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Measures of Dispersion

Measures of dispersion are non-negative real numbers that help to gauge the spread of data about a central value. These measures help to determine how stretched or squeezed the given data is. There are five most commonly used measures of dispersion. These are range, variance, standard deviation, mean deviation, and quartile deviation.

The most important use of measures of dispersion is that they help to get an understanding of the distribution of data. As the data becomes more diverse, the value of the measure of dispersion increases. In this article, we will learn about measures of dispersion, their types along with examples as well as various important aspects related to these measures.

1.	What is Measure of Dispersion?
2.	Types of Measures of Dispersion
3.	Absolute Measures of Dispersion
4.	Relative Measures of Dispersion
5.	Measures of Dispersion Formula
6.	Measures of Dispersion and Central Tendency
7.	FAQs on Measures of Dispersion

What is Measure of Dispersion in Statistics?

Measures of dispersion help to describe the variability in data. Dispersion is a statistical term that can be used to describe the extent to which data is scattered. Thus, measures of dispersion are certain types of measures that are used to quantify the dispersion of data.

Measures of Dispersion Definition

Measures of dispersion can be defined as positive real numbers that measure how homogeneous or heterogeneous the given data is. The value of a measure of dispersion will be 0 if the data points in a data set are the same. However, as the variability of the data increases the value of the measures of dispersion also increases.

Measures of Dispersion Example

Suppose we have two data sets A = {3, 1, 6, 2} and B = {1, 5, 9, 10}. The variance(population) of A is 3.5 and the variance(population) of B is 12.68. This implies that data set B is more variable than data set A. Thus, the variance helps to draw a comparison between the two data sets A and B on the basis of variability.

Types of Measures of Dispersion

The measures of dispersion can be classified into two broad categories. These are absolute measures of dispersion and relative measures of dispersion. Range, variance, standard deviation and mean deviation fall under the category of absolute measures of deviation. These measures have the same unit as the data that is being scrutinized. Coefficients of dispersion are relative measures of deviation. Such dispersion measures are always dimensionless. The upcoming sections will further elaborate on these measures.

Types of Measures of Dispersion

Absolute Measures of Dispersion

If the dispersion of data within an experiment has to be determined then absolute measures of dispersion should be used. These measures usually express variations in a data set with respect to the average of the deviations of the observations. The most commonly used absolute measures of deviation are listed below.

Range: Given a data set, the range can be defined as the difference between the maximum value and the minimum value.

Variance: The average squared deviation from the mean of the given data set is known as the variance. This measure of dispersion checks the spread of the data about the mean.

Standard Deviation: The square root of the variance gives the standard deviation. Thus, the standard deviation also measures the variation of the data about the mean.

Mean Deviation: The mean deviation gives the average of the data's absolute deviation about the central points. These central points could be the mean, median, or mode.

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

Relative Measures of Dispersion

If the data of separate data sets have different units and need to be compared then relative measures of dispersion are used. The measures are expressed in the form of ratios and percentages thus, making them unitless. Some of the relative measures of dispersion are given below:

Coefficient of Range: It is 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: It is the ratio of the standard deviation to the mean of the data set. It is expressed in the form of a percentage.

Coefficient of Mean Deviation: This can be defined as the ratio of the mean deviation to the value of the central point from which it is calculated.

Coefficient of Quartile Deviation: It is the ratio of the difference between the third quartile and the first quartile to the sum of the third and first quartiles.

Measures of Dispersion Formula

Measures of dispersion are used when we want to find the scattering of data about a central point such as the mean. The general formulas used to calculate the various measures of dispersion are given in the tables below:

Absolute Measures of Dispersion

Absolute Measures of Dispersion	Formulas
Range	H - S where H is the largest value and S is the smallest value in a data set.
Variance	Population Variance: \(\sigma ^{2}\) = \(\sum_{1}^{n} \frac{(X_{i} - \overline{X})^{2}}{n}\) Sample Variance: s² = \(\sum_{1}^{n} \frac{(X_{i} - \overline{X})^{2}}{n-1}\) where n is the number of observations and \(\overline{X}\) is the mean
Standard Deviation	Population Standard Deviation: S.D. = \(\sqrt{Variance}\) = \(\sigma\) Sample Standard Deviation: S.D. = s
Mean Deviation	\(\sum_{1}^{n}\frac{\|X - \overline{X}\|}{n}\) where \(\overline{X}\) is the central value and denotes the mean, median or mode.
Quartile Deviation	\(\frac{Q_{3}-Q_{1}}{2}\) where \(Q_{3}\) and \(Q_{1}\) are the third and first quartiles respectively.

Relative Measures of Dispersion

Relative Measures of Dispersion	Formulas
Coefficient of Range	(H - S) / (H + S)
Coefficient of Variation	(S.D. / Mean) * 100
Coefficient of Mean Deviation	Mean Deviation / \(\overline{X}\) where, \(\overline{X}\) is the central point about which the mean deviation is calculated.
Coefficient of Quartile Deviation	\(\frac{Q_{3}-Q_{1}}{Q_{3}+Q_{1}}\)

Measures of Dispersion and Central Tendency

Both measures of dispersion and measures of central tendency are used to describe data. The table given below outlines the difference between the measures of dispersion and central tendency.

Measures of Dispersion	Central Tendency
When we want to quantify the variability of data we use measures of dispersion.	Measures of central tendency help to quantify the data's average behavior.
Measures of dispersion include variance, standard deviation, mean deviation, quartile deviation, etc.	Measures of central tendency are mean, median, and mode.

Related Articles:

Important Notes on Measures of Dispersion

Measures of dispersion are used to determine the spread of data. They are measured about a central value.
Measures of dispersion can be classified into two types, i.e., absolute and relative measures of dispersion.
Absolute measures of deviation have the same units as the data and relative measures are unitless.
Range, variance, standard deviation, quartile deviation and mean deviation are absolute measures of deviation
Coefficients of dispersion are relative measures of deviation

Examples on Measures of Dispersion

Example 1: Find the population standard deviation of the data set {1, 3, 6, 7, 12}.
Solution: Standard deviation is a measure of dispersion given by the formula \(\sqrt{\frac{\sum_{1}^{n}(X - \overline{X})^{2}}{n}}\).
n = 5
\(\overline{X}\) = (1 + 3 + 6 + 7 + 12) / 5 = 5.8
S.D = \(\sqrt{\frac{(1 - 5.8)^{2} + (3 - 5.8)^{2} + (6 - 5.8)^{2} + (7 - 5.8)^{2} + (12 - 5.8)^{2}}{5}}\) = 3.76
Answer: Standard deviation = 3.76
Example 2: Find the sample variance of the data set {2, 6, 12, 15}
Solution: Variance is a measure of dispersion given by \(\sum_{1}^{n} \frac{(X_{i} - \overline{X})^{2}}{n-1}\)
n = 4
\(\overline{X}\) = (2 + 6 + 12 + 15) / 4 = 8.75
Variance = \(\frac{(2 - 8.75)^{2} + (6 - 8.75)^{2} + (12 - 8.75)^{2} + (15 - 8.75)^{2}}{3}\) = 34.25
Answer: Variance = 34.25
Example 3: Find the range and coefficient of range of the data set {8, 12, 5, 6, 8, 2,15}
Solution: Range is a measure of dispersion given by Highest value (H) - Smallest value(S)
H = 15, S = 2
Range = 15 - 2 = 13
Coefficient of Range = (H - S) / (H + S) = 13 / 17 = 0.76
Answer: Range = 13, Coefficient of Range = 0.76

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Practice Questions on Measures of Dispersion

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FAQs on Measures of Dispersion

What is Meant By Measures of Dispersion in Statistics?

In statistics, measures of dispersion refer to positive real numbers that help to measure the variability of data about a central point.

What are the 5 Measures of Dispersion?

The absolute measures of dispersion are variance, standard deviation, mean deviation, quartile deviation, and range.

Why Do We Calculate Measures of Dispersion?

Two distinct data sets can have the same measure of central tendency, i.e., they can have the same mean or median. However, their levels of variability might be completely different. Measures of dispersion are required to determine this variability level.

What are Absolute and Relative Measures of Dispersion?

Absolute and relative measures of dispersion are used to check the spread of data. Absolute measures of dispersion have the same units as the data itself while relative measures of dispersion are dimensionless.

What is the Best Measure of Dispersion?

Standard deviation is the best and the most commonly used measure of dispersion. This is because it has most of the qualities that an ideal measure of dispersion should consist of.

What are the Objectives of Measures of Dispersion?

Given below are the objectives of measures of dispersion:

They help to find the average distance of data points from the average of the data set.
Identify the variability in data.
Compare two or more data sets on the basis of variability.

What are the Advantages and Disadvantages of Measures of Dispersion?

The advantages and disadvantages of the measures of dispersion are listed below:

Advantages

They help to identify the reliability of the average value of the data set.
They help to quantify the variability or dispersion of the data points in a data set.

Disadvantages

The computation process of certain measures of dispersion can be lengthy and complicated.
They cannot give an idea of symmetricity.

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