Measure of dispersion mathematics

Introduction

The measure of dispersion is used to indicate how the data is spread out. It indicates how the data differs from one another and provides a clear picture of how it is distributed. An observation’s dispersion measures how homogeneous or diverse the distribution of the data is.

Distribution dispersion refers to how far a given distribution’s values deviate from the average. Individual things can be compared to each other and a central value to see how much they differ.

Some of the common forms of statistical dispersion are:

• Mean deviation
• Standard deviation
• Variance
• Range & Interquartile range

This measure of dispersion study material discusses measures of dispersion in detail. Read on. 

What qualities should a good measure of dispersion have? 

• A measure of dispersion needs to be straightforward to calculate and understand.
• It should be on the basis of the series’ observations.
• A measure of dispersion needs to be defined in a precise manner.
• Extreme values should have no effect on it. 
• Sampling fluctuations shouldn’t have a significant impact on a measure of dispersion.
• Further mathematical and statistical analysis should be possible.

What is the need for a measure of dispersion computation?

Comparative research:

• To get an idea of how consistent or uniform the distribution is, dispersion measures are used. We may compare different distributions by using this single number.
• The stronger the consistency or uniformity, the lesser the magnitude (value) of dispersion.

The average’s dependability:

• A small correlation value between individual observations and the average is a sign of poor dispersion. As a result, the average is a pretty accurate representation of what we’ve observed.
• Dispersion is a measure of how much variation there is in the data. In this instance, the average is unreliable and should not be used to measure success.

Controlling the variability:

• Dispersion measurements give us information about variability from various perspectives, and this information can be useful in reducing the amount of variation.
• Especially in commercial and medical financial analysis, these dispersion measurements can be highly beneficial.

A foundation for more statistical investigation:

• Correlation, regression, hypothesis testing, etc., may all be calculated using a measure of dispersion.

Absolute measure of dispersion

Dispersion can be measured in a variety of ways, including the following:

Range:

• It is the simplest way to measure dispersion.
• Difference between the largest and smallest items in a particular distribution is defined as the range
• The Range (S) = Biggest item (L)- Smallest item (S) 

The Interquartile Range:

• The interquartile range is the difference between a distribution’s upper and lower quartiles.
• Interquartile range is the difference between the upper (Q3) and lower quartiles (Q1)

(Interquartile range= Q3–Q1)

Quartile Deviation:

• Half of the difference between the upper quartile and lower quartile is the Semi Inter Quartile Range 
• Quartile Deviation = (Q3–Q1)/2

Where Q3=Upper Quartile, and Q1= Lower Quartile

Mean Deviation:

• The arithmetic mean (average) of the deviations D of observations from a central value Mean or Median is the term “mean deviation.”

Variance: 

• Variance is a measure of the degree of variability. It is calculated by averaging the squared deviations from the mean. Variance is a measure of the degree to which your data set is spread out. The more dispersed the data are, the greater the variation from the mean. Variance can be calculated by the formula= σ2=1ni=1n(xi–x)2

Standard Deviation

• For example, the square root of the means of the squared deviations is called standard deviation.

Lorenz Curve:

• When calculating dispersion, the Lorenz Curve can be used as a visual representation.
• This line is frequently employed to depict the widening gap between the rich and the rest of society’s citizens.

Different relative measures of dispersion:

Comparisons of distributions of two or more data sets are made using relative measures of dispersion. This metric compares values without regard for units. Among the most often used methods of relative dispersion are the following:

Coefficient of Range

• It’s the difference between the sum of the two most extreme elements in the distribution.
• Coefficient of Range is derived by: (largest item–smallest item)/ (largest item+ smallest item)

Coefficient of Quartile Deviation:

• It is the ratio of the difference between the Upper and Lower Quartiles of a distribution to the sum of their values.
• The Coefficient of Quartile Deviation is obtained by Q3–Q1/Q3+Q1.

Coefficient of Variation:

• It’s used to determine the stability of two sets of data (or uniformity or homogeneity or consistency).
• It is a percentage-based representation of the connection between the arithmetic mean and standard deviation.
• Coefficient of Variation (C.V.) =σ/ x̄ * 100
• Where, C.V. = Coefficient of Variation; σ= Standard Deviation; x̄= Arithmetic Mean

Uses of Measures of Dispersion

• A spread measure, also known as a measure of dispersion, is used to define the variability within a sample or population.
• Dispersion measures are used to estimate a dataset’s “normal” values; they are also useful for describing the spread of the data or its fluctuation around a central value.
• It is frequently used in conjunction with a measure of central tendency, like the mean or median, to provide an overview of a collection of data.

Conclusion

We have thus understood the measure of dispersion, its applications, qualities and various forms. We have also learned a few formulas of ranges, mean deviation and standard deviation according to the measure of dispersion study material. Moreover, we have also understood its needs and importance in statistics. As a result, a few significant concepts of this module have been covered here.