Question

Calculate, by hand, the variance of the data set, which is a sample of a population.Round your answer to the nearest tenth.12, 14, 19, 11, 8, 21, and 13

Answer 1

- The formula for finding the variance of the sample dataset is given below:

`\begin{gathered} \sum ^n_(i=1)\frac{(x_i-\bar{x})^2}{n-1} \\ \\ \text{where,} \\ \bar{x}=\text{The mean of the sample} \\ x_i=\text{ The individual data points in the dataset} \\ n=\text{The number of data points in the sample} \end{gathered}`

- The data points have been given to be 12, 14, 19, 11, 8, 21, and 13.

- The formula for finding the Mean is

`\begin{gathered} \sum ^n_(i=1)(x_i)/(n) \\ \text{where,} \\ x_i=\text{The individual data point} \\ n=\text{The number of data points in the sample} \end{gathered}`

- Thus, we can simply apply the formula given above to solve the question. We shall follow these steps to solve this question:

1. Find the Mean.

2. Calculate the Variance

1. Find the Mean

`\begin{gathered} \bar{x}=(12+14+19+11+8+21+13)/(7) \\ \\ \bar{x}=14 \end{gathered}`

2. Calculate the Variance:

`\begin{gathered} s^2=\sum ^n_(i=1)\frac{(x_i-\bar{x})^2}{n-1} \\ \\ =((12-14)^2+(14-14)^2+(19-14)^2+(11-14)^2+(8-14)^2+(21-14)^2+(13-14)^2)/(7-1) \\ \\ =((-2)^2+0^2+5^2+(-3)^2+(-6)^2+7^2+(-1)^2)/(6) \\ \\ =(4+0+25+9+36+49+1)/(6)=(124)/(6) \\ \\ s^2=20(4)/(6)=20.666\ldots\approx20.7\text{ (To the nearest tenth)} \end{gathered}`

Final Answer

The value of the variance is

`s^2=20.7`