SKEWNESS AND KURTOSIS, Biostatistics

1. AN OVERVIEW OF SKEWNESS
AND KURTOSIS

2. INTRODUCTION
Skewness :- In statistics, Skewness is a measure of the asymmetry of the
probability distribution of a real-valued random variable about its mean.
The skewness value can be positive, zero or negative.
Kurtosis :- In statistics, Kurtosis is a measure of the “tailedness” of the
probability distribution of a real-valued random variable.

3. GRAPHICAL REPRESENTATION OF
SKEWNESS
MEAN<MEDIAN<MODE MEAN=MEDIAN=MODE MEAN>MEDIAN>MODE

4. FORMULA OF SKEWNESS
where,
x = individual data value of a set
x
̄ = mean of the data set
n = number of observation
S = standard deviation
f = frequency
Skewness : Σ(x – x
̄ )3
(n-1).S3
Σf(x – x
̄ )3
(n-1).S3

5. FORMULA OF KURTOSIS
where,
x = individual data value of a set
x
̄ = mean of the data set
n = number of observation
S = standard deviation
f = frequency
Kurtosis : Σ(x – x̄)4
(n-1).S4
Σf(x – x
̄ )4
(n-1).S4

6. EXAMPLE OF A NUMERICAL DEALING
WITH SKEWNESS AND KURTOSIS
Q. Calculate the Skewness and Kurtosis with the help of the table given below:
CLASS FREQUENCY
2-4 3
4-6 4
6-8 2
8-10 1

7. CLASS MID
VALUE (X)
f f.x x-x̄ f.(x-x̄)2 f.(x-x̄)3 f.(x-x̄)4
2-4 3 3 9 (3-5.2)
= -2.2
3.(-2.2)2
= 14.52
14.52*(-2.2)
= -31.94
-31.94*(-2.2)
= 70.27
4-6 5 4 20 (5-5.2)
= -0.2
4.(-0.2)2
= 0.16
0.16*(-0.2)
= -0.032
-0.032*(-0.2)
= 0.0064
6-8 7 2 14 (7-5.2)
= 1.8
2.(1.8) 2
= 6.48
6.48*1.8
= 11.66
11.66*1.8
= 20.99
8-10 9 1 9 (9-5.2)
= 3.8
1.(3.8) 2
= 14.44
14.44*3.8
= 54.87
54.87*3.8
= 208.51
Σ 10 52 35.6 34.56 299.77
x̄ = Σfx => 52 => 5.2
Σfn 10
SOLUTION:

8. 1. S.D : √ Σ(xi –
x̄)2
(n-1)
= √35.6 = 1.98
10-1
2. Skewness : Σ(x – x
̄ )3
(n-1).S3
= 34.56 = 0.49
9*(1.98)3
3. Kurtosis : Σ(x – x̄)4
(n-1).S4
= 299.77 = 2.17
9* (1.98)4

9. APPLICATION OF SKEWNESS AND
KURTOSIS
Skewness can be used to obtain approximate probabilities of distributions.
With the help of skewness we can know or understand whether deviations
from the mean are going to be positive or negative.
It also indicates the direction and relative magnitude of a distribution’s
deviation from the normal distribution.
Kurtosis is a useful method to check whether there is a problem with the
outliers in a particular dataset.

10. THANK YOU