**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.

**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.,

Q_{3}
– Median = Median – Q_{1} or Q_{3} + Q_{1} – 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., Q_{3}
will be farther from median than Q_{1} is from median and the reserve
for negative skewness. Hence a possible measure is:

SK_{B}
= (Q_{3} + Q_{1} – 2 Median) / (Q_{3} – Q_{1})

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.

Moment |
What it
measures |

1. First moment about origin. 2. Second moment about the mean. 3. Third moment about the mean. 4. Fourth moment about the mean. |
Mean. Variance. Skewness. Kurtosis. |

**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.

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