Question

calculate the variance of this sample of scores: 2, 4, 7, 4, 8, 5, 1, 4, 4, 5. 4.27 3.84 4.4

Explanation:

Step1: Calculate the mean

The formula for the mean \(\bar{x}\) of a sample \(x_1,x_2,\cdots,x_n\) is \(\bar{x}=\frac{\sum_{i = 1}^{n}x_i}{n}\).
Here, \(n = 10\) and \(\sum_{i=1}^{10}x_i=2 + 4+7 + 4+8 + 5+1 + 4+4 + 5=44\).
So, \(\bar{x}=\frac{44}{10}=4.4\).

Step2: Calculate the sum of squared - deviations

The formula for the sum of squared - deviations \(\sum_{i = 1}^{n}(x_i-\bar{x})^2\) is:
\((2 - 4.4)^2+(4 - 4.4)^2+(7 - 4.4)^2+(4 - 4.4)^2+(8 - 4.4)^2+(5 - 4.4)^2+(1 - 4.4)^2+(4 - 4.4)^2+(4 - 4.4)^2+(5 - 4.4)^2\)
\(=(- 2.4)^2+(-0.4)^2+(2.6)^2+(-0.4)^2+(3.6)^2+(0.6)^2+(-3.4)^2+(-0.4)^2+(-0.4)^2+(0.6)^2\)
\(=5.76 + 0.16+6.76+0.16 + 12.96+0.36+11.56+0.16+0.16+0.36\)
\(=38.4\)

Step3: Calculate the sample variance

The formula for the sample variance \(s^2\) is \(s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}\).
Since \(n = 10\) and \(\sum_{i = 1}^{n}(x_i-\bar{x})^2 = 38.4\), then \(s^2=\frac{38.4}{9}\approx4.27\)

Answer:

4.27