statistics measures of dispersion symmetry and skewness of frequency distributuion what are the different ways to determine the presense of skewness int the data? how percentiles be used to find skewness

1. WELCOME!!

2. Introduction to Statistics
Instructor:
Muhammad Hassam Shakil Siddiqui
Email: hassam@iqra.edu.pk
Lecture 07
Week # 07
All the text, examples, diagrams are taken from “Elementary Statistics, 10th ed, Allan G.Bluman”.

3. Learning Outcomes
After completion of this lecture, students will be able to demonstrate
knowledge regarding the following:
– Empirical relation between mean, median and mode
– Skewness
– Co-efficient of skewness
– Kurtosis

4. Types of Distribution
A distribution in which the values
of mean, median and
mode coincide(i.e. mean = median
= mode) is known as a
symmetrical distribution or
normal distribution.
Conversely, when values of mean,
median and mode are not equal
the distribution is known as
asymmetrical or skewed
distribution.
Empirical Relation Between Mean, Median And Mode

5. • Symmetry and Skewness refer to the shape of a distribution.
• The Normal Distribution is an example of a symmetric distribution.
• When graphed, a symmetrical distribution at the middle can be
divided such that each half is a mirror image of the other.
• Symmetric distribution happens when the values of the variables
occur at normal intervals and at the same point the mean, median
and mode occur.
Skewness is a measure of a distribution’s asymmetry.
Skewness

6. Negative Skewness
A Negatively Skewed distribution
is skewed to the left. It has a
longer “tail” on the left side.
The mean and the median are to
the left of the mode. Negatively
skewed data generally have the
mean to the left of the median
𝑀𝑒𝑎𝑛 < 𝑀𝑒𝑑𝑖𝑎𝑛 < 𝑀𝑜𝑑𝑒

7. A Positively Skewed distribution
is skewed to the right. It has a
longer “tail” on the right side. The
mean and the median are to the
right of the mode.
Positively skewed data generally
have the mean to the right of the
median.
Positive Skewness
(Positive Skewness)
𝑀𝑜𝑑𝑒 < 𝑀𝑒𝑑𝑖𝑎𝑛 < 𝑀𝑒𝑎𝑛

8. We can also measure the skewness of the data using Karl Pearson’s
measure of skewness, he developed two methods to find skewness in
a sample.
The first method uses mode and it’s formula is:
Moreover, Pearson’s second formula of skewness uses the median and
is denoted by:
𝑆𝑘 =
𝑀𝑒𝑎𝑛 − 𝑀𝑜𝑑𝑒
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝑆𝑘 =
3(𝑀𝑒𝑎𝑛 − 𝑀𝑒𝑑𝑖𝑎𝑛)
𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
Coefficient of Skewness

9. Karl Pearson’s coefficient of skewness
Karl Pearson’s coefficient of skewness lies between -3 and +3.
• If 𝑆𝑘 = 0 then we can say that the frequency distribution is normal
and symmetrical.
• If 𝑆𝑘 < 0 then we can say that the frequency distribution is negatively
skewed.
• If 𝑆𝑘 > 0 then we can say that the frequency distribution is positively
skewed.

10. Bowley’s coefficient of skewness
Bowley’s coefficient of skewness lies between -1 and +1.
Bowley’s and Pearson’s method both give the similar conclusion
regarding the skewness of the data.
𝑆𝑘 𝐵 =
𝑄3 + 𝑄1 − 2𝑄2
𝑄3 − 𝑄1
If 𝑆𝑘 = 0 then we can say that the frequency distribution is normal and
symmetrical.
If 𝑆𝑘 < 0 then we can say that the frequency distribution is negatively skewed.
If 𝑆𝑘 > 0 then we can say that the frequency distribution is positively skewed.

11. Consider the following Box Plot
Here Q1 = 4.3, Q2= 7.2 and Q3= 15.4
so Bowley’s Coefficient of Skewness is equal to 0.4774774774…
Bowley’s coefficient of skewness

12. In statistics, a moment is a basic quantitative measure of a function's shape.
The third central moment is the calculation of the distribution's lopsidedne
ss; if established, any symmetric distribution will have a third central
moment of zero. The normal third central moment is called skewedness.
Generic Formula
(Moment)
3rd Moment
(To Find Skewness)
Ungroup Data
Grouped Data
3
1
3
)
(
ns
x
x
skewness
n
i
i




Third Moment about the Mean

13. • Kurtosis measures how peaked the histogram is
• The kurtosis of a normal distribution is 0
• Kurtosis is based on the size of a distribution's tails.
– Distributions with relatively large tails are called “leptokurtic”.
– Distributions with small tails are called “platykurtic”.
• A distribution with the same kurtosis as the normal distribution is called
"mesokurtic."
Kurtosis
3
)
(
4
4




ns
x
x
kurtosis
n
i
i

14. • The following two distributions have the same variance, approximately
the same skew but differ markedly in kurtosis.
Source: http://davidmlane.com/hyperstat/A53638.html
Kurtosis

15. Review Questions
Explain the following:
• Measures of Dispersion
• Symmetry and Skewness of Frequency Distribution
• What are the different ways to determine the presence of
skewness in the data?
• How percentiles be used to find skewness?

16. THANK YOU