How to Measures of Dispersion?

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A step-by-step guide to measures of dispersion

Measures of dispersion can be defined as positive real numbers that measure the degree of homogeneity or heterogeneity of the given data.

A measure of dispersion will have a value of \(0\) if the data points in a data set are the same. However, as the variability of the data increases, so does the value of dispersion measures.

Types of measures of dispersion

The measures of dispersion can be classified into two general categories. These are absolute measures of dispersion and relative measures of dispersion.

Range, variance, standard deviation, and average deviation are included in the category of absolute deviation measures.

These measures have the same unit as the data being examined. Coefficients of dispersion are relative measures of deviation. Such dispersion measures are always dimensionless.

Absolute measures of dispersion

The most common absolute measures of deviation are:

Range: Given a set of data, the range can be defined as the difference between the maximum and the minimum value.

Variance: The average squared deviation from the mean of the given data set is known as a variance. This measure of dispersion examines the spread of the data about the mean. The formula of variance is:

\(\color{blue}{σ^2=\frac{1}{n}\sum _{i=1}^n\:\left(x_i-x\:̅\:\right)^2\:}\)

where \(n\) is the number of observations and \(x¯\) is the mean.

Standard deviation: It shows the square root of the variance of the standard deviation. Therefore, the standard deviation also measures the variation of the data about the mean. The formula of standard deviation is:

\(\color{blue}{σ=\sqrt{σ^2}}\)

Mean Deviation: Mean deviation shows the mean of the data’s absolute deviation about the central points. These center points can be the mean, median, or mode. The formula for mean deviation is:

\(\color{blue}{\frac{1}{n}\:\sum _{i=1}^n|x_i-x\:̅\:|\:}\)

Where \(x¯\) is the central value and denotes the mean, median, or mode.

Measures of Dispersion – Example 1:

Find the population standard deviation of the data set. \({1, 4, 8, 7, 14}\)

Solution: Standard deviation is a measure of dispersion given by the formula \(σ=\sqrt{σ^2}\):

\(x¯=\frac{1+4+8+7+14}{5}=6.8\)

\(σ^2=\frac{\left(1\:-\:6.8\right)^2+\left(4\:-\:6.8\right)^2+\left(8\:-\:6.8\right)^2+\left(7\:-\:6.8\right)^2+\left(14\:-\:6.8\right)^2}{5}=18.96\)

\(σ=\sqrt{18.96}= 4.35\)