skewness

PRESENTED BYGET PLACED

MEASURES OF SKEWNESS AND INTERPRETATION

On01/03/2011

PGDM@IAMEE www.iamee.in

Institute of Advanced Management Education and Entrepreneurship, Hyderabad © 2010. IAMEE

http://www.iamee.in/index.html


OVERVIEW

1. What is meant by Skewness.2. What are the different types of

Skewness Distributions.3. Skewness Distributions in graphical

representation.4. Calculation of Skewness Coefficient

in different (two types) methods.





Measures of Skewness

The statistical technique to indicate the direction and extent of skewness in the distribution of numerical values in the data set.

A frequency distribution of the set of values that is not Symmetrical(normal) is called asymmetrical or skewed.





Skewed distributions

There are three types: Symmetrical distribution A.M = Median = Mode

Positively skewed distribution A.M > Median > Mode

Negatively skewed distribution.A.M < Median < Mode





Skewed distribution Curves





Measures of Skewness

For an asymmetrical distribution, the distance b/w Mean and Mode may be used to measure the degree of skewness because the Mean is equal to Mode in a symmetrical distribution. Thus,

Absolute, Sk = Mean – Mode

= Q3 + Q1 – 2 Median.For Skewed distribution, if Mean>Mode,

the Skewness becomes +ive. Otherwise –ive.





Variation tells us about the amount of the variation where as Skewness tells about the direction of variation.

In business and economic series, measures of variation have greater practical application than measures of skewness.





Relative measures of Skewness

Karl Pearson’s Co-efficient of Skewness,

The value of SKp various b/w ±3. For moderately skewed distribution,But moderately skewed distribution, the value of

SKp lies between ±1.





Relative measures of Skewness

Bowley’s Coefficient of Skewness,

Kelly’s coefficient of Skewness,





Calculation of SKp

1. Data of rejected items during a production process is as follows:

Calculate the mean, standard deviation, and coefficient of skewness and comment on the results.

No of rejects(per operator)

21-25

26-30 31-35 36-40

41-45 46-50

51-55

No of operators

5 15 28 42 15 12 3





Solution:Class Mid-

value(m)Frequenc

y(f)d=(m-A)/h

d² fd fd²

21-25 23 5 -3 9 -15 45

26-30 28 15 -2 4 -30 60

31-35 33 28 -1 1 -28 28

36-40 38 42 0 0 0 0

41-45 43 15 1 1 15 15

46-50 48 12 2 4 24 48

51-55 53 3 3 9 9 27

N=120 d=0 ∑fd = -25 ∑fd²=223





Assume mean, A=38, then

By inspection, mode lies in the class 36-40.

=





Answers:

X = 36.96 rejects / operator.𝜎 = 6.736 rejects / operator.Mode = 37.70Coefficient of Skewness = -0.109Interpretation :Distribution is Negatively Skewed and hence the concentration of the rejects/operator is more on the left side.





Calculation of Bowley's coefficient

2) The data on the profits(in Rs lakhs) earned by 60 companies is as follows:

Obtain the limits of profits of the central 50 percent companies. And also calculate Bowley’s coefficient of skewness.

Profits Below 10

10-20 20-30 30-40 40-50

50 and above

No. of companies

5 12 20 16 5 2





Solution:Profits (in Rs lakhs) Frequency(f) Cumulative frequency

(c.f)

Below 10 5 5

10-20 12 17

20-30 20 37

30-40 16 53

40-50 5 58

50 and above 2 60

N = 60





Q1 = size of (N/4)th observation. And Q3 = size of (3N/4)th observation. Therefore Q1 = 15th observation, Q3 = 45th observation.





Hence, the profit of central 50 percent companies lies b/w Rs.35 lakhs and 18.33 lakhs.

Therefore, Coefficient of quartile deviation,

Q.D = (Q3 – Q1) / (Q3 + Q1) =____________

Median=size of (N/2)th observation =(60/2)th observation. Thus, median lies in the class interval 20-30, and





Median,

Coefficient of Skewness, SKb = 0.02.

The positive value of SKb indicates that the distribution in positively skewed.





Answers:

Q1 = 18.33 lakhsQ3 = 35 lakhs.Q.D = 0.313Median (Me) = 26.5Coefficient of Skewness = 0.02.

Interpretation:The distribution is positively skewed and

therefore, concentration of larger value on the right side of the distribution.





THE END

MEASURNESS OF SKEWNESS AND INTERPRETATION

THANK YOU





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Try it…

The following table gives the distribution of weekly wages of 500 workers in a factory:

a) Obtain the limits of income of the central 50% of the observed workers.

b) Calculate Bowley's coefficient of skewness.

ANS: Negatively Skewed.(SKb = -0.102).

Weekly Wages (Rs).

Below 200

200-250

250-300

300-350

350-400

400 and

above

No. of Workers

10 25 145 220 70 30



Keeping your

work organized

will help a lot!!



skewness

Documents

extent of skewness

degree of skewness

kellys coefficient of

asymmetrical distribution

median mode

frequency distribution

mean mode

median lies