Measure of Skewness

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.