Measures of Skewness
Introduction
The objects of Descriptive Statistics are mainly – condensation by frequency distribution, and description of the frequency distribution by suitable statistical measures. The raw data which by its mere bulk is unintelligible, should be replaced by a relatively few quantities which will adequately describe the whole situation. It may be observed here that frequency distributions not only condense the bulk of observations, but it also facilitates the calculation of the statistical measures which seek to describe the characteristics of frequency distributions. Although, the mean and standard deviation describing the two characteristics of a frequency distribution specify completely a class of symmetrical distributions, called ‘normal distribution’, which we often come across in both theoretical and applied statistics, there are other distributions which will not be adequately described by these two measures only. Two frequency distributions may have the same mean and standard deviation and yet may differ with respect to another characteristics – the Skewness or, asymmetry of the distribution.
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Definition
The word Skewness refers to lack of symmetry. Non-normal or asymmetrical distribution is called skew distribution. Any measure of Skewness indicates the difference between the manner in which items are distributed in a particular distribution compared with a normal distribution. Lack of symmetry or Skewness in frequency distributions is due to the existence of a longer tail on one side (either to the left or the right), which has no counterpart on the other side. If the larger tail is on the right, we say that the distribution is positively skewed; whereas if the longer tail is on the left side, we say that the distribution is negatively skewed.
What are the tests of Skewness?
To find out whether the distribution is normal or skewed, the following tests are applied:
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Relationship between averages: If the values of mean, median and mode are not equal the distribution is skewed.
Total of Deviations: The distribution is skewed when the sum of the positive deviations from the median is not equal to the sum of the negative deviations.
The distance of Partition Values from the Median: In a skewed distribution quartiles are not equidistant from the median.
The Curve: When the data are plotted on the graph paper they do not give the normal bell-shaped curve. In other words, when the curve is cut along a vertical line through the centre the two halves are not equal.
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The frequencies on either side of the mode: In a skewed distribution frequencies are not equally distributed at points of equal deviation from mode.