Statistics for Business: Decision Making and Analysis, 3rd Edition

... condition. The condition requires one statistic from the sample: the kurtosis K₄. The sample kurtosis detects the presence of skewness as well as outliers. If the data are normally distributed, then K₄ ≈ 0. Large, positive values of K₄ indicate that data are not normally distributed. K₄ is calculated from the z-scores of the sample, $z_i=\left(x_i-\overline{x}\right)\text{/}s.$

Sample Size Condition

Unless it is known that the data are a sample from a normally distributed population, find the kurtosis, K₄ (Chapter 12).

$$K_4=\frac{z_1^4+z_2^4+\dots +z_{\mathrm{n}}^4}{n}-3$$

A normal model provides an adequate approximation to the sampling distribution of $\overline{X}$ if the sample size n is larger than 10 times the absolute value of the kurtosis,

$$n>10\left|K_4\right|$$

As an example, consider the HALT scores of a sample of 20 ...