Question

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

Explanation:

Step1: Calculate the squared differences

For each data point \(x_i\), calculate \((x_i - \bar{x})^2\).
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 \(x_5 = 268\): \((268 - 247)^2=21^2 = 441\)

Step2: Compute the variance

The formula for variance \(\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n}\) (assuming this is a population, \(n = 5\)).
\(\sum_{i=1}^{5}(x_i - \bar{x})^2=16 + 529+100 + 64+441=1150\)
\(\sigma^{2}=\frac{1150}{5}\)

Answer:

\(230\)