this pdf provides a brief summary of measures of skewness / asymmetry.

2. INTRODUCTION
• Central tendency gives a value
around which many other items of
the data congregate.
• Dispersion or variability tells us how
much the items deviate from central
tendency.
• The measure of skewness tells us the
direction and the extent of skewness.
• Skewness is an important statistical
technique that helps to determine
asymmetrical behavior than of the
frequency distribution.
• Skewness means lack of symmetry.
• If the distribution is not symmetric, the
frequencies will not be uniformly
distributed about the center of the
distribution.

3. SYMMETRIC DISTRIBUTION
• A symmetric distribution is one in which frequency
distribution is the same on the sides of the center point
of the frequency curve.
• In this distribution, Mean = Median = Mode.
• There is no skewness in a perfectly symmetrical
distribution.
• In this distribution, the two sides of the distribution are
a mirror image of each other.
• The normal distribution is a symmetrical, bell-shaped
distribution in which the mean, median and mode are
all equal.

4. SYMMETRIC DISTRIBUTION
• Unimodal distribution: distribution
in which there is only one peak due
to one mode in data set.
• Bimodal distribution: distribution
which has two peaks due to two
modes in data set.
• Multimodal: distribution which has
multiple peaks due to three or more
modes in data set.

5. UNIFORM SYMMETRIC DISTRIBUTION
• If the distribution is symmetric, it is roughly bell-shaped, or
has a discrete shape.
• Uniform distribution is a type of probability distribution
where all the outcomes are equally likely.
• It is also called as rectangular distribution where
 The height of each rectangle is approximately the
same.
The data is spread equally across the range.
There are no clear peaks

6. ASYMMETRIC DISTRIBUTION
• A asymmetrical or skewed distribution is one in
which the spread of the frequencies is different on
both the sides of the center point or the frequency
curve is more stretched towards one side or value
of mean.
• Median and mode falls at different points.
• The extreme data values are higher in a positive
skew distribution, which increases the mean
value of the data set.
• In a positive skew distribution, the right tail is
longer than the left tail.

7. ASYMMETRIC DISTRIBUTION
• Mean > median > mode in positive
skewness.
• The extreme data values are smaller in
negative skewness distribution, which
lowers the dataset’s mean value.
• In a negative skew distribution, the left tail
is longer than the right tail.
• In negative skewness, mean < median <
mode.

8. MEASURES OF ASYMMETRY
• Measures of skewness or asymmetry help
us to know to what degree and in which
direction the frequency distribution has a
departure from symmetry.
• These measures can be either-
i. absolute
or
ii. relative.
• Absolute measures tell us the extent of
asymmetry and whether it is positive or
negative.
• The first absolute measure of skewness is
based on the difference between mean and
mode or mean and median.
• Symbolicilly,
i. Absolute skewness (Sk) = Mean - Mode
or
ii. Absolute skewness (Sk) = Mean – Median
• If the value of mean is greater than the mode
or median, skewness is positive, otherwise it is
negative.

9. MEASURES OF ASYMMETRY
• The second absolute measure of
skewness is based on quartiles.
• When
Q3 – Median = Median – Q1, the
distribution is symmetric.
Q3 – Median > Median – Q1, the
distribution is positive asymmetric.
Q3 – Median < Median – Q1, the
distribution is negative asymmetric.
• Coefficient of skewness is relative measure of
skewness.
• Coefficient of skewness is obtained by dividing
the skewness by any measure of dispersion.
• The following are the three important methods
of measuring relative skewness:
i. Karl Pearson's coefficient of skewness
ii. Bowley’s coefficient of skewness
iii. Kelly’s coefficient of skewness

10. KARL PEARSON'S CO-EFFICIENT OF SKEWNESS
• The Karl Pearson’s Co-efficient of Skewness is calculated by dividing the difference between mean and
mode by the standard deviation of the distribution.
• Co-efficient of skewness (J)= (Mean – Mode) / Standard deviation
• Symbolically,
𝒙 𝑴𝒐
𝝈
• Karl Pearson's coefficient of skewness is equal to zero in symmetric distribution.
• Karl Pearson's coefficient of skewness is greater than zero in a positively skewed
distribution.
• Karl Pearson's coefficient of skewness is less than zero in a negative skewed distribution.

11. KARL PEARSON'S CO-EFFICIENT OF SKEWNESS
• Skewness is positive or negative accordingly as the co-efficient of skewness value is
positive or negative.
• When mode is ill defined,
• Co-efficient of skewness (J)= 3 (Mean – Median) / Standard deviation
• Symbolically,
̅ 𝐞
• Value of this co-efficient ranges from -3 to +3.
• This is because the mean is always within one standard deviation of any median.

12. BOWLEY’S COEFFICIENT OF SKEWNESS
• Bowley’s coefficient of skewness is based upon the relationship between median and two quartiles.
• Absolute measure of skewness is measured by the difference (Q3 - Me) - (Me - Q1).
• This measure is not satisfactory since the quartiles and median are expressed in the same units as the
original data.
• (Q3 - Me) - (Me - Q1) is divided by the sum of these two expressions. This result in what is termed as
Bowley’s coefficient of skewness.
• Symbolically,
𝐽 =
𝑄 − 𝑀 − 𝑀𝐞 − 𝑄
𝑄 − 𝑀𝐞 + 𝑀𝐞 − 𝑄

13. BOWLEY’S COEFFICIENT OF SKEWNESS
• The value of Bowley’s coefficient of skewness varies between -1 and +1.
• When the median coincides with the lower quartile, it results in the value of the co-
efficient being equal to +1.
• On the other hand, if the median is equal to the upper quartile, the value of the
coefficient of skewness is -1.

14. KELLY’S COEFFICIENT OF SKEWNESS
• Kelly’s measure of skewness is based upon the relationship among median, 10th
percentile and 90th percentile of a distribution.
• According to Kelly’s measure,
Skewness (Sk) =
Coefficient of Skewness (J) =
• Kelly’s measure is also known as ‘percentile measure of skewness’.

15. MEASURES OF SKEWNESS / ASYMMETRY
KARL PEARSON’S MEASURE
• Based on mean, median, and
standard distribution
Or
• Based on mean, median and
standard distribution.
• Based on mean and median can
range between -3 and +3.
• Uses the entire set of values.
BOWELY’S MEASURE
• Based on the location of
median and two quartiles.
• Can range between -1 and +1.
• Is based only on the part of
distribution.
KELLY’S MEASURE
• Based upon median along
with 10th percentile and 90th
percentile.
• Is used very sparingly.

16. REFERENCES
 NDVohra, Business Statistics (McGraw Hill Education, 2019)
https://www.igntu.ac.in/eContent/IGNTU-eContent-467281593500-B.Com-4-
Prof.ShailendraSinghBhadouriaDean&-BUSINESSSTATISTICS-All.pdf
https://egyankosh.ac.in/bitstream/123456789/19499/1/Unit-6.pdf
https://www.geeksforgeeks.org/