Measure of dispersion part I (Range, Quartile Deviation, Interquartile deviation, Mean deviation)

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By
Narender Sharma

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Topic covered in this tutorial
S.No. Topic Page No.
1
Measure of Variability or Dispersion
 Introduction
 Objective of Measuring Dispersion
 Absolute or Relative Measure
 Types of Measure of Dispersion
3-6
2
Range
 Introduction
 Coefficient of Range
 Method of calculation
7-8
3
Interquartile Deviation and Quartile Deviation
 Introduction
 Interquartile Range,
 Quartile Deviation,
 Coefficient of Quartile Deviation
 Method of calculation
9-12
4
Mean Deviation
 Introduction
 Mean Deviation from Mean
 Mean Deviation from Median
 Coefficient of Mean Deviation from Mean
 Coefficient of Mean Deviation from Median
13-17
5
Excel Commands
 Minimum Value
 Maximum Value
 Range
 1st Quartile
 2nd Quartile
 3rd Quartile
 Interquartile Range
 Quartile Deviation
 Mean Deviation
18

Measures of Variability or Dispersion
What is measure of 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 of measuring 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.
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 or relative measure of dispersion
Measures of dispersion may be either absolute or relative.
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
Following are the types of measure of dispersion or variability as shown above in the fig.
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;
Calculation of Range
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.

Interquartile Range and Quartile Deviation
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.
Formula 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.
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.
Quartile deviation, also called Semi-interquartile Range is half of the difference between
the upper and lower quartile i.e. half of the interquartile range. Symbolically
Coefficient of quartile deviation (The relative measure of quartile deviation)
Calculation of Interquartile Range, Q.D., Coefficient of Q.D.
Individual Series
Example 4 Find interquartile range, quartile deviation and coefficient of quartile deviation from the
following data;
28, 18, 20, 24, 27, 30, 15
Solution: Arrange the data in ascending order;
15, 18, 20, 24, 27, 28, 30
[ ] [ ]
Discrete Series
Example 5 Calculate interquartile range, quartile deviation and the coefficient of quartile deviation from
the following data;
Earnings (₹) 10 20 30 40 50 60
No. of People 2 8 20 35 42 20
Solution : Calculation of Q.D.

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Measure of dispersion part I (Range, Quartile Deviation, Interquartile deviation, Mean deviation)

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Measure of dispersion part I (Range, Quartile Deviation, Interquartile deviation, Mean deviation)