Module 1. Descriptive statistics

Lesson 5

MEASURES OF SKEWNESS AND KURTOSIS

5.1 Introduction

In the preceding lessons, we have discussed the measures of central tendency and dispersion in case of frequency distribution. The measures of central tendency tell us about the concentration of the observations about the middle of the distribution and the measure of dispersion gives us an idea about the spread or scatter of the observations about some measure of central tendency. These measures, however, don’t adequately describe a frequency distribution in the sense that there could be two or more distributions with the same mean and standard deviation but still different from each other with regard to shape or pattern of distribution. Thus these two measures of central tendency and dispersion are inadequate to characterize a distribution completely and must be supported and supplemented by two more measures viz. skewness and kurtosis which we shall discuss in this lesson.

5.2 Skewness

Literal meaning of skewness is “lack of symmetry”. It measures the degree of departure of a distribution from symmetry and reveals the direction of scatterdness of the items.

A frequency distribution is said to be symmetrical when values of the variables equidistant from their mean have equal frequencies. If a frequency distribution is not symmetrical, it is said to be asymmetrical or skewed. Any deviation from symmetry is called skewness.

According to Morris Humberg “Skewness refers to the asymmetry or lack of symmetry in the shape of a frequency distribution”.

According to Croxton & Cowden “When a series is not symmetrical it is said to be asymmetrical or skewed”.

According to Simpson & Kafka “Measures of skewness tell us the direction and the extent of skewness. In a symmetrical distribution the mean, median and mode are identical. The more we move away from the mode, the larger the asymmetry or skewness”.

In the words of Riggleman and Frisbee “Skewness is the lack of symmetry when a frequency distribution is plotted on a chart, skewness present in the items tends to be dispersed more on one side of the mean than on the other”.

Thus the above definitions make it clear that the word skewness refers to the lack of symmetry. If a distribution is normal there would be no skewness in it and the curve drawn from the distribution would be symmetrical. In case of skewed distributions the curve drawn would be elongated either to the left or to the right. The concept of skewness gains importance from the fact that statistical theory is often based upon the assumption of the normal distribution. A measure of skewness is, therefore necessary in order to guard against the consequences of the assumption. The following three figures would give an idea about the shape of symmetrical and asymmetrical curves.

5.2.1 Symmetrical curve

The figure 5.1, given below, presents the shape of a symmetrical curve which is bell shaped having no skewness. The value of mean (M), median (M_d) and mode (M_o) for such a curve would be identical.

Fig. 5.1 Symmetrical distribution

In a symmetrical distribution the values of mean, median and mode coincide. The spread of the frequencies is the same on both sides of the centre point of the curve. For a symmetrical distribution Mean = Median = Mode.

5.2.2 Positively skewed curve

A positively skewed curve has a longer tail towards the higher values of X i.e. the frequency curve gradually slopes down towards the higher values of X. In a positively skewed distribution the mean is greater than the median and then mode and the median lies in between mean and mode. The frequencies are spread over a greater range of values on the high value end of the curve (the right hand side) as is clear from the Fig 5.2.

Fig. 5.2 Positively skewed distribution

For a positively skewed distribution Mean > Median > Mode.

5.2.3 Negatively skewed curve

A negatively skewed curve has a longer tail towards the lower values of X i.e. the frequency curve gradually slopes down towards the lower values of X as shown in Fig. 5.3.

Fig. 5.3 Negatively skewed distribution

In the negatively skewed distribution the mode is the maximum and mean is the least. The median lies in between mean and mode. The elongated tail in negatively skewed distribution is on the left hand side as would be clear from Fig 5.3. For a negatively skewed distribution, Mean < Median < Mode.

5.3 Measures of Skewness

Measures of skewness are meant to give an idea about the extent of asymmetry in a series. A distribution is said to be skewed if

· The frequency curve of the distribution is not a symmetric bell shaped curve but stretched more to one side than to the other.

· The values of mean (M), median (M_d) and mode (M_o) fall at different points i.e they don’t coincide.

· Quartiles Q₁ and Q₃ are not equidistant from the median.

· The corresponding pairs of deciles and percentiles are not equidistant from the median.

If a particular distribution is found to be skewed, the next problem that arises is to measure the extent of skewness. To find out the direction and extent of asymmetry in a series statistical measures of skewness are employed.

These measures can be absolute or relative. The absolute measures of skewness tell us the extent of asymmetry and whether it is positive or negative. The absolute skewness is based on the difference between mean and mode. Symbolically,

Absolute skewness = Mean – Mode

Skewness will be positive, if the value of mean is greater than the mode and skewness will be negative, if the value of mean is less than the mode. The difference between the mean and the mode, whether positive or negative, indicates the distribution is positively skewed or negatively skewed. However, such an absolute measure of skewness is not adequate because it cannot be used for comparison of skewness in two distributions, if they are in different units, since difference between the mean and mode will be in terms of the units of distribution. Thus for comparison purposes we use the relative measures of skewness known as co-efficient of skewness.

5.3.1 Karl pearson’s coefficient of skewness

The first coefficient of skewness as defined by Karl Pearson is

This measure is based on the fact that the mean and the mode are drawn widely apart. Skewness will be positive if mean > mode and negative if mean < mode. There is no limit to this measure in theory and this is a slight drawback. But in practice the value given by this formula is rarely very high and its value usually lies between -1 and +1.

It may also be written as as Mode = 3 Median – 2 Mean

This coefficient is a pure number without units since both numerator and denominator have the same dimensions. The value of this coefficient lies between -3 and +3.

5.3.2 Bowley’s Coefficient of Skewness

Prof. A.L. Bowley’s Coefficient of Skewness is based on quartiles and is given by:

This is also known as Coefficient of Skewness based on quartiles and is especially useful in situations where quartiles and median are used viz.

· When the mode is ill-defined and extreme observations are present in the data.

· When the distribution has open end classes or unequal class intervals.

This coefficient is a pure number without units since both numerator and denominator have the same dimensions. The value of this coefficient lies between -1 and +1.

5.3.3 Kelly’s Coefficient of Skewness

The drawback of Bowley’s Coefficient of Skewness is that it ignores the 50% of the data which can be partially removed by taking two deciles or percentiles equidistant from the median value. The refinement was suggested by Kelly.

5.3.4 Coefficient of Skewness based on moments

5.3.4.1 Moments

Moments are the general statistical measure used to describe and analyse the characteristics of a frequency distribution viz. central tendency, dispersion, skewness and kurtosis. Let us consider the variable X having a frequency distribution as given below:

X	X₁	X₂	X₃	----	---	X_n
f	f₁	f₂	f₃	----	---	f_n

Then, is the arithmetic mean.

5.3.4.2 Moments about Mean

The r^th moment about Mean is denoted by µ_r and also known as central moment and is defined as

………(Eq. 5.1)

Putting r=0 in equation (5.1) we get

………( Eq. 5.2)

Putting r=1 in equation (5.1) we get ………(Eq. 5.3)

because the algebraic sum of deviations of a given set of observations from their mean is zero. Thus the first moment about mean is always zero.

Again taking r=2 we get ………(Eq. 5.4)

Hence second moment about mean gives the variance of the distribution.

………(Eq. 5.5)

………(Eq. 5.6)

5.3.4.3 Moments about an arbitrary point A

The r^th moment about any point A denoted by µ_r^’ are also known as raw moment and is defined as

………(Eq. 5.7)

Putting r=0 and r=1 in equation (5.7) we get respectively

………(Eq. 5.8)

………(Eq. 5.9)

where is the first moment about the arbitrary point A ………(Eq. 5.10)

Taking r=2, 3, 4 in (5.7) we get respectively

Second moment about the point A ………(Eq. 5.11)

Third moment about the point A ………(Eq. 5.12)

Fourth moment about the point A ………(Eq. 5.13)

5.3.4.4 Relation between moments about mean and moments about an arbitrary point A

In this section we shall try to obtain the expression for , the r^th moment about mean as defined in (5.1) in terms of , the r^th moment about any arbitrary point A as defined in (5.7).

………(Eq. 5.14)

From (5.9) we get , substituting in (5.14) we get

Now by binomial theorem

Putting r = 2, 3, 4 respectively, we get

5.3.4.5 Pearson’s β and γ coefficients

Karl Pearson gave the following four coefficients calculated from the moments about mean and defined them as follows:

These coefficients are pure numbers independent of units of measurement and as such can be conveniently used for comparative studies.

In terms of these coefficients, the coefficient of skewness is given exactly by

The importance of these coefficients lies in the fact that they give some idea about the shape of the curve obtained from the frequency distribution. For symmetrical distribution, all the moments of odd order about the mean vanish or (are zero) and therefore µ₃ = 0 and hence β₁ = 0. Thus, β₁ gives a measure of departure from symmetry i.e. of skewness. Thus, the sign of skewness depends upon µ3. If µ₃ is positive we get positive skewness and if µ₃ is negative, we get negative skewness.

5.4 Kurtosis

The expression Kurtosis is used to describe the peakedness of a curve. Kurtosis is a Greek word means ‘bulginess’. In statistics kurtosis refers to the degree of flatness or peakedness in the region about the mode of a frequency curve. The degree of Kurtosis of a distribution is measured relative to the peakedness of normal curve. If we know the measures of central tendency, dispersion and skewness, we cannot still form a complete idea about the distribution as is clear from the figure 5.4.

Fig. 5.4 Type of kurtosis

All the three curves are symmetrical about mean and have same variation (range). In order to identify a distribution completely we need one more measure which Prof. Karl Pearson called ‘convexity of the curve’ or ‘kurtosis’. It is measured by β₂ and γ₂ given as under

Curve of type B which is neither flat nor peaked is known as normal curve and shape of its hump is accepted as a standard one. Curves with humps of the form of normal curve are said to have normal kurtosis and are termed as Mesokurtic .The curve of type A , which are more peaked than the normal curve are known as Leptokurtic and are said to lack kurtosis or to have negative kurtosis. Curves of type C , which are flatter than the normal curve are called Platykurtic and they are said to posses kurtosis in excess or have positive kurtosis.

Example 1: Find different measures of skewness and kurtosis taking data given in example 1 of Lesson 3, using different methods.

Solution: Prepare the following table to calculate different measures of skewness and kurtosis using the values of Mean (M) = 1910, Median (M_d) = 1890.8696, Mode (M_o) = 1866.3636, Variance σ² = 29500, Q1 = 1772.1053 and Q₃ = 2030 as calculated earlier.

Calculation of moments about an arbitrary constant 2080

Class Interval	Mid-value (X_i)	frequency (f_i)
1630-1730	1680	17	-400	-6800	2720000	-1088000000	435200000000
1730-1830	1780	19	-300	-5700	1710000	-513000000	153900000000
1830-1930	1880	23	-200	-4600	920000	-184000000	36800000000
1930-2030	1980	16	-100	-1600	160000	-16000000	1600000000
2030-2130	2080	14	0	0	0	0	0
2130-2230	2180	7	100	700	70000	7000000	700000000
2230-2330	2280	2	200	400	80000	16000000	3200000000
2330-2430	2380	2	300	600	180000	54000000	16200000000
Total		100	-400	-17000	5840000	-1724000000	647600000000

First moment about the point A=2080

Second moment about the point A

Third moment about the point A

Fourth moment about the point A

Compute the central moments using raw moments as follows:

= (-17240000) – 3(58400)(-170) + 2(-170)³

=2718000

=6476000000 – 4(-17240000)(-170) + 6(58400)(-170)² – 3(-170)⁴

= 2373730000

Calculation of moments about mean

Class Interval	Mid-value (X_i)	frequency (f_i)
1630-1730	1680	17	-230	-3910	899300	-206839000	47572970000
1730-1830	1780	19	-130	-2470	321100	-41743000	5426590000
1830-1930	1880	23	-30	-690	20700	-621000	18630000
1930-2030	1980	16	70	1120	78400	5488000	384160000
2030-2130	2080	14	170	2380	404600	68782000	11692940000
2130-2230	2180	7	270	1890	510300	137781000	37200870000
2230-2330	2280	2	370	740	273800	101306000	37483220000
2330-2430	2380	2	470	940	441800	207646000	97593620000
Total		100		0	2950000	271800000	237373000000