Measures of Dispersion

Introduction 

The measures of dispersion are ways to gauge how spread apart the values in a dataset are from the mean of that dataset. When we think of the measures of central tendency, we first tend to think of mean, median, and mode. However, these are not appropriate when we need to measure data that is fluctuating or even takes two values at once. For this, we need the measures of dispersion. There are four basic such measures- Range, Interquartile range, Standard deviation, and Coefficient of variation.

Absolute measures

A popular measure of statistical dispersion, the absolute method is used for systems measured in the same unit. For example, an absolute measure of dispersion would be expressed as standard deviations or ranges. This method explores variations through the deviation found in averages using means and averages. That includes quartile ranges, standard deviation, etc.

Absolute measures of dispersion show the assortment of data; in terms of units of observations. For example, when rainfall information is made available for different days in mm, any absolute measure of dispersion estimates the irregularity from day to day in mm.

The following are the absolute measures of dispersion:

Range: Range is one of the simplest ways to measure the spread or dispersion of a data set. It is defined as the highest value of your data minus the lowest value.

Example: 2,4,6,8,10=> Range: 10-2=8

Quartile Deviation: Let’s learn about how quartiles can be used to understand the dispersion of a set of data. Quartiles divide a data set into four equal parts and always contain an equal number of observations. There are three quartiles in a distribution: Q1, the lower quartile; Q2, the median; and Q3, the upper quartile. Absolute measures of quartile deviation represent the difference between Q1 and Q3 quartiles.

Standard Deviation: The standard deviation plays a core role in the study of data variation. It is widely used to analyse data, and because many important statistical inferences rely on the standard deviation as a starting point, it’s very crucial for any company that processes large amounts of statistics and makes decisions based solely on them. The positive square root of the variance is the standard deviation.

S.D. = √σ

Mean Deviation: The mean deviation is a measure of dispersion, and it has been most commonly used in the context of central tendency where it is typically seen as one type of alternative to the variance or standard deviation. The arithmetic means deviation is the average number of deviating observations.

Mean Deviation = [Σ |X – µ|]/N

Variance: Calculating variance involves isolating the mean of an entire set of values and then squaring it and adding these numbers together, as well as dividing that figure by the total number of values within the data set.

Variance: (σ2)=∑(X−μ)2/N

 Relative measures

The only drawback of absolute measures of dispersion is that they do not accurately display the dispersion of data in a given series when the units of measurement are different. The concept of relative measures largesses over this limitation by employing an index to calculate the number. And since it removes scaling issues, we can use a relative measure to compare dispersion for series that are in different units.

The following are the absolute measures of dispersion:

Coefficient of Range: The coefficient of Range represents the extent by which values in a data set vary. This can be calculated by subtracting the lowest value from the highest and then dividing this amount by the sum of all values in the series.

Coefficient of Range: (H-L)÷(H+L)

H= The highest value

L= The lowest value

Coefficient of quartile deviation: The following formula is used to compute the coefficient of quartile deviation:

(Q3-Q1)÷(Q3+Q1)

Q3= The third quartile

Q1= The first quartile.

The coefficient of mean deviation: The coefficient of mean deviation can be calculated for the mean, median, or mode.

• The coefficient of mean deviation about the mean is equal to the mean deviation about the mean divided by the arithmetic mean
• The coefficient of mean deviation about the median is equal to the mean deviation about the median divided by the median
• The Coefficient of mean deviation about the Mode is equal to mean deviation about mode divided by the mode

The Coefficient of Standard Deviation: The standard deviation coefficient is simply the ratio of a series’ standard deviation to its arithmetic mean.

Mathematically, formula: σ/Mean

σ: Standard Deviation of the series

Coefficient of variation: The coefficient of variation is one-hundred times the coefficient of standard deviation.

Mathematically, formula: (coefficient of standard deviation x 100)

Interquartile Range

The interquartile range is a statistical concept that represents the centre 50% of values in data distribution. It can be calculated by finding the difference between the third and first quartile, i.e., Q3 and Q1 respectively.

Mathematically, formula: (Q3-Q1)

Conclusion

In statistics, we need to measure the dispersion of a dataset.  It is vital to identify whether a dataset is concentrated at a few points or is spread out evenly through the range of values. Understanding the dispersion can help us understand how the data is distributed. Each measure of dispersion has a different way of describing the variation. The measures of dispersion used most often in statistics are the range, mean deviation, variance, standard deviation, and interquartile range.

We hope this blog post has helped you learn more about a few different ways in which terms like standardised deviation, variance, and absolute deviation are used.