Question

find the variance of the data.
16, 13, 20, 16, 12, 17, 18
$overline{x} = 16$
variance ($sigma^2$) = ?
round to the nearest tenth.

Explanation:

Step1: Recall variance formula

The formula for population variance is $\sigma^2=\frac{\sum_{i = 1}^{n}(x_i-\overline{x})^2}{n}$, where $x_i$ are the data points, $\overline{x}$ is the mean, and $n$ is the number of data points. Here, $n = 7$, $\overline{x}=16$, and the data points are 16, 13, 20, 16, 12, 17, 18.

Step2: Calculate each $(x_i - \overline{x})^2$

• For $x_1 = 16$: $(16 - 16)^2=0^2 = 0$
• For $x_2 = 13$: $(13 - 16)^2=(-3)^2 = 9$
• For $x_3 = 20$: $(20 - 16)^2=4^2 = 16$
• For $x_4 = 16$: $(16 - 16)^2=0^2 = 0$
• For $x_5 = 12$: $(12 - 16)^2=(-4)^2 = 16$
• For $x_6 = 17$: $(17 - 16)^2=1^2 = 1$
• For $x_7 = 18$: $(18 - 16)^2=2^2 = 4$

Step3: Sum the squared differences

$\sum_{i = 1}^{7}(x_i-\overline{x})^2=0 + 9+16 + 0+16 + 1+4=46$

Step4: Divide by $n$ to get variance

$\sigma^2=\frac{46}{7}\approx6.6$ (rounded to the nearest tenth)

Answer:

$6.6$