Measure of Skewness andKurtosis

Dr. Romario Bhong P. Ybaez

Research Director

St. Paul University Dumaguete

Measure of Skewness and Kurtosis

Types of Skewness1. Skewed to the right distribution tapers more to the right longer tail to the rightmore concentration of values below than above more concentration of values below than above

the meanMean > Md > Momost skewed curves encountered in the social

sciences are skewed to the right

Monthly Income of Households in Barangay Maligaya, 2003

6

8

10

12

14

16

18

Nu

mb

er

of

Ho

useh

old

s

0

2

4

6

2.0 an d below

2.1 - 7 .0

7.1 - 12.0

12.1 - 17.0

17.1 - 22.0

22.1 - 27.0

27.1 - 32.0

Monthly Income (in 10,000 Pesos)

Nu

mb

er

of

Ho

useh

old

s

Examples: frequency distributions ofwages or salaries, use of electricity,weights of adult male human beings,family income

2. Skewed to the leftAges at Death of Inventors in Makulay Barangay

6

8

10

12

Nu

mb

er o

f In

v en

tors

0

2

4

45 1nd b elow

46 - 5051 - 55

56 - 6061 - 65

66 - 7071 and ab ove

Age at Death (in years)

Nu

mb

er o

f In

v en

tors

Formulas

1.

s

MoXSK

2.

s

MdxSK

)(3

Interpretation of SK: SK>0: positively skewed since the

mean > mode SK < 0: negatively skewed since the SK < 0: negatively skewed since the

mean < mode SK=0: symmetric since the mean =

mode

Example

Given the following scores: 80,92,85, 87, 94, 83, 88, 91, 92, and80.

Solution:Solution:

Mean =87.2, Median = 87.5, s= 5.09

SK=3(87.287.5)/5.09 = -0.18

Kurtosis

Measures of kurtosis look at how sharply the distribution rises to a peak and then drops away

MEASURE OF KURTOSIS Peakedness or flatness

1. Leptokurtic

2. Platykurtic2. Platykurtic

3. Mesokurtic

Formulas:

1

dataungroupedfor)(

1

41

4

4 S

XXn

n

n

ii

datagroupedfor

)(1

4

1

4

4 S

fXXf

n

iii

i

Computational formula

1090

13

2)(PP

QQ

Kkurtosis

Interpretation:K=3: mesokurticK>3: leptokurticK

Example:

Q3 = 91.75 Q1 = 83.5

P90 = 92.2 P10 = 80

5.8375.91 34.0

802.922

5.8375.91

)(

K