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Measure of Dispersion

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Measure of Dispersion is the numbers that are used to represent the scattering of the data. These are the numbers that show the various aspects of the data spread across various parameters. There are various measures of dispersion that are used to represent the data that includes,

  • Standard Deviation
  • Mean Deviation
  • Quartile Deviation
  • Variance
  • Range, etc

Dispersion in the general sense is the state of scattering. Suppose we have to study the data for thousands of variables there we have to find various parameters that represent the crux of the given data set. These parameters are called the measure of dispersion.

In this article, we will learn about, the measure of dispersion, its various examples, formulas, and others related to it.

What is the Measure of Dispersion in Statistics?

Measures of Dispersion measure the scattering of the data, i.e. how the values are distributed in the data set. In statistics, we define the measure of dispersion as various parameters that are used to define the various attributes of the data.

The image added below shows the measure of dispersion of various types.

measure-of-dispersion

These measures of dispersion capture variation between different values of the data.

Measures of Dispersion Definition

Measures of Dispersion is a non-negative real number that gives various parameters of the data. The measure of dispersion will be zero when the dispersion of the data set will be zero. If we have dispersion in the given data then, these numbers which give the attributes of the data set are the measure of dispersion.

Example of Measures of Dispersion

We can understand the measure of dispersion by studying the following example, suppose we have 10 students in a class and the marks scored by them in a Mathematics test are 12, 14, 18, 9, 11, 7, 9, 16, 19, and 20 out of 20. Then the average value scored by the student in the class is,

Mean (Average) = (12 + 14 + 18 + 9 + 11 + 7 + 9 + 16 + 19 + 20)/10

= 135/10 = 13.5

Then, the average value of the marks is 13.5

Mean Deviation = {|12-13.5| + |14-13.5| + |18-13.5| + |9-13.5| + |11-13.5| + |7-13.5| + |9-13.5| + |16-13.5| + |19-13.5| + |20-13.5|}/10 = 34.5/10 = 3.45

Types of Measures of Dispersion

Measures of dispersion can be classified into two categories shown below:  

  • Absolute Measures of Dispersion
  • Relative Measures of Dispersion

These measures of dispersion can be further divided into various categories. The measures of dispersion have various parameters and these parameters have the same unit.

Measure-of-Depresion

Let’s learn about them in detail.

Absolute Measures of Dispersion

These measures of dispersion are measured and expressed in the units of data themselves. For example – Meters, Dollars, Kg, etc. Some absolute measures of dispersion are: 

Range: Range is defined as the difference between the largest and the smallest value in the distribution.

Mean Deviation: Mean deviation is the arithmetic mean of the difference between the values and their mean.

Standard Deviation: Standard Deviation is the square root of the arithmetic average of the square of the deviations measured from the mean.

Variance: Variance is defined as the average of the square deviation from the mean of the given data set.

Quartile Deviation: Quartile deviation is defined as half of the difference between the third quartile and the first quartile in a given data set.

Interquartile Range: The difference between upper(Q3 ) and lower(Q1) quartile is called Interterquartile Range. The formula for Interquartile Range is given as Q3 – Q1

Relative Measures of Dispersion

Suppose we have to measure the two quantities that have different units than we used relative measures of dispersion to get a better idea about the scatter of the data. Various relative measures of the dispersion are,

Coefficient of Range: The coefficient of range is defined as the ratio of the difference between the highest and lowest value in a data set to the sum of the highest and lowest value.

Coefficient of Variation: The coefficient of Variation is defined as the ratio of the standard deviation to the mean of the data set. We use percentages to express the coefficient of variation.

Coefficient of Mean Deviation: The coefficient of the Mean Deviation is defined as the ratio of the mean deviation to the value of the central point of the data set.

Coefficient of Quartile Deviation: The coefficient of the Quartile Deviation is defined as the ratio of the difference between the third quartile and the first quartile to the sum of the third and first quartiles.

Now let’s learn more about some of Absolute Measures of Dispersion in detail.

Range of Data Set

The range is the difference between the largest and the smallest values in the distribution. Thus, it can be written as

R = L – S

where

  • L is the largest value in the Distribution
  • S is the smallest value in the Distribution

A higher value of range implies higher variation. One drawback of this measure is that it only takes into account the maximum and the minimum value which might not always be the proper indicator of how the values of the distribution are scattered. 

Example: Find the range of the data set 10, 20, 15, 0, 100.

Solution:

  • Smallest Value in the data = 0
  • Largest Value in the data = 100 

Thus, the range of the data set is,

R = 100 – 0

R = 100

Note: Range cannot be calculated for the open-ended frequency distributions. Open-ended frequency distributions are those distributions in which either the lower limit of the lowest class or the higher limit of the highest class is not defined. 

Range for Ungrouped Data

The range of the data set for the ungrouped data set is first we have to find the smallest and the largest value of the data set by observing and the difference between them gives the range of ungrouped data. This is explained by the following example:

Example: Find out the range for the following observations, 20, 24, 31, 17, 45, 39, 51, 61.

Solution:

  • Largest Value = 61
  • Smallest Value = 17

Thus, the range of the data set is

Range = 61 – 17 = 44

Range for Grouped Data

The range of the data set for the grouped data set is found by studying the following example,

Example: Find out the range for the following frequency distribution table for the marks scored by class 10 students. 

Marks Intervals Number of Students
0-10 5
10-20 8
20-30 15
30-40 9

Solution:

  • For Largest Value: Taking the higher limit of Highest Class = 40 
  • For Smallest Value: Taking the lower limit of Lowest Class = 0

Range = 40 – 0 

Thus, the range of the given data set is,

Range = 40 

Mean Deviation

Range as a measure of dispersion only depends on the highest and the lowest values in the data. Mean deviation on the other hand measures the deviation of the observations from the mean of the distribution. Since the average is the central value of the data, some deviations might be positive and some might be negative. If they are added like that, their sum will not reveal much as they tend to cancel each other’s effect. For example, 

Consider the data given below, -5, 10, 25

Mean = (-5 + 10 + 25)/3 = 10

Now a deviation from the mean for different values is,

  • (-5 -10) = -15
  • (10 – 10) = 0
  • (25 – 10) = 15

Now adding the deviations, shows that there is zero deviation from the mean which is incorrect. Thus, to counter this problem only the absolute values of the difference are taken while calculating the mean deviation.

So the formula for the mean deviation is,

Mean Deviation (MD)\frac{|(x_1 - \mu)| + |(x_2 - \mu)| + ....|(x_n - \mu)|}{n}

Mean Deviation for Ungrouped Data

For calculating the mean deviation for ungrouped data, the following steps must be followed: 

Step 1: Calculate the arithmetic mean for all the values of the dataset.

Step 2: Calculate the difference between each value of the dataset and the mean. Only absolute values of the differences will be considered. |d|

Step 3: Calculate the arithmetic mean of these deviations using the formula,

M.D = \frac{\sum|d|}{n}

This can be explained using the example.

Example: Calculate the mean deviation for the given ungrouped data, 2, 4, 6, 8, 10

Solution: 

Mean(μ) = (2+4+6+8+10)/(5)

μ = 6

M. D = \frac{\sum|d|}{n}

⇒ M.D = \frac{|(2 - 6)| + |(4 - 6)| + |(6 - 6)| + |(8 - 6)| + |(10 - 6)|}{5}

⇒ M.D = (4+2+0+2+4)/(5)

⇒ M.D = 12/5 = 2.4

Measures of Dispersion Formula

Measures of Dispersion Formulas are the formulas that are used to tell us about the various parameters of the data. Various formulas related to the measures of dispersion are discussed in the table below.

The table added here is for the Absolute Measure of Dispersion.

Absolute Measures of Dispersion

Related Formulas

Range

H – S

where,

  • H is the Largest Value
  • S is the Smallest Value
Variance

Population Variance(σ2)

σ2 = Σ(xi-μ)2 /n

Sample Variance(S2)

S2 = Σ(xi-μ)2 /(n-1)

where,

  • μ is the mean
  • n is the number of observation
Standard Deviation S.D. = √(σ2)
Mean Deviation

 μ = (x – a)/n

where,

  • a is the central value(mean, median, mode)
  • n is the number of observation
Quartile Deviation

 (Q3 – Q1)/2

where,

  • Q3 = Third Quartile
  • Q1 = First Quartile

The table added here is for the Related Measure of Dispersion.

Relative Measures of Dispersion Related Formulas
Coefficient of Range (H – S)/(H + S)
Coefficient of Variation (SD/Mean)×100
Coefficient of Mean Deviation

(Mean Deviation)/μ

where,

μ is the central point for which the mean is calculated

Coefficient of Quartile Deviation (Q3 – Q1)/(Q3 + Q1)

Co-Efficient of Dispersion

Coefficients of dispersion are calculated when two series are compared, which have great differences in their average. We also use co-efficient of dispersion for comparing two series that have different measurements. It is denoted using the letters C.D.

Measures of Dispersion and Central Tendency

Measures of Dispersion and Central Tendency both are numbers that are used to describe various parameters of the data. The differences between Measures of Dispersion and Central Tendency are added in the table below,

Central Tendency

Measure of Dispersion

Central Tendency is the numbers that are used to quantify the properties of the data set.

Measure of Distribution is used to quantify the variability of the data of dispersion.

Measure of Central tendency include,

Various parameters included for the measure of dispersion are,

  • Range
  • Variance
  • Standard Deviation
  • Mean Deviation
  • Quartile Deviation

Read More,

Examples on Measures of Dispersion

Examples 1: Find out the range for the following observations. {20, 42, 13, 71, 54, 93, 15, 16}

Solution:

Given,

  • Largest Value of Observation = 71
  • Smallest Value of Observation = 13

Thus, the range of the data set is,

Range = 71 – 13

Range = 58

Example 2: Find out the range for the following frequency distribution table for the marks scored by class 10 students. 

Marks Intervals Number of Students
10-20 8
20-30 25
30-40 9

Solution:

Given,

  • Largest Value: Take the Higher Limit of the Highest Class = 40 
  • Smallest Value: Take the Lower Limit of the Lowest Class = 10

Range = 40 – 10

Range = 30

Thus, the range of the data set is 30.

Example 3: Calculate the mean deviation for the given ungrouped data {-5, -4, 0, 4, 5}

Solution: 

Mean(μ) = {(-5)+(-4)+(0)+(4)+(5)}/5

μ = 0/5 = 0

M. D = \frac{\sum|d|}{n}

⇒ M.D = \frac{|(-5 - 0)| + |(-4 - 0)| + |(0 - 0)| + |(4 - 0)| + |(5 - 0)|}{5}

⇒ M.D = (5+4+0+4+5)/5

⇒ M.D = 18/5

⇒ M.D = 3.6

FAQs on Measures of Dispersion

1. What are Measures of Dispersion in Mathematics?

Measure of Dispersion is the positive real numbers that are used to define the variability of the data set about any central point.

2. What are the Types of Measures of Dispersion?

Measure of Dispersion is categorized into two categories,

  • Absolute Measures of Dispersion
  • Relative Measures of Dispersion

3. What are Absolute and Relative Measures of Dispersion?

Absolute and Relative Measures of Dispersion are the various parameters that are used to quantify the spread of the dispersion of the data.

4. How to Calculate Dispersion?

Dispersion is calculated by using various formulas for mean, standard deviation, variance, etc.


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Last Updated : 06 Sep, 2023
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