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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## 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 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 which express the scattering of observation in terms of distances i.e., range, quartile deviation.
- The measure which 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.

## Range

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, (Q_{1}) is the middle number between the smallest number and the median of the data. The second quartile, (Q_{2}) is the median of the data set. The third quartile, (Q_{3}) is the middle number between the median and the largest number.

Quartile deviation or semi-inter-quartile deviation is

Q = ½ × (Q_{3} – 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 x_{1}, x_{2}, … , 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 σ_{1}, σ_{2 }are two standard deviations of two series of sizes n_{1} and n_{2 }with means ȳ_{1} and ȳ_{2}. The variance of the two series of sizes n_{1} + n_{2 }is:

σ ^{2 }= (1/ n_{1} + n_{2}) ÷ [n_{1} (σ_{1} ^{2 }+ d_{1} ^{2}) + n_{2} (σ_{2} ^{2} + d_{2} ^{2})]

where, d_{1} = ȳ^{ }_{1} − ȳ^{ }, d_{2} = ȳ^{ }_{2 }− ȳ^{ }, and ȳ^{ } = (n_{1} ȳ_{ }_{1}^{ }+ n_{2 }ȳ_{ }_{2}) ÷ ( n_{1} + n_{2}).

**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 = (Q
_{3}– Q1) ⁄ (Q_{3}+ 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.

- Which of the company has a larger wage bill?
- Calculate the coefficients of variations for both of the companies.
- 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 = n_{1} = 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 = n_{2} = 1000, and average daily wages = ȳ_{2} = 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 = σ_{1}^{2} = 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 = σ_{2}^{2} = 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^{ } = (n_{1 }ȳ_{ }_{1}^{ }+ n_{2 }ȳ_{ }_{2})⁄( n_{1} + n_{2}) = (900 × 250 + 1000 × 220) ÷ (900 + 1000) = 445000⁄1900 = Rs. 234.21

The combined variance, σ^{2} = (1/ n_{1} + n_{2}) ÷ [n_{1} (σ_{1} ^{2 }+ d_{1} ^{2}) + n_{2} (σ_{2} ^{2} + d_{2} ^{2})]

Here, d_{1} = ȳ_{1} − ȳ^{ } = 250 – 234.21 = 15.79, d_{2} = ȳ_{2 }− ȳ^{ } = 220 – 234.21 = – 14.21.

Hence, σ^{2} = [900 × (100 + 15.79^{2}) + 1000 × (144 + – 14.21^{2})] ⁄ (900 + 1000)

or, σ^{2} = (314391.69 + 345924.10) ⁄ 1900 = 347.53.

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