Question

Example 4 Suppose we have the following data: 2, 3, 5, 6,7 What is the mean of these data? What is the median? What is the first quartile? What is the third quartile? 26

Answer 1

Solutions :

$2,\quad\:3,\quad\:5,\quad\:6,\quad\:7$

$\bold{Mean \::}$

$\mathrm{Mean = \frac{Sum ~ of ~ terms}{Number ~ of ~ terms}}=\frac{2 + 3 + 5 + 6 + 7 }{5}=\frac{23}{5}=4.6$

$\mathrm{So,\:the\:mean\:is\:\:\bold{4.6}}$

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$\bold{Median \::}$

$\mathrm{The\:median\:is\:the\:middle\:number\:in\:a\:sorted\:list\:of\:numbers.\:So,\:to\:find\:the\:median,\:we\:need\:to\:place\:the\:numbers\:in\:value\:order\:and\:find\:the\:middle\:number.}$

$\mathrm{Ordering\:the\:data\:from\:least\:to\:greatest,\:we\:get\::}$

$2\quad\:3\quad\:\bold{5}\quad\:6\quad\:7$

$\mathrm{So,\:the\:median\:is\:\:\bold{5}}$

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$\bold{First\:quartile\::}$

$\mathrm{The\:first\:quartile\:\left(or\:lower\:quartile\:or\:25th\:percentile\right)\:is\:the\:median\:of\:the\:bottom\:half\:of\:the\:numbers.\:So,\:to\:find\:the\:first\:quartile,\:we\:need\:to\:place\:the}$

$\mathrm{\:numbers\:in\:value\:order\:and\:find\:the\:bottom\:half.}$

$\bold{2}\quad\:\bold{3}\quad\:5\quad\:6\quad\:7$

$\mathrm{So,\:the\:bottom\:half\:is}$

$\bold{2}\quad\:\bold{3}$

$\mathrm{The\:median\:of\:these\:numbers\:=\:\frac{2+3}{2}=\frac{5}{2}=2.5}$

$\mathrm{Hence,\quad\:the\:first\:quartile\:of\:the\:data\:set\:is\:\:\:\bold{2.5}}$

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$\bold{Second\:quartile\::}$

$\mathrm{The\:third\:quartile\:\left(or\:upper\:quartile\:or\:75th\:percentile\right)\:is\:the\:median\:of\:the\:upper\:half\:of\:the\:numbers.\:So,\:to\:find\:the\:third\:quartile,\:we\:need\:to\:place\:the}$

$\mathrm{\:numbers\:in\:value\:order\:and\:find\:the\:upper\:half.}$

$2\quad\:3\quad\:5\bold{\quad\:6\quad\:7}$

$\mathrm{So,\:the\:upper\:half\:is}$

$\bold{\quad\:6\quad\:7}$

$\mathrm{The\:median\:of\:these\:numbers\:=\:\frac{6+7}{2}=\frac{13}{2}=6.5\:\:}$

$\mathrm{Hence,\quad\:the\:first\:quartile\:of\:the\:data\:set\:is\:\:\:\bold{6.5}.}$