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How to calculate skewness and kurtosis

This is an excerpt from Statistics in Kinesiology 6th Edition by Joseph P. Weir,Anthony B. Ciccone,Jacob A. Siedlik,William J. Vincent.

The process of statistical inference is addressed in detail in chapter 7. A major assumption of statistical inference is that the characteristics of the normal curve can be applied. Consequently, it is critical that we know whether the data deviate from normality. Skewness is a measure of the bilateral symmetry of the data, and kurtosis is a measure of the relative heaviness of the tails of the curve of the data.

By observing a graph of the data and identifying the three measures of central tendency, we can get a general idea of the skewness of the data; however, this method is not exact (see figure 4.1). Using Z scores, we can obtain a numerical value that indicates the amount of skewness or kurtosis in any set of data.

Because Z scores are a standardized measure of the deviation of each raw score from the mean, we can use Z scores to determine whether the raw scores are equally distributed around the mean. When the data are completely normal, or bilaterally symmetrical, the sum of the Z scores above the mean is equal but opposite in sign to the sum of the Z scores below the mean. The positive and negative values cancel each other out, and the grand sum of the Z scores is zero.

If we take the third moment (the cube of the Z scores, or Z³), we can accentuate the extreme values of Z, but the signs of the Z values remain the same. This places greater weight on the extreme scores and permits a numeric evaluation of the amount of skewness. Computing the average of the Z³ scores produces a raw score value for skewness. The formula for calculating the raw value for skewness is

where N is the sample size.

When the Z³ mean is zero, the data are normal. When the Z³ mean is positive, the data are skewed positive, and when the Z³ mean is negative, the data are skewed negative. This effect can be seen by examining the data presented in table 6.1. Notice that the data are skewed negative. When these data are graphed (see figure 6.7), the skewness is easily observed.

Kurtosis may also be calculated from Z scores. By taking the fourth moment Z⁴ of the Z scores, the extreme Z values are again accentuated, but the signs are all converted to positive. When the average of the Z⁴ value is 3.0, the curve is normal. To make the units equal for both skewness and kurtosis, the mean of Z⁴ is typically reduced by 3.0. The formula for calculating the raw value for kurtosis is (AndersonBell, 1989, p. 173; Spiegel, 1961, p. 91)

A score of 0 indicates complete normal kurtosis, or a mesokurtic curve, just as a score of 0 for skewness indicates complete bilateral symmetry. When the raw score for kurtosis is greater than 0.0, the curve is leptokurtic (heavier tails than normal), and when the raw score is less than 0.0, the curve is platykurtic (thinner tails than normal) (Westfall, 2014). The heavier tails of leptokurtic curves suggest there is a higher incidence of outliers.

Raw skewness and kurtosis scores are not easily interpreted because a raw score alone does not indicate a position on a known scale. But when raw scores are converted to Z scores, they are easy to interpret. To convert the raw scores for skewness (equation 6.04) or kurtosis (equation 6.05) to Z scores for skewness or kurtosis, we divide the raw scores by a factor called the standard error. Standard error is a type of standard deviation. (We explore standard error in more detail in subsequent chapters.) Therefore, when we divide the raw skewness and kurtosis scores by the appropriate standard error, the result is a Z score for skewness and kurtosis. According to Dixon (1990, p. 137), the standard error (SE) for skewness is

and the standard error for kurtosis is

If we divide the raw scores for skewness or kurtosis by the appropriate standard error, we obtain a Z_skew or Z_kurt value as follows:

These values may be interpreted as Z scores (i.e., values greater than 1.96 or less than 1.96 exceed p = .05, and values greater than 2.58 or less than −2.58 exceed p = .01). Typically, data are considered to be within acceptable limits of skewness or kurtosis if the Z values do not exceed ±2.0.

Using the data from table 6.1, we can find Z_skew in the following manner:

Z_kurt can be found as follows:

From these values (Z_skew = −1.07 and Z_kurt = −0.37), we can determine that the data in table 6.1 and figure 6.7 are skewed slightly negative and slightly platykurtic; however, neither value approaches significance (±2.0). Therefore, we may conclude that the data are within acceptable ranges of normality. Data sets with small values of N may appear to be significantly skewed when graphed (see figure 6.7), but the true evaluation of the degree of skewness must be made by Z score analysis.

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