IGNOU EEC 13 2017 2018 - Explain the concepts of skewness and kurtosis. Briefly describe how th

Question

Explain the concepts of skewness and kurtosis. Briefly describe how these can be measured.

08 Mar 2018

Answer :

Word Count : 1365

Skewness and Kurtosis

A fundamental task in many statistical analyses is to characterize the location and variability of a data set. A further characterization of the data includes skewness and kurtosis.

Skewness is a measure of symmetry, or more precisely, the lack of symmetry. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point.

Kurtosis is a measure of whether the data are heavy-tailed or light-tailed relative to a normal distribution. That is, data sets with high kurtosis tend to have heavy tails, or outliers. Data sets with low kurtosis tend to have light tails, or lack of outliers. A uniform distribution would be the extreme case.

The histogram is an effective graphical technique for showing both the skewness and kurtosis of data set.

Definition of Skewness

For univariate data Y₁, Y₂, ..., Y_N, the formula for skewness is:

g 1 = \sum N i = 1 (Y i - Y ¯) 3 / N s 3

where Y¯ is the mean, s is the standard deviation, and N is the number of data points. Note that in computing the skewness, the s is computed with N in the denominator rather than N - 1.

The above formula for skewness is referred to as the Fisher-Pearson coefficient of skewness. Many software programs actually compute the adjusted Fisher-Pearson coefficient of skewness

G 1 = N (N - 1) - - - - - - - - \sqrt N - 2 \sum N i = 1 (Y i - Y ¯) 3 / N s 3

This is an adjustment for sample size. The adjustment approaches 1 as N gets large. For reference, the adjustment factor is 1.49 for N = 5, 1.19 for N = 10, 1.08 for N = 20, 1.05 for N = 30, and 1.02 for N = 100.

The skewness for a normal distribution is zero, and any symmetric data should have a skewness near zero. Negative values for the skewness indicate data that are skewed left and positive values for the skewness indicate data that are skewed right. By skewed left, we mean that the left tail is long relative to the right tail. Similarly, skewed right means that the right tail is long relative to the left tail. If the data are multi-modal, then this may affect the sign of the skewness.

Some measurements have a lower bound and are skewed right. For example, in reliability studies, failure times cannot be negative.

It should be noted that there are alternative definitions of skewness in the literature. For example, the Galton skewness (also known as Bowley's skewness) is defined as

Galton skewness = Q 1 + Q 3 - 2 Q 2 Q 3 - Q 1

where Q₁ is the lower quartile, Q₃ is the upper quartile, and Q₂ is the median.

The Pearson 2 skewness coefficient is defined as

S k 2 = 3 (Y ¯ - Y ~) s

where Y~ is the sample median.

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