Question

we have calculated $(x - \bar{x})^2$ for each data value in the data set.

x	1	2	5	7	9
$x - \bar{x}$	-3.8	-2.8	0.2	2.2	4.2
$(x - \bar{x})^2$	14.44	7.84	0.04	4.84	17.64

now use the defining formula for sample standard deviation $s = \sqrt{\frac{\sum(x - \bar{x})^2}{n - 1}}$ with $n = 5$. recall that the sum is taken over all.
$s = \sqrt{\frac{\sum(x - \bar{x})^2}{n - 1}}$
$=\sqrt{\frac{14.44 + 7.84 + 0.04+\square+ 17.64}{5 - 1}}$
$=\sqrt{\frac{\square}{4}}$
$=\square$

Explanation:

Step1: Identify sum of squared - deviations

We know the values of \((x - \bar{x})^2\) are \(14.44\), \(7.84\), \(0.04\), \(4.84\), \(17.64\). The sum \(\sum(x - \bar{x})^2=14.44 + 7.84+0.04 + 4.84+17.64\).

Step2: Calculate sum

\(\sum(x - \bar{x})^2=14.44+7.84 + 0.04+4.84+17.64 = 44.8\).

Step3: Apply sample - standard deviation formula

The formula for sample standard deviation is \(s=\sqrt{\frac{\sum(x - \bar{x})^2}{n - 1}}\), with \(n = 5\). Substitute \(\sum(x - \bar{x})^2 = 44.8\) and \(n=5\) into the formula: \(s=\sqrt{\frac{44.8}{5 - 1}}=\sqrt{\frac{44.8}{4}}\).

Step4: Calculate standard deviation

\(\sqrt{\frac{44.8}{4}}=\sqrt{11.2}\approx3.35\).

Answer:

\(3.35\)