Measures of Dispersion, Meaning, Characteristics, Classifications, Absolute and Relative

Measures of dispersion describe the extent to which data values vary or spread around a central value (like the mean or median). While measures of central tendency provide a single summary value, dispersion tells us how consistent or variable the data is. It helps in understanding the reliability, comparability, and risk associated with data.

Dispersion is important in fields like business, economics, psychology, and engineering to analyze stability, identify outliers, and assess performance.

Suppose you have four datasets of the same size and the mean is also same, say, m. In all the cases the sum of the observations will be the same. Here, the measure of central tendency is not giving a clear and complete idea about the distribution for the four given sets.

Characteristics of Measures of Dispersion:

• Measures the Spread of Data

Dispersion quantifies how much the data points deviate from a central value like the mean or median. It shows the range or variability within a dataset, helping to understand the consistency or inconsistency in the values. A low dispersion indicates closely grouped values, while a high dispersion reflects widely scattered data. This measurement is essential for interpreting the reliability of averages and making informed statistical comparisons.

• Complements Measures of Central Tendency

While measures like mean, median, and mode summarize data with a single value, they don’t reveal how much data values vary around that point. Measures of dispersion fill this gap by providing insights into data consistency. For example, two datasets may have the same mean but very different variabilities. Dispersion allows a more comprehensive analysis by highlighting differences that central tendency measures alone may conceal.

• Sensitive to Outliers and Extreme Values

Some dispersion measures, like the range and standard deviation, are affected by extreme values or outliers in the dataset. This characteristic makes them useful for identifying unusual variations or anomalies. However, it can also distort the understanding of typical spread. Hence, in cases with skewed data, more robust measures like interquartile range or median absolute deviation are preferred, as they offer a clearer picture by minimizing the effect of outliers.

• Uses All or Part of the Data

Different dispersion measures consider different amounts of data. For instance, the range uses only the highest and lowest values, while standard deviation and variance incorporate all data points. Mean deviation and interquartile range lie somewhere in between. This characteristic determines the level of detail and accuracy each measure provides, with more comprehensive methods offering more reliable insights into the true variability in a dataset.

• Expressed in Same or Related Units

Measures like range, standard deviation, and mean deviation are expressed in the same units as the original data (e.g., rupees, kilograms, marks). This helps in meaningful interpretation and comparison. However, variance, being the square of standard deviation, is expressed in squared units, which can be difficult to interpret directly. To overcome this, the square root of variance is taken to obtain standard deviation in original units.

• Helps in Comparison of Consistency

Measures of dispersion, especially the coefficient of variation, allow comparison between datasets even when they differ in units or scale. This characteristic is vital in business, economics, and experiments, where comparing the variability between products, markets, or processes is required. A dataset with lower dispersion is considered more consistent and reliable, making these measures essential for decision-making and performance evaluation.

• Foundation for Advanced Statistical Analysis

Measures of dispersion form the basis for many complex statistical tools such as correlation, regression, hypothesis testing, and probability distributions. Understanding how data varies is critical in these techniques, as it influences confidence levels, error margins, and risk analysis. Dispersion provides the groundwork for predicting outcomes, understanding relationships among variables, and validating statistical models.

• Applicable to Both Individual and Grouped Data

Dispersion measures can be applied to raw (individual) data as well as grouped or classified data. Whether dealing with discrete scores or frequency tables, there are specific formulas and methods to compute dispersion accordingly. This adaptability makes them widely usable across various fields, including education, industry, economics, and healthcare, ensuring statistical insights remain relevant regardless of data format.

Classification of Measures of Dispersion:

Measures of dispersion are broadly classified into two categories:

1. Absolute Measures of Dispersion

These are expressed in original units of the data (e.g., kilograms, rupees, marks) and indicate the extent of spread within the dataset only. They do not allow comparison between datasets with different units.

Types of Absolute Measures:

(a) Range

Difference between the highest and lowest values.

Formula:

Range = Maximum Value − Minimum Value

(b) Quartile Deviation (Semi-Interquartile Range)

Measures spread of the middle 50% of data.

Formula:

Q.D. = (Q3 − Q1) / 2​​

(c) Mean Deviation (Average Deviation)

Average of the absolute deviations from mean/median.

Formula:

M.D. = ∑∣X − A∣ / N​

(where $A$ is the mean or median)

(d) Standard Deviation (SD)

Square root of the average of squared deviations from the mean.​​

(e) Variance

Square of the standard deviation.​

2. Relative Measures of Dispersion

These express variability as a ratio or percentage, allowing for comparison between datasets, even with different units or scales. They are unit-free.

Types of Relative Measures:

(a) Coefficient of Range

Formula:

Coefficient of Range = (Max − Min) / (Max + Min)​

(b) Coefficient of Quartile Deviation

Formula:

Coefficient of Q.D. = (Q3−Q1) / (Q3+Q1)​​

(c) Coefficient of Mean Deviation

Formula:

Coefficient of M.D. = M.D. / Mean or Median​

(d) Coefficient of Variation (CV)

Formula:

CV = (σ / Xˉ) × 100

Used to compare consistency of two or more datasets.

Absolute Dispersion

Absolute Dispersion refers to the actual spread or variability of data values in a dataset, expressed in the same units as the original data (e.g., kilograms, rupees, centimetres). It quantifies how much values deviate from a central point such as the mean, median, or mode without considering relative size or proportion.

It helps measure the extent of variation in raw terms and is useful when analyzing data within the same unit or scale.

Common Measures of Absolute Dispersion:

1. Range

Formula: Range = Maximum Value − Minimum Value 

Explanation: It shows the total spread between the smallest and largest observations. It’s the simplest measure but affected heavily by outliers.

2. Quartile Deviation (Semi-Interquartile Range)

Formula: Q.D. = (Q3 − Q1) / 2​​

Explanation: Measures dispersion of the middle 50% of data. Less affected by extreme values and suitable for skewed distributions.

Characteristics of Absolute Dispersion:

• Expressed in same unit as the original data.

• Measures actual variation, not relative to the mean.

• Useful for descriptive analysis of single datasets.

• Can’t be used to compare datasets with different units or scales.

Relative Dispersion

Relative Dispersion refers to the ratio or proportion of absolute dispersion (like standard deviation or range) relative to a central tendency such as the mean or median. Unlike absolute dispersion, which is expressed in actual units, relative dispersion is unit-free, allowing for comparison between datasets with different units, magnitudes, or scales.

It is extremely useful for evaluating consistency, reliability, and relative variability across diverse datasets.

Common Measures of Relative Dispersion:

1. Coefficient of Range

Formula: Coefficient of Range = (Maximum−Minimum) / (Maximum+Minimum) ​

Use: Helps compare range across datasets with different units.

2. Coefficient of Quartile Deviation

Formula: Coefficient of Q.D. = (Q3−Q1) / (Q3+Q1)​​

Use: Useful when median and interquartile range are more appropriate due to skewed distributions.

3. Coefficient of Mean Deviation

Formula: Coefficient of M.D. = Mean Deviation / Mean (or Median)​

Use: Gives the average absolute deviation in proportion to the central value.

4. Coefficient of Standard Deviation (also known as Coefficient of Variation)

Formula: Coefficient of SD = σ / Xˉ,  or as percentage: CV = (σ / Xˉ) × 100

Most common and powerful relative measure—used to compare variability regardless of units.

Features of Relative Dispersion:

• Unit-free: Makes cross-comparison possible

• Proportional: Shows variation relative to central value

• Normalized: Works even when datasets have different means or scales

• Useful in benchmarking, risk analysis, and decision-making

Applications of Relative Dispersion:

• Finance: Compare risk of investments using coefficient of variation.
• Education: Assess relative performance of students in different subjects.
• Healthcare: Analyze variability in treatment outcomes across hospitals.
• Manufacturing: Benchmark machine performance across units or locations.
• Economics: Study price variation between regions or time periods.

Limitations of Relative Dispersion:

• Not meaningful if the central tendency (mean) is zero — leads to division by zero or undefined results.

• Less informative if data is extremely skewed or has many outliers.

• Interpretation depends on understanding the context of variation.

Coefficient of Dispersion

Whenever we want to compare the variability of the two series which differ widely in their averages. Also, when the unit of measurement is different. We need to calculate the coefficients of dispersion along with the measure of dispersion. The coefficients of dispersion (C.D.) based on different measures of dispersion are

• Based on Range = (X _max – X _min) ⁄ (X _max + X _min).
• C.D. based on quartile deviation = (Q₃ – Q1) ⁄ (Q₃ + Q1).
• Based on mean deviation = Mean deviation/average from which it is calculated.
• For Standard deviation = S.D. ⁄ Mean

Coefficient of Variation

100 times the coefficient of dispersion based on standard deviation is the coefficient of variation (C.V.).

C.V. = 100 × (S.D. / Mean) = (σ/ȳ ) × 100.