OVERVIE W
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.
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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
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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,
S
k = Mean – Mode= Q3 + Q1 – 2 Median.
For Skewed distribution, if
Mean>Mode, the Skewness becomes +ive. Otherwise –ive.
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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.
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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 operato r)
21-25 26-30 31-35 36-40 41-45 46-50 51- 55
No of
operators 5 15 28 42 15 12 3
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Solution :
Class Mid-
value(m) Frequency(
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.
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Answers :
X = 36.96 rejects / operator.
= 6.736 rejects / operator.
Mode = 37.70
Coefficient of Skewness = -0.109
Interpretation :
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
compani es
5 12 20 16 5 2
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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.
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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
Media n,
Coefficient of Skewness, SKb = 0.02.
The positive value of SKb indicates that the distribution in positively skewed.
Institute of Advanced Management Education and Entrepreneurship, Hyderabad © 2010. IAMEE
Answers :
Q1 = 18.33lakhs Q3 = 35 lakhs.
Q.D = 0.313 Median (Me)
= 26.5
Coefficient of Skewness
= 0.02.
Interpretati on:
The distribution is positively skewed and
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THE END
MEASURNESS OF SKEWNESS AND INTERPRETATION
THANK YOU
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to O
go & read today
otherwise
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Try
The following table gives the
it…
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.
Weekly
Wages (Rs).
Below
200 200-250 250-300 300-350 350-400 400 and abov e
No. of
Workers 10 25 145 220 70 30
