The measure of dispersion shows the scatterings of the data. It explains how the data differs and provides a clear picture of the data distribution. Assume you’ve been handed a data set. Someone approaches you and asks you to share intriguing facts regarding this data set. How are you going to do it? You can state that you can find the mean, median, or mode of this data series and explain its distribution by finding the mean, median, or mode.

Is it, however, the only option? Is it true that the central trends are the only means to learn about the focus of the observation? We’ll learn about another measure in this part to better understand the data. We will learn about the measures of dispersion in this section. Let’s get this party started.

Measures of Dispersion or Variability

The measure of dispersion, as the name implies, depicts the data scatterings. It explains how the data differs from one another and provides a clear picture of the data distribution. Assume you have four datasets of the same size, each with the same mean, \(m\). The sum of the observations will be the same in all circumstances. The measure of central tendency, in this case, does not provide a clear and complete picture of the distribution for the four sets.

Can we obtain a sense of the distribution by looking at the dispersion of observations within and between datasets? The basic goal of the dispersion measure is to figure out how widely the data are dispersed. It depicts how far the data deviate from their mean value. Dispersion can be classified as follows:

1. An absolute measure of dispersion:
   Range and quartile deviation are examples of measurements that express the scattering of observations in terms of distances. The average of deviations of observations, such as mean deviation and standard deviation, is the metric that expresses variance in terms of the average of deviations of data.
2. A relative measure of dispersion:
   When comparing the distributions of two or more data sets and for unit-comparison, we utilise a relative measure of dispersion. The coefficient of variance, the coefficient of mean deviation, the coefficient of quartile deviation, the coefficient of variation, and the coefficient of standard deviation are the coefficients of range, mean deviation, quartile deviation, variation, and standard deviation, respectively.

Learn the Concepts of Standard Deviation

The degree to which values in a distribution deviate from the average of the distribution is called dispersion. There are certain measures of dispersion, namely:
(i) Range
(ii) Quartile Deviation
(iii) Mean Deviation
(iv) Variance & Standard Deviation

1. Range: The difference between the highest and lowest values among the given set of data is called range.
\(\rm{Range} = \rm{Maximum}\,\rm{value}\, – \rm{Minimum}\,\rm{value}\)
Merits of Range:
a. It is the most basic of the dispersion measures.
b. The calculation is simple.
c. Simple to comprehend
d. independent of the change in origin
Demerits of Range:
a. It is founded on two extreme observations. As a result, you will be affected by fluctuations.
b. A range is not a good indicator of dispersion.
c. Changes in scale will have an impact.

2. Quartile Deviation: A data collection is divided into quarters by quartiles. The first quartile \((Q_1)\) is the number in the centre of the data set between the smallest and the median. The median of the data set is the second quartile \((Q_2)\). The third quartile \((Q_3)\) is the value that falls halfway between the median and the largest.
Quartile deviation or semi-inter-quartile deviation is
\(Q = \frac{{{Q_3} – {Q_1}}}{2}\)
Merits of Quartile Deviation:
a. The quartile deviation compensates for all of the range’s flaws.
b. Half of the data is used.
c. Changes in origin are unaffected
Demerits of Quartile Deviation:
a. It disregards half of the data.
b. Depending on the scale adjustment
c. Not a good indicator of dispersion.

3. Mean Deviation: A statistical measure of the average deviation of data in a sample from the centre value is mean deviation.
For grouped data:
Mean deviation \(= M.D.\left( A \right) = \frac{{\mathop \sum \nolimits_{i = 0}^n |{x_i} – A|}}{n}\)
Where,
\(M.D.(A) =\) Mean deviation about \(A\)
\(A =\) Mean/Median/Mode
\(n =\) number of data or observation
Merits of Mean Deviation:
a. Based on all of the data
b. When the deviations from the median are taken, it yields a minimal value.
c. Changes in origin are unaffected
Demerits of Mean Deviation:
a. It is difficult and time-consuming to calculate
b. Depending on the scale adjustment
c. Negative sign ignorance results in artificiality and renders further mathematical treatment meaningless.

4. Variance & Standard Deviation: The variance of data points is a measure of how they differ from the mean. The higher the value of variance, the more scattered the data is from its mean, and the lower the value of variance, the less scattered the data is from its mean.
The expectation of a random collection of data’s squared variation from its mean value is called variance.
Variance \( = Var\left( X \right) = {\sigma ^2} = E\left[ {{{\left( {X – \mu } \right)}^2}} \right] = \frac{{\mathop \sum \nolimits_{i = 0}^n {{\left( {{x_i} – \mu } \right)}^2}}}{n}\)
Where,
\(\mu = E\left( X \right) = \) mean of the given data
\({x_i} = {i^{th}}\) data or observation
\(n =\) number of data or observation
There is one more formula of variance on simplification
Hence,
Variance \( = Var\left( X \right) = {\sigma ^2} = E\left( {{X^2}} \right) – {\left( \mu \right)^2} = \frac{{\mathop \sum \nolimits_{i = 0}^n {{\left( {{x_i}} \right)}^2}}}{n} – {\left( {\frac{{\mathop \sum \nolimits_{i = 0}^n {x_i}}}{n}} \right)^2}\)
The standard deviation is the positive square root of variance. The formula used to find the standard deviation is
\(S.D. = \sigma = \sqrt {\frac{{\sum {{{\left( {{x_i} – \mu } \right)}^2}} }}{N}} \)
\(N =\) Total number of frequency
Merits of Standard Deviation:
a. The disadvantage of ignoring signals in mean deviations is overcome by squaring the deviations.
b. It’s a good choice for more mathematical treatment
c. Changes in origin are unaffected
d. If all of the observations are the same, the standard deviation is \(0\)
Demerits of Standard Deviation:
a. The calculation is difficult.
b. For a layman, it’s difficult to grasp.
c. Changes in scale are dependent.

Coefficient of Dispersion

Based on several metrics of dispersion, the coefficients of dispersion (C.D.) are calculated.
1. C.D. based on Range \( = \frac{{{X_{{\text{max}}}} – {X_{\min }}}}{{{X_{\max }} + {X_{\min }}}}\)
2. C.D. based on Quartile deviation \( = \frac{{{Q_3} – {Q_1}}}{{{Q_3} + {Q_1}}}\)
3. C.D. based on Mean deviation \( = \frac{{{\text{Mean}}\,{\text{Deviation}}}}{A}\)
4. C.D. based on Standard deviation \( = \frac{{S.D.}}{{{\text{Mean}}}}\)
5. Coefficient of Variation \( = \frac{{S.D.}}{{{\text{Mean}}}} \times 100\)

Solved Examples

Q.1. What will be the range of the following data:
\(29,\,10,\,3,\,5,\,1,\,11,\,32,\,23,\,12,\,7\)
Ans: \({\text{Range}} = {\text{Maximum}}\,{\text{value}} – {\text{Minimum}}\,{\text{value}} = 29 – 1 = 28.\)

Q.2. Find the quartile deviation for the following data.
\(20,\,25,\,29,\,30,\,35,\,39,\,41,\,48,\,51,\,60,\,70\)
Ans: For quartile deviation \((Q.D.)\), we need to find values of \(Q_3\) and \(Q_1\).
\(Q_1\) is the size of \({\left( {\frac{{n + 1}}{4}} \right)^{{\rm{th}}}}\) value.
\(n\) being \(11,\,Q_1\) is the size of \({3^{{\rm{rd}}}}\) value. Since values are already arranged in ascending order, it can be seen that \(Q_1\), the \({3^{{\rm{rd}}}}\) value is \(29\). Similarly, \(Q_3\) is the size of \(\frac{{3{{\left( {n + 1} \right)}^{{\rm{th}}}}}}{4}\) value; i.e. \(9^{th}\) value which is \(51\). Hence \(Q_3 = 51.\)
\(Q = \frac{{{Q_3} – {Q_1}}}{2} = \frac{{51 – 29}}{2} = 11\)
\(Q.D\) is the average difference of the Quartiles from the median.

Q.3. Calculate the mean deviation about median and coefficient of mean deviation

Size of the item \((X)\)	\(6\)	\(12\)	\(18\)	\(24\)	\(30\)	\(36\)	\(42\)
Frequency \((f)\)	\(4\)	\(7\)	\(9\)	\(18\)	\(15\)	\(10\)	\(5\)

Ans:

\(X\)	\(f\)	\(cf\)	\(|D|\)	\(f|D|\)
\(6\)	\(4\)	\(4\)	\(18\)	\(72\)
\(12\)	\(7\)	\(11\)	\(12\)	\(84\)
\(18\)	\(9\)	\(20\)	\(6\)	\(54\)
\(24\)	\(18\)	\(38\)	\(0\)	\(0\)
\(30\)	\(15\)	\(53\)	\(6\)	\(90\)
\(36\)	\(10\)	\(63\)	\(12\)	\(120\)
\(42\)	\(5\)	\(68\)	\(18\)	\(90\)
	\(\sum f = 68\)		\(\sum {|D|} = 72\)	\(\sum {f |D|} = 510\)

Median \(=\) Size of \({\left( {\frac{{N + 1}}{2}} \right)^{th}}\) item
\(=\) size of \({\left( {\frac{{68 + 1}}{2}} \right)^{th}}\) item
\( = {34.5^{th}}\) item
M.D. about median \( = \frac{{{{\Sigma }}f|D|}}{N} = \frac{{510}}{{68}} = 7.5\)
Coefficient of M.D. \( = \frac{{M.D.}}{{{\text{Median}}}} = \frac{{7.5}}{{24}} = 0.312.\)

Q.4. Find the mean deviation from the mean for the following data:
\(4,\;7,\;8,\;9,\;10,\;12,\;13,\;17\)
Ans: Let \(\overline x \) be the mean of the given data.
\(\overline x = \frac{{4 + 7 + 8 + 9 + 10 + 12 + 13 + 17}}{8} = 10\)

\(x_i\)	\(\left| {{d_i}} \right| = \left| {{x_i} – \overline x } \right|\)
\(4\)	\(6\)
\(7\)	\(3\)
\(8\)	\(2\)
\(9\)	\(1\)
\(10\)	\(0\)
\(12\)	\(2\)
\(13\)	\(3\)
\(17\)	\(7\)
Total	\(24\)

We know, \(MD = \frac{1}{n}\sum _{i = 1}^n\left| {{d_i}} \right|\)
\(\therefore \,MD = \frac{1}{8} \times 24 = 3.\)

Q.5. Calculate the standard deviation and variance for the following data:

\(x:\)	\(3\)	\(8\)	\(13\)	\(18\)	\(23\)
\(f:\)	\(7\)	\(10\)	\(15\)	\(10\)	\(6\)

Ans:

\(x_i\)	\(f_i\)	\(f_i x_i\)	\(f_i x_i^2\)
\(3\)	\(7\)	\(21\)	\(63\)
\(8\)	\(10\)	\(80\)	\(640\)
\(13\)	\(15\)	\(195\)	\(2535\)
\(18\)	\(10\)	\(180\)	\(3240\)
\(23\)	\(6\)	\(138\)	\(3174\)
	\(N = 48\)	\(\sum {{f_i}{x_i} = 614} \)	\(\sum {{f_i}{x_i}^2 = 9652}\)

Standard deviation \((\sigma ) = \sqrt {\frac{{\sum {f_i}{x_i}^2}}{N} – {{\left( {\frac{{\sum {f_i}{x_i}}}{N}} \right)}^2}} \)
\(S.D.\,\sigma = \sqrt {201.08 – {{(12.79)}^2}} \)
\( = \sqrt {201.08 – 163.58} = \sqrt {37.49} = 6.12\)
Variance \(\left( {{\sigma ^2}} \right) = 37.49.\)

Summary

In this article, we have learnt about the measures of dispersion. Dispersion refers to how many values in a distribution deviate from the average. It tells us how different individual items are from one another and from the centre value.

Range, mean deviation, quartile deviation, standard deviation, and variance are examples of statistical dispersion measurements. The standard deviation is regarded as a very good measure of series dispersion because it is a measure of average deviations from the average.

Learn About Measures of Central Tendency

FAQs

Q.1. What are the 4 measures of dispersion?
Ans:  Dispersion measures describe how widely the data is spread out. Range, quartile deviation, Mean deviation, and standard deviation are the 4 measures of dispersion.

Q.2. What do you mean by measures of dispersion?
Ans: The scattering of data is measured by dispersion. It describes how data differs from one another and provides a clear picture of their distribution. The measure of dispersion illustrates and informs us about the variety and core value of a single object.
In other words, dispersion refers to how many values in a distribution deviate from the average. It tells us how different individual items are from one another and from the centre value.

Q.3. What is the measure of dispersion in research?
Ans: The standard deviation (along with various related measures such as variance, coefficient of variation, and so on) is mostly employed in research investigations and is regarded as a very good measure of series dispersion. Because the algebraic signs are not neglected in its calculation, it is accessible to mathematical manipulation (as we ignore in the case of mean deviation).
It is less impacted by sampling fluctuations. Because of these benefits, standard deviation and its coefficient are widely used to assess the scatteredness of a series. It’s commonly used in the context of hypothesis estimate and testing.

Q.4. What are the important measures of dispersion?
Ans: The standard deviation is regarded as a very good measure of series dispersion because it is a measure of average deviations from the average.
The range is also the most commonly used measure of dispersion.

Q.5. What is an example of dispersion?
Ans: Data variability is measured using measures of dispersion. Range, mean deviation, quartile deviation, standard deviation, and variance are examples of statistical dispersion measurements. The choice of measure is determined by the skewness of the data collection.

We hope this detailed article on the measures of dispersion helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!