Measures of Dispersion

Suppose you are given a data series. Someone asks you to tell some interesting facts about this data series. How can you do so? You can say you can find the mean, the median or the mode of this data series and tell about its distribution. But is it the only thing you can do? Are the central tendencies the only way by which we can get to know about the concentration of the observation? In this section, we will learn about another measure to know more about the data. Here, we are going to know about the measure of dispersion. Let’s start.

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What Is Dispersion?

Dispersion refers to the ‘distribution’ of objects over a large region. The degree to which numerical data are dispersed or squished around an average value is referred to as dispersion in statistics. It is, in a nutshell, the dispersion of data. A vast amount of data will always be widely dispersed or firmly packed. Data that is widely dispersed – 0, 30, 60, 90, 120, With tiny data grouped densely – 1, 2, 2, 3, 3, 4, 4….

Understanding Dispersion

The term “dispersion” refers to how dispersed a set of data is. The measure of dispersion is always a non-negative real number that starts at zero when all the data is the same and rises as the data gets more varied. The homogeneity or heterogeneity of the scattered data is defined by dispersion measures. It also refers to how data differs from one another.

Measures of Dispersion

As the name suggests, the measure of dispersion shows the scatterings of the data. It tells the variation of the data from one another and gives a clear idea about the distribution of the data. The measure of dispersion shows the homogeneity or the heterogeneity of the distribution of the observations.

Browse more Topics under Measures Of Central Tendency And Dispersion

• Arithmetic Mean
• Median and Mode
• Partition Values or Fractiles
• Harmonic Mean and Geometric Mean
• Range and Mean Deviation
• Quartiles, Quartile Deviation and Coefficient of Quartile Deviation
• Standard deviation and Coefficient of Variation

Suppose you have four datasets of the same size and the mean is also the same, say, m. In all the cases the sum of the observations will be the same. Here, the measure of central tendency is not giving a clear and complete idea about the distribution for the four given sets.

Can we get an idea about the distribution if we get to know about the dispersion of the observations from one another within and between the datasets? The main idea about the measure of dispersion is to get to know how the data are spread. It shows how much the data vary from their average value.

Characteristics of Measures of Dispersion

• A measure of dispersion should be rigidly defined
• It must be easy to calculate and understand
• Not affected much by the fluctuations of observations
• Based on all observations

Classification of Measures of Dispersion

The measure of dispersion is categorized as:

(i) An absolute measure of dispersion:

• The measures express the scattering of observation in terms of distances i.e., range, quartile deviation.
• The measure expresses the variations in terms of the average of deviations of observations like mean deviation and standard deviation.

(ii) A relative measure of dispersion:

We use a relative measure of dispersion for comparing distributions of two or more data set and for unit free comparison. They are the coefficient of range, the coefficient of mean deviation, the coefficient of quartile deviation, the coefficient of variation, and the coefficient of standard deviation.

Dispersion Measurement Types

The dispersion is constantly dependent on the observations and types of central tendency metrics used. The following are examples of dispersion measures:

1. Range
2. Deviation from the median
3. Deviation from the mean
4. Deviation from the mean

Range

Range refers to the difference between each series’ minimum and maximum values. The range offers us a good indication of how dispersed the data is, but we need other measures of variability to discover the dispersion of data from central tendency measurements. A range is the most common and easily understandable measure of dispersion. It is the difference between two extreme observations of the data set. If X _max and X _min are the two extreme observations then

Range = X _max – X _min

Merits of Range

• It is the simplest of the measure of dispersion
• Easy to calculate
• Easy to understand
• Independent of change of origin

Demerits of Range

• It is based on two extreme observations. Hence, get affected by fluctuations
• A range is not a reliable measure of dispersion
• Dependent on change of scale

Quartile Deviation

The quartiles divide a data set into quarters. The first quartile, (Q₁) is the middle number between the smallest number and the median of the data. The second quartile, (Q₂) is the median of the data set. The third quartile, (Q₃) is the middle number between the median and the largest number.

Quartile deviation or semi-inter-quartile deviation is

Q = ½ × (Q₃ – Q1)

Merits of Quartile Deviation

• All the drawbacks of Range are overcome by quartile deviation
• It uses half of the data
• Independent of change of origin
• The best measure of dispersion for open-end classification

Demerits of Quartile Deviation

• It ignores 50% of the data
• Dependent on change of scale
• Not a reliable measure of dispersion

Mean Deviation

Mean deviation is the arithmetic mean of the absolute deviations of the observations from a measure of central tendency. If x₁, x₂, … , x_n are the set of observation, then the mean deviation of x about the average A (mean, median, or mode) is

Mean deviation from average A = 1⁄n [∑_i|x_i – A|]

For a grouped frequency, it is calculated as:

Mean deviation from average A = 1⁄N [∑_i  f_i |x_i – A|], N = ∑f_i

Here, x_i and f_i are respectively the mid value and the frequency of the i^th class interval.

Merits of Mean Deviation

• Based on all observations
• It provides a minimum value when the deviations are taken from the median
• Independent of change of origin

Demerits of Mean Deviation

• Not easily understandable
• Its calculation is not easy and time-consuming
• Dependent on the change of scale
• Ignorance of negative sign creates artificiality and becomes useless for further mathematical treatment

Standard Deviation

A standard deviation is the positive square root of the arithmetic mean of the squares of the deviations of the given values from their arithmetic mean. It is denoted by a Greek letter sigma, σ. It is also referred to as root mean square deviation. The standard deviation is given as

σ = [(Σ_i (y_i – ȳ) ⁄ n] ^½ =  [(Σ _i y_i ² ⁄ n) – ȳ ²] ^½

For a grouped frequency distribution, it is

σ = [(Σ_i  f_i (y_i – ȳ) ⁄ N] ^½ = [(Σ_i f_i  y_i ² ⁄ n) – ȳ ²] ^½

The square of the standard deviation is the variance. It is also a measure of dispersion.

σ ² = [(Σ_i (y_i – ȳ ) / n] ^½ =  [(Σ_i y_i ² ⁄ n) – ȳ ²]

For a grouped frequency distribution, it is

σ ² = [(Σ_i  f_i (y_i – ȳ ) ⁄ N] ^½ = [(Σ _i f_i  x_i ² ⁄ n) – ȳ ²].

If instead of a mean, we choose any other arbitrary number, say A, the standard deviation becomes the root mean deviation.

Variance of the Combined Series

If σ₁, σ₂ are two standard deviations of two series of sizes n₁ and n₂ with means ȳ₁ and ȳ₂. The variance of the two series of sizes n₁ + n₂ is:

σ ² = (1/ n₁ + n₂) ÷ [n₁ (σ₁ ² + d₁ ²) + n₂ (σ₂ ² + d₂ ²)]

where, d₁ = ȳ ₁ − ȳ , d₂ = ȳ ₂ − ȳ , and ȳ  = (n₁ ȳ ₁ + n₂ ȳ ₂) ÷ ( n₁ + n₂).

Merits of Standard Deviation

• Squaring the deviations overcomes the drawback of ignoring signs in mean deviations
• Suitable for further mathematical treatment
• Least affected by the fluctuation of the observations
• The standard deviation is zero if all the observations are constant
• Independent of change of origin

Demerits of Standard Deviation

• Not easy to calculate
• Difficult to understand for a layman
• Dependent on the change of scale

Coefficient of Dispersion

Whenever we want to compare the variability of the two series which differ widely in their averages. Also, when the unit of measurement is different. We need to calculate the coefficients of dispersion along with the measure of dispersion. The coefficients of dispersion (C.D.) based on different measures of dispersion are

• Based on Range = (X _max – X _min) ⁄ (X _max + X _min).
• C.D. based on quartile deviation = (Q₃ – Q1) ⁄ (Q₃ + Q1).
• Based on mean deviation = Mean deviation/average from which it is calculated.
• For Standard deviation = S.D. ⁄ Mean

Coefficient of Variation

100 times the coefficient of dispersion based on standard deviation is the coefficient of variation (C.V.).

C.V. = 100 × (S.D. / Mean) = (σ/ȳ ) × 100.

Solved Example on Measures of Dispersion

Problem: Below is the table showing the values of the results for two companies A, and B.

1. Which of the company has a larger wage bill?
2. Calculate the coefficients of variations for both of the companies.
3. Calculate the average daily wage and the variance of the distribution of wages of all the employees in the firms A and B taken together.

Solution:

For Company A

No. of employees = n₁ = 900, and average daily wages = ȳ ₁  = Rs. 250

We know, average daily wage = Total wages ⁄ Total number of employees

or, Total wages = Total employees × average daily wage = 900 × 250 = Rs. 225000 … (i)

For Company B

No. of employees = n₂ = 1000, and average daily wages = ȳ₂ = Rs. 220

So, Total wages = Total employees × average daily wage = 1000 × 220 = Rs. 220000 … (ii)

Comparing (i), and (ii), we see that Company A has a larger wage bill.

For Company A

Variance of distribution of wages = σ₁² = 100

C.V. of distribution of wages  = 100 x standard deviation of distribution of wages/ average daily wages

Or, C.V. _A = 100 × √100⁄250 = 100 × 10⁄250 = 4 … (i)

For Company B

Variance of distribution of wages = σ₂² = 144

C.V. _B = 100 × √144⁄220 = 100 × 12⁄220 = 5.45 … (ii)

Comparing (i), and (ii), we see that Company B has greater variability.

For Company A and B, taken together

The average daily wages for both the companies taken together

ȳ  = (n₁ ȳ ₁ + n₂ ȳ ₂)⁄( n₁ + n₂) = (900 × 250 + 1000 × 220) ÷ (900 + 1000) = 445000⁄1900 = Rs. 234.21

The combined variance, σ² = (1/ n₁ + n₂) ÷ [n₁ (σ₁ ² + d₁ ²) + n₂ (σ₂ ² + d₂ ²)]

Here, d₁ = ȳ₁ − ȳ  = 250 – 234.21 = 15.79, d₂ = ȳ₂ −  ȳ  = 220 – 234.21 = – 14.21.

Hence, σ² = [900 × (100 + 15.79²) + 1000 × (144 + – 14.21²)] ⁄ (900 + 1000)

or, σ² = (314391.69 + 345924.10) ⁄ 1900 = 347.53.