Skewness and Kurtosis

Dr. Abhishek Singh [email protected]

Department of MSC, NIT Hamirpur, (H.P.)

June 28, 2020

Basic Statistics Skewness and Kurtosis 1/24 Outline

Skewness Measures of Skewness Coefficients of Skewness

1 Prof. Karl Pearson’s Coefficient of Skewness 2 Prof. Bowley’s Coefficient of Skewness 3 Coefficient of Skewness based upon Moments Pearson’s β and γ Coefficients Kurtosis

Basic Statistics Skewness and Kurtosis 2/24 Skewness

Skewness means lack of symmetry. It gives us an idea about the shape of the curve which we can draw with the help of the given data. A distribution is said to be skewed if—

1 Mean, median and mode fall at different points, i.e., Mean 6= Median 6= Mode; 2 Quartiles are not equidistant from median; and 3 The curve drawn with the help of the given data is not symmetrical but stretched more to one side than to the other.

Basic Statistics Skewness and Kurtosis 3/24 Symmetrical Distribution

A symmetric distribution is a type of distribution where the left side of the distribution mirrors the right side. In a symmetric distribution, the mean, mode and median all fall at the same point. Figure: Symmetrical Distribution

Basic Statistics Skewness and Kurtosis 4/24 Measures of Skewness

The measures of skewness are:

1 Sk = M − Md ,

2 Sk = M − Mo,

3 Sk = (Q3 − Md ) − (Md − Q1),

where M is the mean, Md , the median, Mo, the mode, Q1, the first quartile deviation and Q3, the third quartile deviation of the distribution.

These are the absolute measures of skewness.

Basic Statistics Skewness and Kurtosis 5/24 Coefficients of Skewness

For comparing two series we do not calculate these absolute measures but we calculate the relative measures called the coefficients of skewness which are pure numbers independent of units of measurement.

The following are the coefficients of skewness:

1 Prof. Karl Pearson’s Coefficient of Skewness, 2 Prof. Bowley’s Coefficient of Skewness, 3 Coefficient of Skewness based upon Moments.

Basic Statistics Skewness and Kurtosis 6/24 Prof. Karl Pearson’s Coefficient of Skewness

Definition It is defined as:

(M − M ) S = o , k σ

where σ is the standard deviation of the distribution. If mode is ill-defined, then using the empirical relation, Mo = 3Md − 2M, for a moderately asymmetrical distribution, we have

3(M − M ) S = d k σ

Basic Statistics Skewness and Kurtosis 7/24 Continued

From above two formulas, we observe that Sk = 0 if M = Mo = Md . Hence for a symmetrical distribution, mean, median and mode coincide.

Skewness is positive if M > Mo or M > Md , and

negative if M < Mo or M < Md .

Limits are: |Sk | ≤ 3 or −3 ≤ Sk ≤ 3. However, in practice, these limits are rarely attained.

Basic Statistics Skewness and Kurtosis 8/24 Prof. Bowley’s Coefficient of Skewness

Definition It is also known as Quartile Coefficient of Skewness and is defined by:

(Q3 − Md ) − (Md − Q1) Q3 + Q1 − 2Md Sk = = (Q3 − Md ) + (Md − Q1) Q3 − Q1

where symbols have their usual meaning.

Basic Statistics Skewness and Kurtosis 9/24 Continued

From above formula, we observe that Sk = 0 if Q3 − Md = Md − Q1. This implies that for a symmetrical distribution, i.e., Sk = 0, median is equidistant from the upper and lower quartile.

Skewness is positive if Q3 − Md > Md − Q1 ⇒ Q3 + Q1 > 2Md , and

negative if Q3 − Md < Md − Q1 ⇒ Q3 + Q1 < 2Md .

Limits are: |Sk (Bowley)| ≤ 1 or −1 ≤ Sk (Bowley) ≤ 1. Thus, Bowley’s coefficient of skewness ranges from −1 to + 1.

Basic Statistics Skewness and Kurtosis 10/24 Continued

Sk = +1, if Md − Q1 = 0, i.e., if Md = Q1

Sk = −1, if Q3 − Md = 0, i.e., if Q3 = Md

Basic Statistics Skewness and Kurtosis 11/24 Continued

It is useful in situations where quartiles and medians are used, viz.,

1 When the mode is ill-defined and extreme observations are present in the data. 2 When the distribution has open end classes or unequal class intervals. In these situations Pearson’s coefficient of skewness cannot be used. The values of the coefficients of skewness obtained by Bowley’s formula and Pearson’s formula are not comparable.

Basic Statistics Skewness and Kurtosis 12/24 Continued

In each case, Sk = 0, implies the absence of skewness, i.e., the distribution is symmetrical. It may even happen that one of them gives positive skewness while the other gives negative skewness. In Bowley’s coefficient of skewness, the disturbing factor of variations is eliminated by dividing the absolute measure of skewness, viz., (Q3 − Md ) − (Md − Q1) by the measure of dispersion (Q3 − Q1), i.e., quartile range. It is based only on the central 50% of the data and ignores the remaining 50% of the data towards the extremes.

Basic Statistics Skewness and Kurtosis 13/24 Coefficient of Skewness based upon Moments

Definition It is defined as: √ β1(β2 + 3) Sk = 2(5β2 − 6β1 − 9)

where β0s are Pearson’s Coefficients and defined as:

2 µ3 µ4 β1 = 3 , and β2 = 2 µ2 µ2 0 th where µr s are r moment of a variable x about the meanx ¯, know as the r th central moment, and is defined as:

Basic Statistics Skewness and Kurtosis 14/24 Continued

1 X µ = f (x − x¯)r r N i i i 1 P 0 1 P In particular: µ0 = N fi (xi − x¯) = N fi = 1, and i i 1 P µ1 = N fi (xi − x¯) = 0, being the algebraic sum of i deviations from the mean. Also,

1 X µ = f (x − x¯)2 = σ2 2 N i i i 2 These results, viz., µ0 = 1, µ1 = 0, and µ2 = σ , are of fundamental importance and should be committed to memory.

Basic Statistics Skewness and Kurtosis 15/24 Continued

Sk = 0, if either β1 = 0 or β2 = −3.

Thus Sk = 0, if and only if β1 = 0.

Thus for a symmetrical distribution β1 = 0.

In this respect β1 is taken as a measure of skewness. The coefficient of skewness based upon moments is to be regarded as without sign. The Pearson’s and Bowley’s coefficients of skewness can be positive as well as negative.

Basic Statistics Skewness and Kurtosis 16/24 Positively Skewed Distribution

The skewness is positive if the larger tail of the distribution lies towards the higher values of the variate (the right), i.e., if the curve drawn with the help of the given data is stretched more to the right than to the left. Figure: Positively Skewed Distribution

Basic Statistics Skewness and Kurtosis 17/24 Negatively Skewed Distribution

The skewness is negative if the larger tail of the distribution lies towards the lower values of the variate (the left), i.e., if the curve drawn with the help of the given data is stretched more to the left than to the right. Figure: Negatively Skewed Distribution

Basic Statistics Skewness and Kurtosis 18/24 Pearson’s β and γ Coefficients

Definition Karl Pearson defined the following four moments and coefficients, based upon the first four moments about mean:

2 µ3 p µ4 β1 = 3 , γ1 = β1 and β2 = 2 , γ2 = β2 − 3 µ2 µ2

These coefficients are pure numbers independent of units of measurement.

Basic Statistics Skewness and Kurtosis 19/24 Kurtosis

If we know the measures of central tendency, dispersion and skewness, we still cannot form a complete idea about the distribution. Let us consider the figure in which all the three curves A, B, and C are symmetrical about the mean and have the same range.

Basic Statistics Skewness and Kurtosis 20/24 Continued

Definition Kurtosis is also known as Convexity of the Frequency Curve due to Prof. Karl Pearson. It enables us to have an idea about the flatness or peakedness of the frequency curve.

It is measure by the coefficient β2 or its derivation γ2 given as:

µ4 β2 = 2 , or γ2 = β2 − 3 µ2

Basic Statistics Skewness and Kurtosis 21/24 Continued

Curve of the type A which is neither flat nor peaked is called the normal curve or mesokurtic curve and for such curve β2 = 3, i.e., γ2 = 0. Curve of the type B which is flatter than the normal curve is known as platycurtic curve and for such curve β2 < 3, i.e., γ2 < 0. Curve of the type C which is more peaked than the normal curve is called leptokurtic curve and for such curve β2 > 3, i.e., γ2 > 0.

Basic Statistics Skewness and Kurtosis 22/24 Example

Example

For a distribution, the mean is 10, variance is 16, γ1 is +1 and β2 is 4. Comment about the nature of distribution.

Solution

Since γ1 = +1, the distribution is moderately positively skewed, i.e, if we draw the curve of the given distribution, it will have longer tail towards the right.

Further, since β2 = 4 > 3, the distribution is leptokurtic, i.e., it will be sightly more peaked than the normal curve.

Basic Statistics Skewness and Kurtosis 23/24 Thanks!