Question

find the variance of the data.
56, 52, 46, 49, 57, 64
\\( \overline { x } = 54 \\)
variance \\( ( \sigma ^ { 2 } ) = ? \\)

Explanation:

Step1: Calculate each data - mean difference squared

For \(x = 56\): \((56 - 54)^2=2^2 = 4\)
For \(x = 52\): \((52 - 54)^2=(-2)^2 = 4\)
For \(x = 46\): \((46 - 54)^2=(-8)^2 = 64\)
For \(x = 49\): \((49 - 54)^2=(-5)^2 = 25\)
For \(x = 57\): \((57 - 54)^2=3^2 = 9\)
For \(x = 64\): \((64 - 54)^2=10^2 = 100\)

Step2: Calculate the variance

The formula for variance \(\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}{n}\), where \(n = 6\)
\(\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=4 + 4+64 + 25+9+100=206\)
\(\sigma^{2}=\frac{206}{6}\approx34.3\)

Answer:

\(34.3\)