Question

find the variance of the data.
251, 224, 237, 255, 268
\\(overline{x} = 247\\)
variance (\\(sigma^2\\)) = ?

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, $n = 5$, $\overline{x}=247$, and the data points are 251, 224, 237, 255, 268.

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

• For $x_1 = 251$: $(251 - 247)^2=(4)^2 = 16$
• For $x_2 = 224$: $(224 - 247)^2=(- 23)^2=529$
• For $x_3 = 237$: $(237 - 247)^2=(-10)^2 = 100$
• For $x_4 = 255$: $(255 - 247)^2=(8)^2=64$
• For $x_5 = 268$: $(268 - 247)^2=(21)^2 = 441$

Step3: Sum the squared differences

$\sum_{i = 1}^{5}(x_i-\overline{x})^2=16 + 529+100 + 64+441=1150$

Step4: Divide by $n$ to get variance

$\sigma^2=\frac{1150}{5}=230$

Answer:

230