Skewness for a data distribution

A lack of symmetry is called skewness for data distribution. In other words, a departure from symmetry is called skewness.

A distribution is simply a collection of data, or scores, on a variable. Usually, these scores are arranged in order from smallest to largest and then they can be presented graphically.

Page 6, Statistics in Plain English, Third Edition, 2010.

Skewness

Let’s look at pictures of a Symmetric curve:

The measure of skewness gives the direction and the magnitude of the lack of symmetry.

If the distribution is not symmetric, the frequencies will not be uniformly distributed about the center of the distribution. We will look at pictures of asymmetric distributions shortly.

In mathematics, a figure is called symmetric if there exists a point in it through which if a perpendicular is drawn on the X-axis, it divides the figure into two congruent parts i.e. identical in all respect or one part can be superimposed on the other i.e mirror images of each other.

In Statistics, a distribution is called symmetric if the mean, median, and mode coincide. Otherwise, the distribution becomes asymmetric. 

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.

If the distribution is not symmetric, the frequencies will not be uniformly distributed about the center of the distribution.

Let’s look at pictures of asymmetric distributions:

If the right tail is longer, we get a positively skewed distribution for which mean > median > mode:

If the left tail is longer, we get a negatively skewed distribution for which mean < median < mode:

Skewness gives the direction of variability.

Measures of skewness help us to know to what degree and in which direction (positive or negative) the frequency distribution has a departure from symmetry.

Although positive or negative skewness can be detected graphically depending on whether the right tail or the left tail is longer, we don’t get an idea of the magnitude.

The following are the absolute measures of skewness:

1. Skewness (Sk) = Mean – Median 

2. Skewness (Sk) = Mean – Mode 

3. Skewness (Sk) = (Q3 – Q2) – (Q2 – Q1)

For comparison to series, we do not calculate these absolute measures. We calculate the relative measures which are called the coefficient of skewness. The coefficient of skewness is pure numbers independent of units of measurement. 

Relative Measures of Skewness

Karl Pearson’s Coefficient of Skewness

This method is most frequently used for measuring skewness. The formula for measuring the coefficient of skewness is given by:

Sk =  (Mean-Mode) / standard deviation

The value of this coefficient would be zero in a symmetrical distribution. If the mean is greater than the mode, the coefficient of skewness would be positive otherwise negative. The value of Karl Pearson’s coefficient of skewness usually lies between 1 for moderately skewed distribution.

If the value of mean, median, and mode are the same in any distribution, then the skewness does not exist in that distribution. Larger the difference in these values, the larger the skewness.

If sum of the frequencies are equal on both sides of the mode then skewness does not exist.

If the distance of the first quartile and third quartile are the same from the median then a skewness does not exist. Similarly, if deciles (first and ninth) and percentiles (first and ninety-nine) are at an equal distance from the median. Then there is no asymmetry.

If a graph of data becomes a normal curve and when it is folded in middle and one part overlaps fully with the other one then there is no asymmetry.

Conclusion

Skewness refers to the extent to which the data is asymmetrical or skewed to one side. It helps to identify whether a distribution is symmetric or skewed. Hope this article had helped in shedding some light on “skewness for a data distribution”.

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