Question

find the variance of the data. 78, 45, 73, 66, 61, 67 (overline { x } = 65) variance (( sigma ^ { 2 } ) = )

Explanation:

Step1: Calculate the squared differences

For each data point \(x_i\), calculate \((x_i - \bar{x})^2\).

• For \(x_1 = 78\): \((78 - 65)^2=13^2 = 169\)
• For \(x_2 = 45\): \((45 - 65)^2=(- 20)^2 = 400\)
• For \(x_3 = 73\): \((73 - 65)^2 = 8^2=64\)
• For \(x_4 = 66\): \((66 - 65)^2=1^2 = 1\)
• For \(x_5 = 61\): \((61 - 65)^2=(-4)^2 = 16\)
• For \(x_6 = 67\): \((67 - 65)^2=2^2 = 4\)

Step2: Find the sum of squared differences

\(\sum_{i = 1}^{n}(x_i-\bar{x})^2=169 + 400+64 + 1+16+4=654\)

Step3: Calculate the variance

The formula for variance \(\sigma^{2}=\frac{\sum_{i = 1}^{n}(x_i - \bar{x})^2}{n}\) (assuming this is a population). Here \(n = 6\).
\(\sigma^{2}=\frac{654}{6}=109\)

Answer:

\(109\)