{"id":1666,"date":"2023-04-12T19:43:07","date_gmt":"2023-04-12T19:43:07","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=1666"},"modified":"2024-10-18T20:54:14","modified_gmt":"2024-10-18T20:54:14","slug":"numerical-summaries-of-data-learn-it-3","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/numerical-summaries-of-data-learn-it-3\/","title":{"raw":"Numerical Summaries of Data: Learn It 3","rendered":"Numerical Summaries of Data: Learn It 3"},"content":{"raw":"

Five-Number Summary<\/h2>\r\n

Where standard deviation is a measure of variation based on the mean, quartiles<\/strong> are based on the median.<\/p>\r\n

\r\n
\r\n

quartiles<\/h3>\r\n

Quartiles<\/strong> are values that divide the data in quarters.<\/p>\r\n

 <\/p>\r\n

    \r\n\t
  • The first quartile<\/strong>, also known as [latex]Q1[\/latex], can be thought of as the median of the values that lie below the median for the whole data set. It is the [latex]25[\/latex]th<\/sup> percentile of the data set.<\/li>\r\n\t
  • The third quartile<\/strong>, also known as [latex]Q3[\/latex], can be thought of as the median of the values that lie above the median for the whole data set. It is the [latex]75[\/latex]th<\/sup> percentile of the data set.<\/li>\r\n<\/ul>\r\n

    This divides the data into quarters; [latex]25\\%[\/latex] of the data is between the minimum and [latex]Q1[\/latex], [latex]25\\%[\/latex] is between [latex]Q1[\/latex] and the median, [latex]25\\%[\/latex] is between the median and [latex]Q3[\/latex], and [latex]25\\%[\/latex] is between [latex]Q3[\/latex] and the maximum value.<\/p>\r\n<\/div>\r\n<\/section>\r\n

    While quartiles are not a [latex]1[\/latex]-number summary of variation like standard deviation, the quartiles are used with the median, minimum, and maximum values to form a [latex]5[\/latex]-number summary<\/strong> of the data.<\/p>\r\n

    \r\n
    \r\n

    five-number summary<\/h3>\r\n

    The five-number summary takes this form:<\/p>\r\n

     <\/p>\r\n

    Minimum, [latex]Q1[\/latex], Median, [latex]Q3[\/latex], Maximum<\/p>\r\n<\/div>\r\n<\/section>\r\n

    To find the first quartile, we need to find the data value so that [latex]25\\%[\/latex] of the data is below it. If [latex]n[\/latex] is the number of data values, we compute a locator by finding [latex]25\\%[\/latex] of [latex]n[\/latex]. If this locator is a decimal value, we round up, and find the data value in that position. If the locator is a whole number, we find the mean of the data value in that position and the next data value. This is identical to the process we used to find the median, except we use [latex]25\\%[\/latex] of the data values rather than half the data values as the locator.<\/p>\r\n

    \r\n

    How To: Find the First Quartile, [latex]Q1[\/latex]<\/strong><\/p>\r\n

      \r\n\t
    1. Begin by ordering the data from smallest to largest<\/li>\r\n\t
    2. Compute the locator: [latex]L = 0.25n[\/latex]<\/li>\r\n\t
    3. If [latex]L[\/latex] is a decimal value:\r\n\r\n
        \r\n\t
      • Round up to [latex]L+[\/latex]<\/li>\r\n\t
      • Use the data value in the [latex]L+[\/latex]th position<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t
      • If [latex]L[\/latex] is a whole number:\r\n\r\n
          \r\n\t
        • Find the mean of the data values in the [latex]L[\/latex]<\/em>th and [latex]L+1[\/latex]th positions.<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n

          How To: Find the Third Quartile, [latex]Q3[\/latex]<\/strong><\/p>\r\n

          Use the same procedure as for [latex]Q1[\/latex], but with locator: [latex]L = 0.75n[\/latex]<\/p>\r\n<\/section>\r\n

          To find minimum and maximum values of a data set, find the least and the greatest value in the data set. It might be best to order your data from smallest to the largest because you can can also find the measure of center, median, through the ordered data set as well. Recall: the median is the value that splits the data in half.<\/p>\r\n

          Suppose we have measured [latex]9[\/latex] females, and their heights (in inches) sorted from smallest to largest are:
          \r\n
          [latex]59, 60, 62, 64, 66, 67, 69, 70, 72[\/latex]<\/center>
          \r\nWhat are the first and third quartiles?
          \r\n[reveal-answer q=\"450713\"]Show Solution[\/reveal-answer]
          \r\n[hidden-answer a=\"450713\"]To find the first quartile we first compute the locator: [latex]25\\%[\/latex] of [latex]9[\/latex] is [latex]L = 0.25(9) = 2.25[\/latex]. Since this value is not a whole number, we round up to [latex]3[\/latex]. The first quartile will be the third data value: [latex]62[\/latex] inches.
          \r\n
          \r\nTo find the third quartile, we again compute the locator: [latex]75\\%[\/latex] of [latex]9[\/latex] is [latex]0.75(9) = 6.75[\/latex]. Since this value is not a whole number, we round up to [latex]7[\/latex]. The third quartile will be the seventh data value: [latex]69[\/latex] inches.
          \r\n
          \r\n[\/hidden-answer]\r\n
          \r\n

          Suppose we had measured [latex]8[\/latex] females, and their heights (in inches) sorted from smallest to largest are:<\/p>\r\n

          [latex]59, 60, 62, 64, 66, 67, 69, 70[\/latex]<\/p>\r\n

          What are the first and third quartiles? What is the [latex]5[\/latex] number summary?
          \r\n[reveal-answer q=\"699335\"]Show Solution[\/reveal-answer]
          \r\n[hidden-answer a=\"699335\"]<\/p>\r\n

          To find the first quartile we first compute the locator: [latex]25\\%[\/latex] of [latex]8[\/latex] is [latex]L = 0.25(8) = 2[\/latex]. Since this value is<\/em> a whole number, we will find the mean of the [latex]2[\/latex]nd and [latex]3[\/latex]rd data values: [latex](60+62)\/2 = 61[\/latex], so the first quartile is [latex]61[\/latex] inches.<\/p>\r\n

          The third quartile is computed similarly, using [latex]75\\%[\/latex] instead of [latex]25\\%[\/latex]. [latex]L = 0.75(8) = 6[\/latex]. This is a whole number, so we will find the mean of the [latex]6[\/latex]th and [latex]7[\/latex]th data values: [latex](67+69)\/2 = 68[\/latex], so [latex]Q3[\/latex] is [latex]68[\/latex].<\/p>\r\n

          Note that the median could be computed the same way, using [latex]50\\%[\/latex].<\/p>\r\n

          [\/hidden-answer]<\/p>\r\n

          \r\n

          The [latex]5[\/latex]-number summary combines the first and third quartile with the minimum, median, and maximum values.<\/p>\r\n

          What are the [latex]5[\/latex]-number summaries for each of the previous [latex]2[\/latex] examples?
          \r\n[reveal-answer q=\"190147\"]Show Solution[\/reveal-answer]
          \r\n[hidden-answer a=\"190147\"]<\/p>\r\n

          For the [latex]9[\/latex] female sample, the median is [latex]66[\/latex], the minimum is [latex]59[\/latex], and the maximum is [latex]72[\/latex]. The [latex]5[\/latex] number summary is: [latex]59[\/latex], [latex]62[\/latex], [latex]66[\/latex], [latex]69[\/latex], [latex]72[\/latex].<\/p>\r\n

          For the [latex]8[\/latex] female sample, the median is [latex]65[\/latex], the minimum is [latex]59[\/latex], and the maximum is [latex]70[\/latex], so the [latex]5[\/latex] number summary would be: [latex]59[\/latex], [latex]61[\/latex], [latex]65[\/latex], [latex]68[\/latex], [latex]70[\/latex].<\/p>\r\n

          [\/hidden-answer]<\/p>\r\n<\/section>\r\n

          [ohm2_question hide_question_numbers=1]2084[\/ohm2_question]<\/section>","rendered":"

          Five-Number Summary<\/h2>\n

          Where standard deviation is a measure of variation based on the mean, quartiles<\/strong> are based on the median.<\/p>\n

          \n
          \n

          quartiles<\/h3>\n

          Quartiles<\/strong> are values that divide the data in quarters.<\/p>\n

           <\/p>\n

            \n
          • The first quartile<\/strong>, also known as [latex]Q1[\/latex], can be thought of as the median of the values that lie below the median for the whole data set. It is the [latex]25[\/latex]th<\/sup> percentile of the data set.<\/li>\n
          • The third quartile<\/strong>, also known as [latex]Q3[\/latex], can be thought of as the median of the values that lie above the median for the whole data set. It is the [latex]75[\/latex]th<\/sup> percentile of the data set.<\/li>\n<\/ul>\n

            This divides the data into quarters; [latex]25\\%[\/latex] of the data is between the minimum and [latex]Q1[\/latex], [latex]25\\%[\/latex] is between [latex]Q1[\/latex] and the median, [latex]25\\%[\/latex] is between the median and [latex]Q3[\/latex], and [latex]25\\%[\/latex] is between [latex]Q3[\/latex] and the maximum value.<\/p>\n<\/div>\n<\/section>\n

            While quartiles are not a [latex]1[\/latex]-number summary of variation like standard deviation, the quartiles are used with the median, minimum, and maximum values to form a [latex]5[\/latex]-number summary<\/strong> of the data.<\/p>\n

            \n
            \n

            five-number summary<\/h3>\n

            The five-number summary takes this form:<\/p>\n

             <\/p>\n

            Minimum, [latex]Q1[\/latex], Median, [latex]Q3[\/latex], Maximum<\/p>\n<\/div>\n<\/section>\n

            To find the first quartile, we need to find the data value so that [latex]25\\%[\/latex] of the data is below it. If [latex]n[\/latex] is the number of data values, we compute a locator by finding [latex]25\\%[\/latex] of [latex]n[\/latex]. If this locator is a decimal value, we round up, and find the data value in that position. If the locator is a whole number, we find the mean of the data value in that position and the next data value. This is identical to the process we used to find the median, except we use [latex]25\\%[\/latex] of the data values rather than half the data values as the locator.<\/p>\n

            \n

            How To: Find the First Quartile, [latex]Q1[\/latex]<\/strong><\/p>\n

              \n
            1. Begin by ordering the data from smallest to largest<\/li>\n
            2. Compute the locator: [latex]L = 0.25n[\/latex]<\/li>\n
            3. If [latex]L[\/latex] is a decimal value:\n
                \n
              • Round up to [latex]L+[\/latex]<\/li>\n
              • Use the data value in the [latex]L+[\/latex]th position<\/li>\n<\/ul>\n<\/li>\n
              • If [latex]L[\/latex] is a whole number:\n
                  \n
                • Find the mean of the data values in the [latex]L[\/latex]<\/em>th and [latex]L+1[\/latex]th positions.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

                  How To: Find the Third Quartile, [latex]Q3[\/latex]<\/strong><\/p>\n

                  Use the same procedure as for [latex]Q1[\/latex], but with locator: [latex]L = 0.75n[\/latex]<\/p>\n<\/section>\n

                  To find minimum and maximum values of a data set, find the least and the greatest value in the data set. It might be best to order your data from smallest to the largest because you can can also find the measure of center, median, through the ordered data set as well. Recall: the median is the value that splits the data in half.<\/p>\n

                  Suppose we have measured [latex]9[\/latex] females, and their heights (in inches) sorted from smallest to largest are:<\/p>\n
                  [latex]59, 60, 62, 64, 66, 67, 69, 70, 72[\/latex]<\/div>\n

                  \nWhat are the first and third quartiles?<\/p>\n