Measures of Dispersion Measures of Dispersion

# Measures of Dispersion

## Concept of Measures of Dispersion

Dispersion means scatteredness, variability, diversity, deviations, or fluctuation. Thus, dispersion quantifies the deviations or scatter of the items from the mean. It investigates the scatteredness of various given items and their variation from the central value. The greater the variability, the greater the dispersion, and vice versa.

In general, dispersion refers to the degree to which data is scattered or spread out from a central value, such as the mean or median. In statistics, dispersion refers to measures that quantify the degree of variability or spread in a data set, such as the range, variance, or standard deviation.

Dispersion is also sometimes used to refer to the spatial distribution of objects or events, such as the dispersion of particles in a gas or a disease across a population.

In finance and investing, the dispersion can refer to the degree to which the returns on different assets or investments deviate from their average or expected value, which can affect the risk and diversification of a portfolio.

### Objectives of Dispersion

• The followings are the objectives or importance of dispersion:
• To determine the reliability of an average,
• To control variability,
• To compare the variability of the series, and
• To facilitate the use of other statistical measures.

### Characteristics of ideal measure of dispersion

The main properties of a good measure of dispersion are listed below:
1. It should be rigidly defined.
2. It should be based on all observations.
3. It should be easy to calculate.
4. It should be applicable for further algebraic treatment.
5. It should be simple to understand.
6. Extreme values (observations) shouldn't have an impact on it.

### Types of Dispersion

Measures of dispersion are of two types:

#### A. Absolute measures:

Absolute measures of dispersion are given in the same unit as the original data, such as rupees, kilograms, inches, and so on. If data sets are expressed in the same units and have the same average size, these measures can be used to compare the variations in two distributions. Absolute measurements can't be used to compare two data sets whose average sizes are very different and whose measurements are given in different units.

#### B. Relative measures:

Measures of relative dispersion should be used if the data sets are expressed in different units and the average size is very different. The absolute measure of dispersion is divided by a measure of central tendency to yield a measure of relative dispersion (appropriate average). Because it is expressed in pure numbers such as ratios or percentages, it is sometimes referred to as a coefficient of dispersion. It can be used to compare two distributions expressed in different units because it is expressed in pure numbers.

### Various methods of measures of dispersion

The following are the important methods of absolute and relative measures of dispersion.

#### A. Absolute Measures

Absolute measures of dispersion are measures of variability that are given in the same units as the data. They show how much the data values differ from a measure of central tendency. Some common absolute measures of dispersion include:
• Range: This is the difference between a data set's maximum and minimum values. It is a simple measure of variability that is easy to calculate and interpret but sensitive to outliers.
• Mean Absolute Deviation (MAD): This is the average of the absolute deviations of the data values from the mean. It's a better way to measure dispersion than range because it's less affected by extreme values.
• Variance: Variance is the average squared deviation of the data values from the mean. It is a common way to measure how different things are and gives a standard way to measure dispersion.
• Standard Deviation: This is the square root of the variance. It is a widely used measure of variability useful for describing a data set's spread.
• Mean Squared Error (MSE): This is the average of the squared differences between the data values and a predicted value. It is often used in statistical modeling and machine learning to test a model's prediction accuracy.
• Root Mean Squared Error (RMSE): This is the square root of the MSE. It measures the typical size of the prediction errors in the same units as the data.
These absolute measures of dispersion give different ways to determine how to spread out or variable a data set is. Some of these measures may be better than others, depending on the nature of the data and the research question.

#### B. Relative Measures

Relative measures of dispersion are ways to determine how different things are by comparing them to the mean or another measure of central tendency. Some common relative measures of dispersion include:
• Coefficient of Variation (CV): This is the ratio of the standard deviation to the mean, given as a percentage. It is a helpful measure for comparing the variability of two or more data sets with different units of measurement.
• Relative Range: This is the ratio of the range to the mean, shown as a percentage. It is a simple measure of variability that is easy to calculate and interpret.
• Quartile Coefficient of Dispersion (QCD): This is the percentage difference between the third and first quartiles divided by the sum of the third and first quartiles. It is useful for comparing the variability of skewed data sets.
• Interquartile Range (IQR): The difference between the third and first quartiles. It is a reliable measure of variability that is unaffected by outliers.
• Range-to-Mean Ratio (RMR): This is the ratio of the range to the mean. It is a simple measure of variability that is easy to calculate and interpret.
• Gini Coefficient: This is a measure of inequality often used to describe income distribution, but it can also be applied to other data. It ranges from 0 (perfect equality) to 1 (perfect inequality).
• Lorenz Curve: This graph of the Gini coefficient compares the cumulative distribution of a variable to the cumulative percentage of the population or sample.
These relative measures of dispersion are different ways to spread out a different data set. They may be more or less useful depending on the type of data and the research question being asked.