Skewness, Moments and Kurtosis | Business Statistics Notes | B.Com Notes Hons & Non Hons | CBCS Pattern

BUSINESS STATISTICS NOTES
B.COM 2ND AND 3RD SEM NEW SYLLABUS (CBCS PATTERN)
SKEWNESS, MOMENTS AND KURTOSIS

MEANING OF SKEWNESS

There are two other comparable characteristics called skewness and kurtosis that help us to understand a distribution. Two distributions may have the same mean and standard deviation but may differ widely in their overall appearance as can be seen from the following:

In both these distributions the value of mean and standard deviation is the same (Mean = SD = 5). But it does not imply that the distributions are alike in nature. The distribution on the left-hand side is a symmetrical one whereas the distribution on the right-hand side is asymmetrical or skewed. Measures of skewness help us to distinguish between different types of distributions. Some definitions of skewness are as follows:

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

2)    “Skewness refers to the asymmetry or lack of symmetry in the shape of a frequency distribution.” – Morris Hamburg.

The analysis of above definitions shows that the term ‘SKEWNESS’ refers to lack of symmetry, i.e., when a distribution is not symmetrical (or is asymmetrical) it is called a skewed distribution. Any measure of skewness indicates the difference between the manners in which items are distributed in a particular distribution compared with a symmetrical (or normal) distribution. If, for example, skewness is positive, the frequencies in the distribution are spread out over a greater range of values on the high-value end of the curve (the right-hand side) than they are on the low value end. If the curve is normal spread will be the same on both sides of the centre point and the mean, median and mode will all have the same value. 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 this assumption.

Difference between Dispersion and Skewness:

Dispersion is concerned with the amount of variation rather than with its direction. Skewness tells us about the direction of the variation or the departure from Symmetry. In fact, measures of skewness are dependent upon the amount of dispersion.

It may be noted that although skewness is an important characteristic for defining the precise pattern of a distribution, it is rarely calculated in business and economic series. Variation is by far the most important characteristic of a distribution.

Requisites of a Good Measure of Skewness

A good measure of skewness should have three properties. It should:

1)    Be a pure number in the sense that its value should be independent of the units of the series and also of the degree of variation in the series.

2)    Have a zero value, when the distribution is symmetrical and

3)    Have some meaningful scale of measure so that we could easily interpret the measured value.

MEASURES OF SKEWNESS

Measures of skewness tell us the direction and extent of asymmetry in a series, and permit us to compare two or more series with regard to these. They may either be absolute or relative.

Absolute Measures of Skewness:

Skewness can be measured in absolute terms by taking the difference between mean and mode. Symbolically:

Absolute Skewness = Mean - Mode

If the value of mean is greater than mode skewness will be positive, i.e., we shall get a plus sign in the above formula. Conversely, if the value of mode is greater than mean, we shall get a minus sign meaning thereby that the distribution is negatively skewed.

Relative Measures of Skewness:

There are four important measures of relative skewness, namely,

1.    The Karl Pearson’s coefficient of skewness.

2.    The Bowley’s coefficient of skewness.

3.    The Kelly’s coefficient of skewness.

4.    Measure of skewness based on moments.

Karl Pearson’s Coefficient of Skewness

This method of measuring skewness, also known as Pearson an Coefficient of Skewness, was suggested by Karl Pearson (1857 -1936), a great British Biometrician and Statistician. It is based upon the difference between mean and mode. This difference is divided by standard deviation to give a relative measure. This formula thus becomes:

SK_P = (Mean – Mode)/Standard Deviation. Here, SK_P = Karl Pearson’s Coefficient of skewness.

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 usually lies between +1.

ALSO READ: CHAPTER WISE BUSINESS STATISTICS NOTES

Unit 1: Statistical Data and Descriptive Statistics

a. Nature and Classification of data

b. Measures of Central Tendency

c. Measures of Variation

d. Skewness, Moments and Kurtosis

Unit 2: Probability and Probability Distributions

a. Theory of Probability

b. Probability distributions:Binomial-Poisson-Normal

Unit 3: Simple Correlation and Regression Analysis

a. Correlation Analysis

b. Regression Analysis

Unit 4: Index Numbers

Unit 5: Time Series Analysis

UNIT 6: Sampling Concepts, Sampling Distributions and Estimation

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Bowley’s Coefficient of Skewness

An alternative measure of skewness has been proposed by late Professor Bowley. Bowley’s measure is based on quartiles. In an a symmetrical distribution the third quartile is the same distance over the median as the first quartile is below it, i.e.,

Q₃ – Median = Median – Q₁ or Q₃ + Q₁ – 2Median = 0

If this distribution is positively skewed the top 25 per cent of the values will tend to be farther from median than the bottom 25 per cent, i.e., Q₃ will be farther from median than Q₁ is from median and the reserve for negative skewness. Hence a possible measure is:

SK_B = (Q₃ + Q₁ – 2 Median) / (Q₃ – Q₁)

SK_B = Bowley’s coefficient of skewness.

It must be remembered that the results obtained by these two measures are not to be compared with one another. Especially, the numerical values are not related to one another since the Bowley’s measure, because of its computational basis, is limited to values between – 1 and + 1, while Pearson’s measure has no such limits.

MOMENTS

‘Moment’ is a familiar mechanical term which refers to the measure of a force with respect of its tendency to provide rotation. The strength of the tendency depends on the amount of force and the distance from the origin of the point at which the force is exerted.

However, the term moment as used in physics has nothing to do with the moment used in statistics, the only analogy being that in statistics we talk of moment of random variable about some point. The moment in statistics are used to describe the various characteristics of a frequency distribution like central tendency, variation, skewness and kurtosis. It can be seen that the formula for a moment coefficient is identical with that for an arithmetic mean. This identity has led statisticians to speak of the arithmetic mean as the “first moment about the origin”.

Purpose of Moments

The concept of moment is of great significance in statistical work. With the help of moments we can measure the central tendency of a set of observations, their variability, their asymmetry and the height of the peak their curve would make. Because of the great convenience in obtaining measures of the various characteristics of a frequency distribution, the calculation of the first four moments about the mean may well be made the first step in the analysis of a frequency distribution. The following is the summary of how moments help in analyzing a frequency distribution.

Kurtosis

In Greek Word, Kurtosis means “bulginess”. In statistics kurtosis refers to the degree of flatness or peakedness in the region about the mode of a frequency curve. Measure of kurtosis tells us the extent to which a distribution is more peaked or flat-topped than the normal curve. If a curve is more peaked than the normal curve, it is called leptokurtic. On the other hand if a curve is less peaked than the normal curve, it is called platykutic. The normal curve itself is called mesokurtic. Kurtosis is the most rarely used tool in statistical analysis.