Question

find the variance of the data.
15, 21, 46, 49, 31, 24
\\(overline{x} = 31\\)
variance (\\(sigma^2\\)) = ?

Explanation:

Step1: Recall the variance formula

The formula for population variance is $\sigma^{2}=\frac{\sum (x_{i}-\overline{x})^{2}}{n}$, where $x_{i}$ are the data points, $\overline{x}$ is the mean, and $n$ is the number of data points. Here, $n = 6$, $\overline{x}=31$, and the data points are 15, 21, 46, 49, 31, 24.

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

• For $x_{1}=15$: $(15 - 31)^{2}=(- 16)^{2}=256$
• For $x_{2}=21$: $(21 - 31)^{2}=(-10)^{2}=100$
• For $x_{3}=46$: $(46 - 31)^{2}=(15)^{2}=225$
• For $x_{4}=49$: $(49 - 31)^{2}=(18)^{2}=324$
• For $x_{5}=31$: $(31 - 31)^{2}=(0)^{2}=0$
• For $x_{6}=24$: $(24 - 31)^{2}=(-7)^{2}=49$

Step3: Sum the squared differences

$\sum (x_{i}-\overline{x})^{2}=256 + 100+225 + 324+0 + 49=954$

Step4: Divide by $n$ to get variance

$\sigma^{2}=\frac{954}{6}=159$

Answer:

159