Symmetry, Skewness and Kurtosis

We consider a random variable x and a data set S = {x1, x2, …, xn} of size n which contains possible values of x. The data set can represent either the population being studied or a sample drawn from the population.

Looking at S as representing a distribution, the skewness of S is a measure of symmetry while kurtosis is a measure of peakedness of the data in S.

Symmetry and Skewness

Definition 1: We use skewness as a measure of symmetry. If the skewness of S = 0 then the distribution represented by S is perfectly symmetric. If the skewness is negative, then the distribution is skewed to the left, while if the skew is positive then the distribution is skewed to the right (see Figure 1 below for an example).

Consistent with Excel we calculate the skewness of S as follows:

Skewness Formula in Excel

where is the mean and s is the standard deviation of S. To avoid division by zero, this formula requires that n > 2.

Observation: When a distribution is symmetric, the mean = median, when the distribution is positively skewed the mean > median and when the distribution is negatively skewed the mean < median.

Excel Function: Excel provides the SKEW function as a way to calculate the skewness of S, i.e. if R is a range in Excel containing the data elements in S then SKEW(R) = the skewness of S.

Excel 2013 Function: There is also a population version of the skewness given by the formula

Skewness population

This version has been implemented in Excel 2013 using the function, SKEW.P.

It turns out that for range R consisting of the data in S = {x1, …, xn}, SKEW.P(R) = SKEW(R)*(n–2)/SQRT(n(n–1)) where n = COUNT(R).

Example 1: Suppose S = {2, 5, -1, 3, 4, 5, 0, 2}. The skewness of S = -0.43, i.e. SKEW(R) = -0.43 where R is a range in an Excel worksheet containing the data in S. Since this value is negative, the curve representing the distribution is skewed to the left (i.e. the fatter part of the curve is on the right). Also SKEW.P(R) = -0.34.

Observation: SKEW(R) and SKEW.P(R) ignore any empty cells or cells with non-numeric values.

Kurtosis

Definition 2: We use kurtosis as a measure of peakedness (or flatness). Positive kurtosis indicates a relatively peaked distribution. Negative kurtosis indicates a relatively flat distribution.

Consistent with Excel we calculate the kurtosis of S as follows:

Kurtosis Formula in Excel

where is the mean and s is the standard deviation of S. To avoid division by zero, this formula requires that n > 3.

Excel Function: Excel provides the KURT function as a way to calculate the kurtosis of S, i.e. if R is a range in Excel containing the data elements in S then KURT(R) = the kurtosis of S.

Example 2: Suppose S = {2, 5, -1, 3, 4, 5, 0, 2}. The kurtosis of S = -0.94, i.e. KURT(R) = -0.94 where R is a range in an Excel worksheet containing the data in S. Since this value is negative, the curve representing the distribution is relatively flat.

Observation: KURT(R) ignores any empty cells or cells with non-numeric values.

Graphic Illustration

We now look at an example of these concepts using the chi-square distribution.

Chi-square distribution

Figure 1 – Example of skewness and kurtosis

Figure 1 contains the graph of two chi-square distributions (with different degrees of freedom df). We study the chi-square distribution elsewhere, but for now note the following values for the kurtosis and skewness:

Comparison of skewness and kurtosis

Figure 2 – Comparison of skewness and kurtosis

The red curve (df = 10) is flatter than the blue curve (df = 5), which is reflected in the fact that the kurtosis value of the blue curve is lower.

Both curves are asymmetric, and skewed to the right (i.e. the fat part of the curve is on the left). This is consistent with the fact that the skewness for both is positive. But the blue curve is more skewed to the right, which is consistent with the fact that the skewness of the blue curve is larger.

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