Formulaic Maths Problems - get another problem: level 1, level 2, level 3 - link to this problem
Loading, please wait...
Find the population standard deviation of the dataset
$x = 74$, $67$, $68$, $75$, $75$, $69$, $76$, $74$, $72$, $73$.
Find the population mean. Hint: The population mean is given by the formula $\mu = \dfrac{\sum x}{n}$.
$\mu = \dfrac{\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{A1}}\hspace{54px}}~}}{\class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{A2}}\hspace{54px}}~}} = \class{inputBox step1}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{AR}}\hspace{54px}}~}$ $\mu = \dfrac{723}{10} = \dfrac{723}{10}$
Complete the table.
$x$$x - \mu$$(x - \mu)^2$
$74$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c0}}\hspace{54px}}~}$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r0c1}}\hspace{80px}}~}$
$67$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c0}}\hspace{54px}}~}$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r1c1}}\hspace{80px}}~}$
$68$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c0}}\hspace{54px}}~}$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r2c1}}\hspace{80px}}~}$
$75$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c0}}\hspace{54px}}~}$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r3c1}}\hspace{80px}}~}$
$75$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r4c0}}\hspace{54px}}~}$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r4c1}}\hspace{80px}}~}$
$69$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r5c0}}\hspace{54px}}~}$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r5c1}}\hspace{80px}}~}$
$76$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r6c0}}\hspace{54px}}~}$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r6c1}}\hspace{80px}}~}$
$74$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r7c0}}\hspace{54px}}~}$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r7c1}}\hspace{80px}}~}$
$72$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r8c0}}\hspace{54px}}~}$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r8c1}}\hspace{80px}}~}$
$73$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r9c0}}\hspace{54px}}~}$$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{r9c1}}\hspace{80px}}~}$
$\class{inputBox step2}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{BR}}\hspace{80px}}~}$
$x$$x - \mu$$(x - \mu)^2$
$74$$\dfrac{17}{10}$$\dfrac{289}{100}$
$67$$-\dfrac{53}{10}$$\dfrac{2809}{100}$
$68$$-\dfrac{43}{10}$$\dfrac{1849}{100}$
$75$$\dfrac{27}{10}$$\dfrac{729}{100}$
$75$$\dfrac{27}{10}$$\dfrac{729}{100}$
$69$$-\dfrac{33}{10}$$\dfrac{1089}{100}$
$76$$\dfrac{37}{10}$$\dfrac{1369}{100}$
$74$$\dfrac{17}{10}$$\dfrac{289}{100}$
$72$$-\dfrac{3}{10}$$\dfrac{9}{100}$
$73$$\dfrac{7}{10}$$\dfrac{49}{100}$
$\dfrac{921}{10}$
Find the population variance. Hint: The population variance is given by the formula $\sigma^2 = \dfrac{\sum (x - \mu)^2}{n}$.
$\sigma^2 = \dfrac{\class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{C1}}\hspace{80px}}~}}{\class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{C2}}\hspace{80px}}~}} = \class{inputBox step3}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{CR}}\hspace{100px}}~}$ $\sigma^2 = \dfrac{\dfrac{921}{10}}{10} = \dfrac{921}{100}$
Therefore, Hint: The standard deviation is the square root of the variance.
$\sigma = \sqrt{\class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{D1}}\hspace{100px}}~}} = \class{inputBox step4}{~\bbox[border:2px solid blue]{\strut\rlap{\class{inputReplace}{DR}}\hspace{100px}}~}$ $\sigma = \sqrt{\dfrac{921}{100}} = 3.03479818$