>\r\nWhile most people have an average body temperature of 98.6\u00b0F, there is actually a distribution of normal body temperature across people. A study of healthy adults revealed the following:\r\n\r\n\"Histogram\r\n\r\nThe numbers along the horizontal temperature axis indicate the middle of each interval.\r\n\r\n1. What are the lower and upper temperature limits of the interval with a middle temperature of [latex]90.0[\/latex]\u00b0F?\r\n

      The distance between middle values is [latex]0.2[\/latex], so [latex]0.1[\/latex] less than the middle value will give the lower limit and [latex]0.1[\/latex] greater than the middle value will give the upper limit. Consequently, the lower limit is [latex]89.9[\/latex]\u00b0F and the upper limit is [latex]90.1[\/latex]\u00b0F.<\/p>\r\n2. How many people had a temperature between [latex]97.9[\/latex]\u00b0F and [latex]98.5[\/latex]\u00b0F?\r\n

      [latex]97.9[\/latex] is the lower limit for the interval marked [latex]98.0[\/latex], and [latex]98.5[\/latex] is the upper limit for the interval labelled [latex]98.4[\/latex].\u00a0The number of people between [latex]97.9[\/latex]\u00b0F and [latex]98.5[\/latex]\u00b0F is [latex]20+34+56=110[\/latex].<\/p>\r\n3. How many people had a temperature between [latex]99.3[\/latex]\u00b0F and [latex]99.9[\/latex]\u00b0F?\r\n

      [latex]90.3[\/latex] is the lower limit for the interval marked [latex]99.4[\/latex], and [latex]99.9[\/latex] is the upper limit for the interval labelled [latex]90.8[\/latex].\u00a0The number of people between [latex]99.3[\/latex]\u00b0F and [latex]99.9[\/latex]\u00b0F is [latex]12+0+2=14[\/latex].<\/p>\r\n4. How many people were surveyed?\r\n

      Number surveyed = [latex]2+5+20+34+56+81+72+38+18+12+2=338[\/latex].<\/p>\r\n5. What is the mode of the data, and what does it represent?\r\n

      The mode is the temperature with the highest frequency. The interval with a midpoint of [latex]98.6[\/latex]\u00b0F is the mode.<\/p>\r\n6. What percent (rounded to the nearest tenth of a percent) of those surveyed had a temperature in the modal interval?\r\n

      [latex]81\\text{ out of }338=0.2396...= 24.0\\text{%}[\/latex]<\/p>\r\n\r\n<\/div>\r\n \r\n

      \r\n

      TRY IT!<\/h3>\r\n\"\"\r\n\r\n1. What was the lowest accumulated snowfall and in which decade did it occur?\r\n\r\n2. What was the highest\u00a0accumulated snowfall and in which decade did it occur?\r\n\r\n3. What is the difference between the highest and lowest accumulated snowfall?\r\n\r\n4. How many decades had an accumulated snowfall greater than 600 inches?\r\n\r\n5. What percent of decades shown on the graph had a snowfall under 500 inches?\r\n\r\n[reveal-answer q=\"89473\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"89473\"]\r\n\r\n1. 370.3 inches from 2010-2020.\r\n\r\n2. 734.2 inches from 1970-1980.\r\n\r\n3. 363.9 inches.\r\n\r\n4. 2\r\n\r\n5. 3 out of 12 = 25%\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
      <\/div>\r\nThe three histograms in figure 2 were constructed using the same set of data.\u00a0Are you surprised by how different the distribution looks in each histogram? When we change the intervals, the data gets grouped differently. The different grouping affects the appearance of the histogram.\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"612\"]\"Three Figure 2[\/caption]\r\n\r\nThe histogram on the left has an interval width of 20. The first interval starts at 40.\u00a0In the middle histogram, the bin width was changed to 10 with the first interval still starting at 40.\u00a0The last histogram, still has an interval width of 10 but the first bin is now at 45.\r\n\r\nThese changes affect our description of the shape, center, and spread of this set of data. For example, in the histogram on the left, the distribution looks symmetric with a central mode. The data in the histogram on the right appear to live mostly in the lower grades. The histogram on the left shows a mode between 60 and 80, while the middle histogram shows a mode between 70 and 80 and the one on the right shows a mode between 65 and 75.\r\n\r\nChanging the size of the interval and the starting point of the first interval will change the look of the histogram even though the data set is the same.\r\n\r\nUse the simulation below to investigate what happens to the histogram as\u00a0the size of the interval and the starting point are changed using the same data set.\r\n\r\n