Course sections

Introduction to Analytics, Lecture 5

Skewness and Kurtosis

Two distributions may have the same mean and variance but may differ widely in their overall appearance. It is this difference which shows the presence of Skewness or the ‘lack of symmetry’. Again, two distributions may vary on the basis of ‘peakedness’ or on the basis of kurtosis. So, we try to characterize the distributions according to their shape.

Skewness

A distribution is known as a skewed distribution if it is asymmetrical. According to Simpson and Kafka “Skewness or asymmetry is the attribute of a frequency distribution that extends further on one side of the class with the highest frequency than on the other”. The idea of Skewness gives us an idea about the nature and extent of concentration of the observations towards the higher or the lower values of the variable.

A distribution is said to be skewed;

  • The frequency curve of the distribution is not a symmetric bell-shaped curve but it is stretched more to one side than to the other. In other words, it has a longer tail to one side (left or right) than to the other. A frequency distribution which has a longer tail towards the right is said to be positively skewed and if the longer tail lies to the left, it is said to be negatively skewed.
  • The mean, median and mode fall at different points, i.e. they do not coincide.
  • Quartiles Q1 and Q3 are not equidistant from the median.
  • The sum of the positive deviations about the median is not equal to the sum of the negative deviations from the median.

Measures of Skewness

Some of the absolute measure of skewness are:

  • Skewness = Mean-Mode = M – M0
  • Skewness = 3 (Mean- Median) =3 (M –Md)
  • Skewness = (Q3-Md) – (Md- Q1)

The absolute measures of Skewness are not much of practical use because of the following reasons:

  • Since the absolute measures of skewness involve the units of measurements, they cannot be used for comparative study of the two distributions measured in different units.
  • Even if the distributions are having the same unit of measurement, the absolute measures are not recommended because we may come across different distributions which have more or less identical skewness but which vary widely in the measures of central tendency.

Therefore, for comparing the two or more distributions for skewness we compute the relative measures of skewness, also known as coefficients of skewness which are pure numbers independent of units. The most commonly used coefficients of Skewness are:


The measures of skewness give us an idea about the spread of the frequency distribution. But, the spread of the distribution also has a relation to the peakedness of the distribution. This peakedness of a frequency distribution is discussed under Kurtosis.

Kurtosis

Kurtosis is the ‘peakedness’ of a frequency distribution. If the frequency curve has long tails and high peak, we call it ‘leptokurtic’ distribution. On the other hand, if the frequency curve has short and thick tails and is flat topped, we call it ‘platykurtic’. A ‘mesokurtic’ distribution describes the situation in between a leptokurtic and platykurtic distribution.

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