Question

Which of the following would be the variance of this population data set: 3, 9,8, 9, 4, 5, 7, 11, 9, 7, 5, 4, 3, 1

Answer 1

GIVEN:

We are given the set of data as indicated below;

`3,9,8,9,4,5,7,11,9,7,5,4,3,1`

Required;

To find the variance of the data set.

Step-by-step solution;

We begin by calculating the mean of the data set as follows;

`\begin{gathered} Mean=(3+9+8+9+4+5+7+11+9+7+5+4+3+1)/(14) \\ \\ Mean=(85)/(14)=6.07142857143 \end{gathered}`

We now round the mean to two decimal places and we have;

`Mean\approx6.07`

Next we subtract the mean from EACH value in the data set. the individual results are the "deviation from the mean."

After that we square each deviation from the mean, and then add them all up.

This is effectively explained by the formula for the variance which is;

`\begin{gathered} variance=s^2 \\ \\ s^2=\frac{\Sigma(x-\bar{x})^2}{n-1} \end{gathered}`

We now have;

`\begin{gathered} (3-6.07)^2+(9-6.07)^2+(8-6.07)^2+...+(3-6.07)+(1-6.07)^2 \\ \\ =9.4249+8.5849+3.7249+8.5849+4.2849+1.1449+0.8649+24.3049 \\ \\ +8.5849+0.8649+1.1449+4.2849+9.4249+25.7049 \\ \\ =110.9286 \end{gathered}`

Now we can refine the formula as shown below;

`\begin{gathered} s^2=\frac{\Sigma(x-\bar{x})^2}{n-1} \\ \\ Where,\text{ }n=14 \\ \\ s^2=(110.9286)/(14-1) \\ \\ s^2=(110.9286)/(13) \\ \\ s^2=8.53296923077 \end{gathered}`

We can round this to 2 decimal places and the variance therefore is;

ANSWER:

`variance\approx8.53`