(a) Prove that for ungrouped observations with arithmetic mean , the variance can be…

MATHEMATICS · Subjective · A Level · Mathematics · Data presentation and interpretation · Data presentation and interpretation

Problem

(a) Prove that for nn ungrouped observations x1,x2,…,xnx_1, x_2, \dots , x_n with arithmetic mean xˉ\bar x, the variance can be written as σ2=1n∑i=1nxi2−xˉ2.\sigma^{2}=\frac{1}{n}\sum_{i=1}^{n}x_i^{2}-\bar x^{2}. (b) Using the result obtained in part (a), calculate the variance and the standard deviation for the observations: 12, 15, 20, 22, 26. (c) Comment on whether the data are closely clustered around the mean based on the value of the standard deviation obtained.

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