Measure of Dispersion: Meaning, objective and Types of dispersion, Absolute and Relative measure of dispersion, Range, Interquartile range, Quartile deviation, Mean deviation, Standard deviation: Meaning and formula, Mathematical properties, Chebyshev’s theorem, The empirical rule, Variance, Combined standard deviation, Coefficient of variation. Illustrative examples.Skewness: Symmetric distribution, meaning of skewness, Skewed distribution, dispersion and skewness, Tests of skewness, Measures of skewness: Karl Pearson method, Bowley’s method, Kelly’s method. Illustrative examples.

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Quantitative Techniques
Volume-3
(Revised)
Measure of Dispersion and Skewness
E-Book Code : QTVOL3
by
Narender Sharma
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Quantitative Techniques
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Click on Contents
Measure of dispersion .................................................................................................................................................................................... 3
Variability or dispersion ...................................................................................................................................................................... 3
Objective to measure dispersion ...................................................................................................................................................... 3
Absolute measure of dispersion ....................................................................................................................................................... 4
Relative measure of dispersion ........................................................................................................................................................ 4
Types of measures of dispersion ...................................................................................................................................................... 4
Range ..................................................................................................................................................................................................................... 4
Interquartile Range(IQR) .............................................................................................................................................................................. 6
Quartile Deviation(Q.D.) ................................................................................................................................................................................ 7
Mean Deviation ................................................................................................................................................................................................. 9
Standard Deviation ....................................................................................................................................................................................... 13
Difference between Mean Deviation and Standard Deviation .......................................................................................... 14
Mathematical Properties of Standard Deviation .................................................................................................................... 15
Chebyshev’s Theorem ........................................................................................................................................................................ 16
The Empirical Rule .............................................................................................................................................................................. 16
Variance ........................................................................................................................................................................................................ 21
Combined Standard Deviation ............................................................................................................................................................ 21
Coefficient of Variation ......................................................................................................................................................................... 22
Skewness........................................................................................................................................................................................................... 24
Symmetric distribution ..................................................................................................................................................................... 24
Meaning of Skewness ......................................................................................................................................................................... 25
Skewed Distribution ................................................................................................................................................................................ 25
Positively(Right hand) Skewed Distribution ........................................................................................................................... 25
Negatively(Left hand) Skewed distribution ............................................................................................................................. 25
Difference between Dispersion and Skewness ............................................................................................................................ 26
Test of Skewness ....................................................................................................................................................................................... 26
Measures of Skewness ................................................................................................................................................................................. 27
Karl Pearson’s Method ........................................................................................................................................................................... 27
Bowley’s Method ( Quartile Method) ............................................................................................................................................... 30
Kelly’s Method ........................................................................................................................................................................................... 31
References: ....................................................................................................................................................................................................... 34
Feedback ........................................................................................................................................................................................................... 34

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Measure of dispersion
Variability or dispersion
Consider the following two data sets.
Set I : 1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11
Set II : 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8
Compute the mean, median and mode for each of the two data sets. We find that the two data sets have the same mean , the same median, and the same mode, all equal to 6.
The two data set also have the same number of observations, i.e. n=12. But the two data sets are different. What is the main difference between them?
The two data sets have the same central tendency (as measured by any of the three measures of centrality) but they have different variability or the dispersion or spread.
In particular, we see that data set I is more variable than data set II. The values in set I are more spread out: they lie farther away from their mean than do those of set II.
Objective to measure dispersion
To determine the reliability of an average
The measures of dispersion help in determining the reliability of an average. It points out as to how far an average is representative of a statistical series. If the dispersion or variation is small, the average will closely represent the individual values and it is highly representative. On the other hand if the dispersion or variation is large, the average will be quite unreliable
To compare the variability of two or more series
The measures of dispersion is useful to determine the consistency or uniformity of the two or more series. It helps to comparing the variability of the variability of two or more series. A high degree of variability means the less consistency in the data and if the series shows high consistency that means the data series has less variability.
For facilitating the use of other statistical measures
Measures of dispersion serve the basis of man other statistical measures such as correlation, regression, testing of hypothesis etc. These measures are based on measures of variation of one kind or another. Graphics: Narender Sharma

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Basic of statistical quality control
The measures of dispersion serve the quality control in the manufacturing or service industries. These help to trace the process variation. Control chart is one of the measure tool to find variation so control the causes of variation in the process.
Absolute measure of dispersion
Absolute measure of dispersion are expressed in the same unit in which data of the series are expressed. They are expressed in same statistical unit, e.g., rupees, kilogram, tons, years, centimeters etc.
Relative measure of dispersion
Relative measure of dispersion refers to the variability stated in the form of ratio or percentage. Thus, relative measure of dispersion is independent of unit of measurement. It is also called coefficient of dispersion. These measure are used to compare two series expressed in different units.
Types of measures of dispersion
1. Range
2. Interquartile Range and Quartile Deviation
3. Mean Deviation
4. Standard Deviation
5. Coefficient of Variation
Range
It is the simplest measure of dispersion. It is defined as the difference between the largest and smallest value in the series. Its formula is;
Where R= Range, L= Largest value in the series, S = Smallest value in the series
The relative measure of range, also called coefficient of range, is defined as
The following examples illustrate the calculation of range;
Graphics: Narender Sharma

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Illustrative Examples
Individual series
Example 1. 10 pcs of a product in manufacturing industry taken from an hourly lot and weighted, the weight (gms) of the product was
10.5, 10.7, 10.3, 10.2, 10.9, 11, 11.1, 11.2, 10.3, 10.9
Find the Range and Coefficient of Range
Solution : L=11.2 and S=10.2 . . . . . . . .
Discrete Series
Example 2. Find the range and coefficient of range from the following data;
Marks
10
20
30
40
50
60
70
No. of Students
15
18
25
30
16
10
9
Solution Here L=70 and S=10 .
Continuous series
Example 3. Find out range and coefficient of range of the following series
Size
5-10
10-15
15-20
20-25
25-30
Frequency
4
9
15
30
40
Solution:
Here, L = Upper limit of the largest class =30, S = Lower limit of the smallest class =5 .
Note : Since the maximum and minimum of the observations are not identifiable for a continuous series, the range is defined as the difference between the upper limit of the largest class and the lower limit of the smallest class.
Merits and demerits of Range
Merits
Demerits
1. It is simple to understand.
2. It is easy to calculate.
3. It is widely used in statistical quality control. Range charts are useful in controlling the quality of the product.
1. It cannot be calculated in case of open ended distribution
2. It is not based on all observations of the series.
3. It is affected by sampling fluctuations.
4. It is affected by extreme values in the series.

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Interquartile Range(IQR)
We know about the median, which divide the whole data set into two equal parts, one part less than the median and other half is greater than median.
In the same manner the quartiles ( , ) divide the data into four parts. First part of the data is less than , second part of the data lies between and the third part of the data is lies between and fourth or the last part of the data is greater than . is called lower quartile and is called the upper quartile.
Methods for calculation of quartiles ( ) [ ] ( ) [ ] ( ) [ ]
Second Quartile ( ) is also called the median because the second quartile or median divides the data into two equal parts.
The Interquartile range and quartile deviation are another measure of dispersion which can be calculated with help of quartiles and defined as below.
Interquartile Range(IQR)
The difference between the upper quartile ( ) and lower quartile ( ) is called the interquartile range. Symbolically,
The interquartile ranges covers dispersion of middle 50% of the items of the series. Graphics: Narender Sharma