Measures of
Skewness
Skewness –refers to the degree of symmetry or
asymmetry of a distribution.
3 Typesof Distribution
- is a distribution with a bell-shaped appearance. In a
normal distribution, the
mean=median=mode
1.NORMAL DISTRIBUTION
Example:Distribution of
Correct answers of 19 Students who participated in a
Math Contest
No of Correct answer
Frequency
1 12 23 44 55 46 27 1
N=19
1
2
3
4
5
81 2 3
4
5 6 7
6
FrequencyNo. of Correct Answers
The distribution has:mean:4.0
median:4.0mode:4.0
standard deviation:1.53
Negatively Skewed
- skewed to the Left -the mean is less than its median
- bulk of the distribution is on the
right
Example:Distribution of Correct answers of 19 Students who participated in a
Math Contest
No of Correct answer
Frequency
1 02 03 14 25 46 97 3
N=19
2
4
6
4
10
81 2 3
8
5 6 7
12
FrequencyNo. of Correct Answers
The distribution has:
mean:5.58median:6.0mode:6.0standard
deviation:1.07
Positively Skewed
- Skewed to the right
-the mean is greater than its median
-the bulk of the distribution is on
the left
Example:Distribution of Correct answers of 19 Students who participated in a
Math Contest
No of Correct answer
Frequency
1 32 93 44 25 16 07 0
N=19
2
4
6
4
10
81 2 3
8
5 6 7
12
FrequencyNo. of Correct Answers
The distribution has:
mean:2.4median:2.0mode:2.0standard
deviation:1.07
The extent of Skewness
The extent of Skewness can be
obtained by getting the coefficient of
skewness with the formula:
Formula:SK=3(mean-median) Standard Deviation
*Where SK is the coefficient of skewness.
Summary of the examples of the
measurements from the three distribution.
NORMAL
Skewed to the Left
Skewed to the Right
Mean 4.00 5.58 2.40
Median 4.00 6.00 2.00
Mode 4.00 6.00 2.00
Standard Deviation
1.53 1.07 1.07
Using the formula to find the coefficient of skewness we have:
1.For Normal Distribution SK=3(4.0-4.0)/1.53 =0
2.For Skewed to the left
SK=3(5.6-6.0)/1.07=-1.12
3.For Skewed to the right
SK=3(2.4-2.0)/1.07=1.12
Notice that if:1.SK=0,it is normal
2.SK<0,it is skewed to the left
3.SK>0,it is skewed to the right
Measures of
Kurtosis
Kurtosis refers to the peakedness
or flatness of a distribution
Mesokurtic is a
normal distributio
n.
Leptokurtic is more peaked
than the normal
distribution.
Platykurtic is flatter than the normal
distribution.
Formulas:
For ungrouped data:
4
4)(
Ns
xxKu
For grouped data:
4
4)(
Ns
xXfKu m
where: samplesizeN
ianceesquareofths
meanX
classmarkX
rawdataX
kurtosisKu
m
var4
A distribution is normal or
mesokurtic if Ku=3, leptokurtic if Ku>3 and platykurtic if
Ku<3.