by Measure of dispersion indicates the scattering of data. It explains the disparity of data from one another, delivering a precise view of the distribution of data. The measure of dispersion displays and gives us an idea about the variation and central value of an individual item.
Characteristics of a Good Measure of Dispersion
- It should be easy to calculate & simple to understand.
- It should be based on all the observations of the series.
- It should be rigidly defined.
- It should not be affected by extreme values.
- It should not be unduly affected by sampling fluctuations.
- It should be capable of further mathematical treatment and statistical analysis.
Relative Measure of Dispersion
Relative Measures of Dispersion express the amount of variation or dispersion in relation to a central value. Unlike absolute measures such as Range, Quartile Deviation, Mean Deviation, and Standard Deviation, relative measures are generally expressed as ratios, coefficients, or percentages. They are especially useful for comparing the variability of two or more datasets that may have different averages or different units.
1. Coefficient of Range
Coefficient of Range is a relative measure based on the largest and smallest observations. It expresses the range in relation to the sum of the extreme values.
Formula: Coefficient of Range = (L − S) / (L + S)
Where L = Largest Value and S = Smallest Value.
Its value generally lies between 0 and 1. A higher coefficient indicates greater relative dispersion, while a lower coefficient indicates greater uniformity. It is simple to calculate but depends only on the two extreme observations.
Computation: The Coefficient of Range is calculated using the largest and smallest observations.
Interpretation: A higher coefficient indicates greater relative dispersion, while a lower coefficient indicates greater uniformity. It is simple and useful for quick comparisons.
2. Coefficient of Quartile Deviation
Coefficient of Quartile Deviation measures relative dispersion using the first quartile (Q₁) and third quartile (Q₃).
Formula: Coefficient of Q.D. = (Q₃ − Q₁) / (Q₃ + Q₁)
It is useful when extreme values may distort the results because it focuses on the middle 50% of observations. It is particularly suitable for skewed distributions and ordinal data. A higher coefficient indicates greater relative variability, while a lower coefficient indicates greater consistency.
Computation: The Coefficient of Quartile Deviation is calculated using the first quartile (Q1) and third quartile (Q3).
Interpretation: It indicates the relative spread of the middle 50% of observations. A higher value represents greater variability, whereas a lower value represents greater consistency. It is particularly useful when extreme values are present.
3. Coefficient of Mean Deviation
Coefficient of Mean Deviation expresses mean deviation relative to the central value from which the deviations are calculated.
Formula: Coefficient of M.D. = Mean Deviation / Average
The average may be the Mean, Median, or Mode, depending on the basis of calculation. This measure allows comparisons between datasets having different magnitudes. Since it is a relative measure, it provides information about dispersion in proportion to the central value rather than in absolute units.
Computation: The Coefficient of Mean Deviation is obtained by dividing Mean Deviation by the selected average from which the deviations are calculated.
Interpretation: A higher coefficient indicates greater relative variability, while a lower coefficient indicates greater consistency. It allows comparison between datasets with different average levels.
4. Coefficient of Variation
Coefficient of Variation (CV) is one of the most widely used relative measures of dispersion. It expresses Standard Deviation as a percentage of the Mean.
Formula: CV = (Standard Deviation / Mean) × 100
A lower CV indicates greater consistency and stability, whereas a higher CV indicates greater relative variability. It is particularly useful for comparing datasets with different means or units and is widely applied in business, economics, finance, investment, and research.
Computation: The Coefficient of Variation (CV) is calculated by dividing Standard Deviation by Arithmetic Mean and multiplying the result by 100.
Interpretation: A lower CV indicates greater consistency and stability, while a higher CV indicates greater relative variability. CV is widely used for comparing business performance, investment risk, production stability, and economic data.
5. Relative Standard Deviation
Relative Standard Deviation (RSD) expresses standard deviation in relation to the mean. It is essentially another way of expressing the Coefficient of Variation.
Formula: RSD = (Standard Deviation / Mean) × 100
RSD is useful when researchers want to understand the size of variation relative to the average observation. A smaller percentage indicates greater precision and consistency, while a larger percentage indicates greater variability. It is frequently used in scientific, financial, laboratory, and statistical analysis.
Computation: Each relative measure is derived from an absolute measure and its corresponding average or characteristic values. Coefficient of Range uses extreme values, Coefficient of QD uses quartiles, Coefficient of MD uses Mean Deviation and an average, while CV uses Standard Deviation and Mean.
Interpretation: Relative measures are especially useful when comparing datasets with different averages, scales, or units. In general, a lower relative measure indicates greater consistency, while a higher value indicates greater variability. Among them, CV is widely preferred for comprehensive statistical comparison.
Absolute Measure of Dispersion
Absolute Measures of Dispersion express the actual amount of variation in a dataset in the same units as the original observations, except Variance, which is expressed in squared units. They show how widely the observations are scattered around a central value. Absolute measures are useful for understanding the actual magnitude of variability within a dataset.
1. Range
Range is the simplest absolute measure of dispersion. It is the difference between the largest and smallest values in a dataset.
Formula:
Range = Largest Value − Smallest Value
A larger range indicates greater dispersion, while a smaller range indicates greater uniformity. Range is easy to calculate and understand, but it depends only on the two extreme observations and is therefore highly affected by extreme values.
2. Quartile Deviation
Quartile Deviation, also known as Semi-Interquartile Range, measures the dispersion of the middle 50% of observations.
Formula:
Q.D. = (Q₃ − Q₁) / 2
Here, Q₁ is the first quartile and Q₃ is the third quartile. It is less affected by extreme observations than range and is useful for skewed distributions. However, it ignores the lowest and highest 25% of observations.
3. Mean Deviation
Mean Deviation is the arithmetic average of the absolute deviations of observations from a central value such as the mean, median, or mode.
Formula:
M.D. = Σ|X − A| / N
Here, A represents the selected average. Mean deviation considers all observations and expresses the average distance from the central value. It is relatively easy to understand but is less suitable for advanced mathematical and statistical analysis than standard deviation.
4. Standard Deviation
Standard Deviation is one of the most important absolute measures of dispersion. It measures the spread of observations around the arithmetic mean by considering squared deviations.
Formula:
σ = √[Σ(X − X̄)² / N]
A low standard deviation indicates that observations are close to the mean, while a high standard deviation indicates greater variability. It is widely used in business, economics, finance, research, and statistical analysis.
5. Variance
Variance is the square of the standard deviation. It measures dispersion by taking the average of squared deviations from the arithmetic mean.
Formula:
σ² = Σ(X − X̄)² / N
Variance gives greater importance to larger deviations because deviations are squared. It is widely used in probability, statistical analysis, finance, research, and risk measurement. Unlike most absolute measures, variance is expressed in squared units rather than the original units.
6. Absolute Nature of Measures
Absolute measures express dispersion in terms of the original measurement units. For example, if observations represent kilograms, range, quartile deviation, mean deviation, and standard deviation are generally expressed in kilograms. This makes them useful for understanding the actual magnitude of variability. However, they may not be suitable for comparing datasets having substantially different units, scales, or averages.
7. Importance of Absolute Measures
Absolute measures are important because they show the actual degree of variation within a dataset. They help determine whether observations are closely concentrated or widely scattered. Businesses can use them to study variations in sales, costs, production, and profits. Researchers can use them to understand the distribution of observations. Therefore, absolute measures provide a direct assessment of the amount of dispersion present in data.
8. Difference from Relative Measures
Absolute measures express the actual amount of dispersion, while relative measures express dispersion in relation to a central value. For example, Standard Deviation is an absolute measure, whereas Coefficient of Variation is a relative measure. Absolute measures are useful for understanding the actual spread of one dataset, while relative measures are more appropriate when comparing the variability of different datasets.
