# Find the sample variance and standard deviation. 20, 16​, 2​, 7​, 12

Find the sample variance and standard deviation. 20, 16​, 2​, 7​, 12
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constanzma
Step 1
Given data set is 20,16,2,7,12
Let the given data set be ${x}_{1},{x}_{2},{x}_{3},{x}_{4},{x}_{5}$
${x}_{1}=20\phantom{\rule{0ex}{0ex}}{x}_{2}=16\phantom{\rule{0ex}{0ex}}{x}_{3}=2\phantom{\rule{0ex}{0ex}}{x}_{4}=7\phantom{\rule{0ex}{0ex}}{x}_{5}=12$
It is clear that, the total number of data points $\left(x\right)=5$
The formula for the sample variance is given by
Sample variance $\left({\sigma }^{2}\right)=\frac{\left({x}_{i}-\overline{x}{\right)}^{2}}{n-1}$ Formula
where $\overline{x}=mean$.
which is given by $\overline{x}=\frac{\text{Sum of all data points}}{\text{Total number of datapoints}}$ Formula.
Calculation of mean $\left(\overline{x}\right)$
We know $mean\left(\overline{x}\right)=\frac{\text{Sum of all data points}}{\text{Total number of data points}}$
$⇒\overline{x}=\frac{{x}_{1}+{x}_{2}+{x}_{3}+{x}_{4}+{x}_{5}}{n}$
Step 2
$⇒\overline{x}=\frac{20+16+2+7+12}{5}$
$⇒\overline{x}=\frac{57}{5}$
$⇒\overline{x}=11.4$
mean $\left(\overline{x}\right)$ of the given data set is $\overline{x}=11.4$
We know $⇒{\sigma }^{2}=\frac{\left({x}_{1}-\overline{x}{\right)}^{2}+\left({x}_{2}-\overline{x}{\right)}^{2}+\left({x}_{3}-\overline{x}{\right)}^{2}+\left({x}_{4}-\overline{x}{\right)}^{2}+\left({x}_{5}-\overline{x}{\right)}^{2}}{5-1}$
$⇒{\sigma }^{2}=\frac{\left(20-11.4{\right)}^{2}+\left(16-11.4{\right)}^{2}+\left(2-11.4{\right)}^{2}+\left(7-11.4{\right)}^{2}+\left(12-11.4{\right)}^{2}}{4}$
$⇒{\sigma }^{2}=\frac{\left(8.6{\right)}^{2}+\left(4.6{\right)}^{2}+\left(-9.4{\right)}^{2}+\left(-4.4{\right)}^{2}+\left(0.6{\right)}^{2}}{4}$
$⇒{\sigma }^{2}=\frac{73.96+21.16+88.36+19.36+0.36}{4}$
$⇒{\sigma }^{2}=\frac{203.2}{4}$
$⇒{\sigma }^{2}=50.8$
Step 3
We obtained
Variance $\left({\sigma }^{2}\right)=50.8$
Standard Deviation $\left(\sigma \right)=\sqrt{Variance}$
$⇒\sigma =\sqrt{50.8}$
$⇒\sigma =7.127$
Standard deviation $\left(\sigma \right)=7.127$
We obtained
Sample variance $\left({\sigma }^{2}\right)=50.8$
Sample standard deviation $\left(\sigma \right)=7.127$

Elsa Brewer
Step 1:
$20+16+2+7+12=57$
Divide this by number of data $=5$
Mean $=57/5=11.4$
Step 2:

Step 3
Sum of all the squares in the third column $=203.20$
Step 4
Sample Variance $=203.20/5-1=203.20/4=50.80$
Step 5
Standard deviation $={\text{sample variance}}^{1/2}={50.80}^{1/2}=7.1274$
So,the standard deviation is 7.1274.