calculating variance and population standard deviation\nthe average estimated hours a person in the united…

Answer:

The variance of this data - set is 498. The population standard deviation for this data - set is 22.

Explanation:

Step1: Calculate the mean

$\bar{x}=\frac{71 + 80+82+78+80+91+107+121+125+131+142}{11}=\frac{1118}{11}\approx101.64$

Step2: Calculate the squared - differences

For each data point $x_i$, calculate $(x_i-\bar{x})^2$. For example, when $x_1 = 71$, $(71 - 101.64)^2=(- 30.64)^2 = 938.81$. Do this for all 11 data points.

Step3: Calculate the variance

The formula for population variance $\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n}$. $\sum_{i = 1}^{11}(x_i-\bar{x})^2=938.81+(80 - 101.64)^2+(82 - 101.64)^2+(78 - 101.64)^2+(80 - 101.64)^2+(91 - 101.64)^2+(107 - 101.64)^2+(121 - 101.64)^2+(125 - 101.64)^2+(131 - 101.64)^2+(142 - 101.64)^2$ $=938.81 + 468.29+385.73+558.81+468.29+113.21+28.73+374.81+545.73+862.01+1629.01$ $=5475.9$ $\sigma^{2}=\frac{5475.9}{11}\approx497.81\approx498$

Step4: Calculate the population standard deviation

The formula for population standard deviation $\sigma=\sqrt{\sigma^{2}}$. $\sigma=\sqrt{497.81}\approx22.31\approx22$