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SELF INSTRUCTIONAL STUDY MATERIAL

MASTER OF DEVELOPMETN STUDIES (M.A) DEGREE PROGRAMME

Course: STATISTICAL METHODS FOR DEVELOPMENT RESEARCH (MADVS2005C04)

UNIT II(PART II)

INSTRUCTOR Prof. (Dr.) KRISHNAN CHALIL

11 APRIL 2020

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UNIT II- PART B

SKEWNESS AND KURTOSIS AND THEIR MEASURES

Objectives of the Unit

After studying this unit, you will be able to:

Understand the meaning of the terms skewness and kurtosis

Familiarize various measures of skewness and Kurtosis

Work out some practical examples

Introduction

By analyzing a frequency distribution, we can measure its central tendency as well

as dispersion. Beyond that if want to understand the other salient features of the distribution,

we have to study two other measures namely Skewness and Kurtosis.

Skewness

Skewness help us to study the shape of distribution., that is, symmetry or asymmetry, while,

Kurtosis indicates the flatness or peakedness of curve. Now let us see the meaning of

symmetry. A frequency distribution is said to be “symmetrical”, if the frequencies are

distributed symmetrically or evenly on either side of an average. When plotted on a graph

paper, such distributions will show a normal or ideal curve. In a normal curve mean, median or

mode coincides. A normal curve is a bell shaped curve in which the values on either side of an

average are symmetrical.

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If general frequency distributions are not symmetrical, they are slightly or highly

asymmetrical. Skewness is opposite to symmetrical. Measures of skewness will not only show

the amount of skewness, but also its direction. A distribution is said to be positively skewed

when it has a long tail towards the higher values of the variable and negatively skewed when

the longer tail is present towards the lower values of the variable. The above figures describe

the meaning of symmetry and asymmetry.

Test of Skewness

1. The values of mean, median and mode would not be the same.

2. When plotted on a graph paper, a skew distribution would not show a bell-shaped curve

Measures of Skewness A measure of skewness gives the extent and direction of skewness of a distribution. As in the case of dispersion, we can define the absolute and the relative measures of skewness. There are three broad categories of measures of skewness. They are: measures of skewness based on, (i) mean, median and mode., (ii) quartiles or percentiles and (iii) moments.

(i) Measure of Skewness based on Mean, Median and Mode This measure was suggested by Karl Pearson. According to this method, the difference between mean and mode can be taken as an absolute measure of skewness in the distribution. Karl Pearson’s first formula: Absolute Skewness = Mean minus Mode., i.e., (Mean –Mode) Alternatively, when mode is ill-defined and the distribution is moderately skewed, the above measure can also be approximately expressed as : 3(Mean – Median) A relative measure of skewness known as Karl Pearson’s Coefficient of Skewness is given by:

Coefficient of Skewness = Mean - Mode Standard Deviation Or 2.

Co-efficient of Skewness = 3(Mean – Median) Standard Deviation

(ii) Measure of Skewness based on Quartiles or Percentiles

(a) Bowley has developed a measure of skewness based on quartiles. This is based

on the fact that for skewed distribution Q1 and Q3 are not equi-distant from

median. Therefore, the difference between Md and Q1 and Q3 and Md gives the

measure of skewness. When this difference is high, skewness is also high. This is

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an absolute measure. The relative measure is obtained by dividing the measure

by Q3 –Q1 and is known as coefficient of skewness.

SkB = (Q3 – Md) – (Md –Q1) = (Q3+Q1-2Md) Q3 – Q1 Q3 – Q1

This formula is known as Bowley’s coefficient of skewness.

(b) As the first decile and ninth decile are equidistant from median in the case of

symmetric distributions, the difference between D9 – Md and Md –D1 give the

measure of skewness. This is an absolute measure. The relative measure is

obtained by dividing by D9 – D1. This is known as Kelley’s coefficient of

skewness. The formula, then is:

SkK = (D9 + D1 -2Md) or (P90 + P10 -2Md)

D9 –D1 P90 – P10

(iii) Moments and Skewness Moments are the mean of various powers of deviations of the items. If the deviations are about the Mean, the moments are called as Central Moments. Whenever, the deviations are taken from values other than the arithmetic mean, the moments are called as raw moments. Particular kind of raw moments are raw moments about origin. That is, the deviations are measured from origin, which is zero. Moments are denoted by Greek alphabet μ (read as mu). The power of the deviations is put as suffix. Raw moments have ‘ (read, dash) sign.

First central moment is the mean of the deviations of the values measured from their arithmetic mean and so is zero always. But, first raw moment about origin is ∑fx/N, the arithmetic mean. μ2 is variance whose positive square root is standard deviation, is the most useful measure of dispersion. μ3 helps in the study of skewness, while μ4 helps to measure Kurtosis. The measure of skewness is based on third moment (μ3). The value of μ3 gives the absolute measure of skewness and the coefficient of skewness is given by:

Skewness = μ3 ÷ √μ2

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KURTOSIS

The three measures namely average, dispersion and skewness that we have so far studied are

not even sufficient for analysis of a frequency distribution completely. So, to analyse a

distribution completely we need one more measure which is known as Kurtosis. While

skewness refers to the extent of lack of symmetry, Kurtosis refers to the extent to which

frequency is peaked. Karl Pearson, in 1905, introduced three types of curves depending upon

the shape of their peaks. These three shapes are known as Mesokurtic, Leptokurtic and

Platykurtic as shown in the given figure.

Measure of Kurtosis

A measure of the coefficient of kurtosis, given by Karl Pearson is:

β2 = μ 4 ÷ μ 22

where, = μ 4 is the fourth moment and μ 2 is the second moment. For a normal distribution (i.e., mesokurtic curve) the value of β2 = 3. When, β2 > 3, the curve is more peaked than the normal and is called leptokurtic curve. Further, when , β2 < 3, the curve is less peaked than the normal and called a platykurtic curve. The measure of kurtosis an also be expressed in terms of γ2 (read as gama two). Since γ2 = β2 - 3, We can write γ2 = 0 for mesokurtic, γ2 > 0 for leptokurtic and γ2 < 0 for platykurtic

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Let us sum up: Skewness is the degree of distortion from the symmetrical bell curve or the normal

distribution. It measures the lack of symmetry in data distribution. The skewness for a normal distribution is zero, and any symmetric data should have a skewness near zero. Negative values for the skewness indicate data that are skewed left and positive values for the skewness indicate data that are skewed right. By skewed left, we mean that the left tail is long relative to the right tail. Similarly, skewed right means that the right tail is long relative to the left tail. If the data are multi-modal, then this may affect the sign of the skewness. Kurtosis is a measure of whether the data are heavy-tailed or light-tailed relative to a normal distribution. That is, data sets with high kurtosis tend to have heavy tails, or outliers. Data sets with low kurtosis tend to have light tails, or lack of outliers. A uniform distribution would be the extreme case.

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Check your progress 1: (a) Calculate Karl Pearson’s coefficient of skewness from the following data:

Size 1 2 3 4 5 6 7 frequency 10 18 30 25 12 3 2

……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………… (hints: Mean = 3.28 σ =1.35 Mode =3 (by inspection) Sk =0.207)

(b) Find Bowley’s measure of skewness from the following data: 15,38,40,28,25,32,10,17,20,45,33

……………………………………………………………………………………………………………………………………………………………………………………………………………………………………….. (Hints:Q1 = 17 Q3 = 38 Median =28 Skewness =-0.048)

(c) From the following table, calculate absolute skewness and also coefficient skewness:

Weekly wages 15 20 25 30 35 40 45 (Rs.)

No. of workers 3 25 19 16 4 5 6 …………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………… (Hints: Mean = 26.41 Mode: 20 Standard Deviation = 7.65 Absolute skewness = 6.41 coefficient of skewness = 0.84)

(d) For a group of 11 items, ∑X = 65., ∑X 2 =493 and mode = 6, find the person’s coefficient of

skewness? …………………………………………………………………………………………………………..……………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………………….

[Hints: Mean = 5.91., Standard deviation = 3.145 coefficient of skewness = -0.029] References Bhardwaj, R.S (2015), Business Statistics, Excel Books, New Delhi Gupta, S.P (2016) Statistical methods, S.Chand, New Delhi