Measures of Dispersion – Definition, Types, Formulas

Measures of Dispersion

Measures of dispersion help us to describe how much spread a data set is. We can define it as the state of data getting scattered, stretched, squeezed or spread out in different categories.

In statistics, data dispersion helps us quickly understand the dataset by classifying them into their own specific scattering criteria like variance, ranging, and standard deviation. 

Types of Measures of Dispersion

The measure of dispersion is classified as:

(i) Absolute measure of dispersion

(ii) A relative measure of dispersion

The Absolute Measure of Dispersion

This measure of dispersion represents the dispersion of observation in terms of distances. We can express the measure of dispersion in units such as Centimetre, Rupees, kilograms, and more quantities depending on the situation.

Let’s see the types of the absolute measure of dispersion.

• Range: Range is a measure of the difference between the maximum and minimum values of the data set. This is the simplest form of measure of dispersion. 

Example: 2, 3, 4, 5, 6, 7 

Range = (Maximum value – Minimum value) 

            = (7 – 2) 

            = 5 

• Mean: Mean is the average of the given terms. To calculate it, add all the values and then divide it with the total number of values. It is represented by μ.

Example: 1, 2, 3, 4, 5, 6, 7 

Mean = (sum of all the values)/(total number of terms) 

               = (1 + 2 + 3 + 4 + 5 + 6 + 7)/7 

              = 28/7

               = 4

• Variance: We can calculate the variance by calculating the sum of the squared distance of each value in the data set from the mean and then dividing it by the number of terms in the data set.  

In simple language, it shows how far a value is from the mean of the entire value. For example, we can calculate how far a student’s mark in an exam is from the mean of the whole class using a variance.

Formula: 

\left(\sigma^{2}\right)=\sum(X-\mu)^{2} / N

• Standard Deviation: Standard Deviation can be expressed as the square root of the variance. To find the standard deviation of any data set, first, we need to find the variance. 

Formula: 

Standard Deviation = √σ

• Quartile: Quartiles divide the data set into quarters. 

• Quartile Deviation: The quartile deviation is half (½) of the distance between the third and the first quartile.

Formula: 

Q=(1 / 2) \times\left(Q_{3}-Q_{1}\right)

• Mean deviation: It is also known as an average deviation. We can calculate the mean deviation using the mean or median of the data. 

Formula: 

Mean Deviation using Mean:\sum|X-M| / N

Mean Deviation using Median: ∑ |X – Xi| / N 

Where X = Mean, M = Median, N= Number of values, Xi = frequency of the ith class interval

Relative Measures of Dispersion

We compare the distributions of two or more data sets using a relative measure of dispersion. They can be – 

•  Coefficient of Range: it is calculated as the ratio of the difference between the maximum and minimum values of the distribution to the sum of the highest and smallest values of the distribution.  

Formula:  (L – S) / (L + S)

where L = maximum value 

S = minimum value 

• Coefficient of Variation: We use the coefficient of variation to compare the two data for homogeneity or consistency.  

Formula: 

C.V = (σ / X) × 100 

X = standard deviation  

σ = mean 

• Coefficient of Standard Deviation: It is the ratio of standard deviation with the mean of the distribution of values.  

Formula:

\sigma=\left(\sqrt{\left(X-X_{1}\right)}\right) /(N-1)

Deviation = (X – X1)  

σ = standard deviation  

N= total number  

• Coefficient of Quartile Deviation: The ratio of the difference between the upper and the lower quartile to the sum of the upper and lower quartile is coefficient of quartile deviation.

Formula: 

\left(Q_{3}-Q_{1}\right) /\left(Q_{3}+Q_{1}\right)

\mathrm{Q}_{3} = Upper Quartile  

Q_{1} = Lower Quartile 

• Co-efficient of Mean Deviation: We can calculate the co-efficient of mean deviation using the mean or median of the data. 

Mean Deviation using mean: ∑ |X – M| / N 

Mean Deviation using median: ∑ |X – Xi| / N 

Where, X = Mean, M = Median, N= Number of values, Xi = frequency of the ith class interval

Note – Relative measures of dispersion don’t have any units.

Examples

 1. Find the range and coefficient of range of the following data: 25, 68, 48, 53, 18, 39, 44.

Maximum value L = 68; Minimum value S =18

Range R = L − S = 68 −18 = 50

Formula for coefficient of range = (L –S) / (L + S)

So, Coefficient of range = (68 – 18 ) / (68 +18) 

                                           = 50/86

                                           = 0.581

2. Find the Variance and Standard Deviation of the Following Numbers: 1, 3, 5, 6, 6, 7, 8, 10.

Mean = (1 + 3 + 5 + 6 + 6 + 7 + 8 + 10)/8 

           = 46/ 8

           = 5.75

Now, to calculate the standard deviation,

Subtract the mean from each individual value

(1 – 5.75), (3 – 5.75), (5 – 5.75), (6 – 5.75), (6 – 5.75), (7 – 5.75), (8 – 5.75), (10 – 5.75)

= -4.75, -2.75, -0.75, 0.25, 0.25, 1.25, 2.25, 4.25

Squaring these values we get, 

22.563, 7.563, 0.563, 0.063, 0.063, 1.563, 10.563, 18.063

Adding the above values we get,

22.563 + 7.563 + 0.563 + 0.063 + 0.063 + 1.563 + 5.063 + 18.063

= 55.504

Number of terms (n) = 8

Therefore, variance \left(\sigma^{2}\right) = 55.504/ 8 = 6.94

So, Standard deviation (σ) = √ 6.94 = 2.63