Coefficient of Dispersion: Definition, Formula and Examples

Collecting data and expressing it in the form of measures of data is an essential concept for us. The measure of the spread shows how much variation is there in data. It shows how the data is spread, and scattered, and what is the deviation, and variance of the data. These values describe the data in a better way and help the analyst to analyze the data in a better way and take out the insights from it. This is one of the fundamentals of statistics which has numerous applications in various domains like data analysis, weather forecast, business, etc.

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This Story also Contains

1. Measures of the Dispersion of the Data
2. Coefficient of Dispersion
3. Solved Examples Based on Coefficient of Dispersion
4. Summary

This article is about the concept Coefficient of Dispersion. This is an important concept which falls under the broader category of Statistics. This is not only important for board exams but also for various competitive exams.

Measures of the Dispersion of the Data

An important characteristic of any set of data is the variation in the data. The degree to which the numerical data tends to vary about an average value is called the dispersion or scatteredness of the data.

The following are the measures of dispersion:

1. Range

2. Mean Deviation

3. Standard deviation and Variance

Range

Range is the difference between the highest and the lowest value in a set of observations.

The range of data gives us a rough idea of variability or scatter but does not tell about the dispersion of the data from a measure of central tendency.

Mean Deviation

Mean deviation measures the deviation of the average mean to the given set of data.

Mean deviation for ungrouped data

Let $n$ observations are $\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \ldots ., \mathrm{x}_{\mathrm{n}}$.

Mean deviation about 'a', M.D. $(a)=\frac{1}{n} \sum_{i=1}^n\left|x_i-a\right|$
Mean deviation about mean, M.D. $(\bar{x})=\frac{1}{n} \sum_{i=1}^n\left|x_i-\bar{x}\right|$
Mean deviation about median, M.D.(Median $) \left.=\frac{1}{n} \sum_{i=1}^n \right\rvert\, x_i-$ Median $\mid$

Standard Deviation

The standard deviation is a number that measures how far data values are from their mean.
The positive square root of the variance is called the standard deviation. The standard deviation is usually denoted by $\sigma$ and it is given by

$
\sigma=\sqrt{\frac{1}{n} \sum_{i=1}^n\left(x_i-\bar{x}\right)^2}
$

Variance

The mean of the squares of the deviations from the mean is called the variance and is denoted by $\sigma^2$ (read as sigma square). Variance is a quantity that leads to a proper measure of dispersion.

The variance of $n$ observations $x_1, x_2, \ldots, x_n$ is given by

$
\sigma^2=\frac{1}{n} \sum_{i=1}^n\left(x_i-\bar{x}\right)^2
$

Coefficient of Dispersion

The measure of variability which is independent of units is called coefficient of dispersion.

An important characteristic of any set of data is the variation in the data. The degree to which the numerical data tends to vary about an average value is called the dispersion or scatteredness of the data.

The following are the measures of dispersion:

1. Range

2. Mean Deviation

3. Standard deviation and Variance

1. Coefficient of Range

The coefficient of the range equals $\frac{x_{\max }-x_{\min }}{x_{\max }+x_{\min }}$
Where $x_{\max }$ is the highest observation, and $x_{\min }$ is the lowest observation

2. Coefficient of Mean Deviation

The coefficient of mean deviation is $=\frac{M D}{\bar{x}}$

where $MD$ is the mean deviation and $\bar{x}$ is the mean of the data.

3. Coefficient of Standard Deviation

The coefficient of standard deviation is $\frac{\sigma}{\bar{x}}$
where $\sigma$ and $\bar{x}$ are the standard deviation and mean of the data respectively.

4. Coefficient of Variance

The mean of the squares of the deviations from the mean is called the variance and is denoted by $\sigma^2$ (read as sigma square).
Variance is a quantity that leads to a proper measure of dispersion.
The variance of $n$ observations $x_1, x_2, \ldots, x_n$ is given by

$
\sigma^2=\frac{1}{n} \sum_{i=1}^n\left(x_i-\bar{x}\right)^2
$

The coefficient of variation is defined as

$
\text { C.V. }=\frac{\sigma}{\bar{x}} \times 100, \bar{x} \neq 0
$

where $\sigma$ and $\bar{x}$ are the standard deviation and mean of the data. It is consistent than the other and thus is considered better.

Recommended Video Based on Coefficient of Dispersion

Solved Examples Based on Coefficient of Dispersion

Example 1: Coefficient of variation of two distributions are $50$ and $60$ , and their arithmetic means are $30$ and $25$ respectively. Difference of their $S.D.$ is?
1) $0$
2) $1$
3) $1.5$
4) $2.5$

Solution
$
\begin{aligned}
& \frac{\sigma_1}{\bar{x}_1} \times 100=50 \Rightarrow \sigma_1=\frac{\bar{x}}{2}=\frac{30}{2}=15 \\
& \frac{\sigma_2}{\bar{x}_2} \times 100=60 \Rightarrow \sigma_1=\frac{60}{4}=15 \\
& \left|\sigma_1-\sigma_2\right|=0
\end{aligned}
$
Hence, the answer is the option 1 .

Example 2: The following are the weights (in kg ) of $8$ students in a class: $62,65,58,60,70,55,68$, and $72$ . What is the coefficient of variation $(CV)$ of the weights?
1) $10.3 \%$
2) $9.9 \%$
3) $23.1 \%$
4) $27.5 \%$

Solution
To calculate the coefficient of variation $(CV)$, we first need to calculate the mean weight and standard deviation of the weights:

$
\begin{aligned}
& \text { Mean weight }=\frac{62+65+58+60+70+55+68+72}{8}=64.5 \\
& \begin{aligned}
\text { Standard deviation } & =\sqrt{\frac{(62-64.5)^2+(65-64.5)^2+(58-64.5)^2+(60-64.5)^2+(70-64.5)^2+(55-64.5)^2+(68-64.5)^2+(72-64.5)^2}{7}} \\
& =6.39
\end{aligned}
\end{aligned}
$
Now we can calculate the coefficient of variation $(CV)$:

$
\frac{6.39}{64.5} \times 100 \%=9.9 \%
$
Hence the answer is option (2)

Example 3: If the coefficient of variation $(CV)$ of a set of data is $0.5$ , what can be said about the variability of the data compared to the mean?
1) The data is highly variable compared to the mean.
2) The data is moderately variable compared to the mean.
3) The data is slightly variable compared to the mean.
4) The variability of the data cannot be determined from the coefficient of variation.

Solution mean, which indicates that the data is highly variable compared to the mean. In general, a $CV$ greater than $1$ indicates high variability, while a $CV$ less than $1$ indicates low variability.
Hence, the answer is option (1).

Example 4: Which of the following is a drawback of using the coefficient of quartile deviation as a measure of dispersion?
1) It does not take into account extreme values.
2) It is sensitive to changes in the units of measurement.
3) It is difficult to calculate.
4) It is affected by the size of the dataset.

Solution

The coefficient of quartile deviation $(CQD)$ is a measure of relative dispersion that is based on the quartiles of a dataset. However, a drawback of $CQD$ is that it does not take into account extreme values, such as outliers, which can have a significant impact on the dispersion of the data. Therefore, $CQD$ may not accurately reflect the dispersion of the dataset if it contains extreme values.
Hence, the answer is option (1).

Example 5: If the mean of the data: $7,8,9,7,8,7, \lambda, 8$ is $8$, then the variance of this data is :
1) $\frac{7}{8}$
2) $1$
3) $\frac{9}{8}$
4) $2$

Solution

$
\begin{aligned}
& \text { mean of data }=\frac{7+8+9+7+8+7+7+8}{8}=8 \\
& \Rightarrow \lambda=10
\end{aligned}
$
Variance

$
\begin{aligned}
& V^2=\frac{(7-8)^2+(8-8)^2+(9-8)^2+(7-8)^2+0^2+(7-8)^2+(10-8)^2+(8-}{8} \\
& =\frac{8}{8}=1
\end{aligned}
$

Variance $=1$
Hence, the answer is the option 2.

Summary

The measure of variability which is independent of units is called coefficient of dispersion. The standard deviation is a number that measures how far data values are from their mean.
$
\sigma=\sqrt{\frac{1}{n} \sum_{i=1}^n\left(x_i-\bar{x}\right)^2}
$
The mean of the squares of the deviations from the mean is called the variance.
The coefficient of variation is defined as

$
\text { C.V. }=\frac{\sigma}{\bar{x}} \times 100, \bar{x} \neq 0
$

where $\sigma$ and $\bar{x}$ are the standard deviation and mean of the data.