Mathematics and Statistics · Ch 11 — Measures of Dispersion
Meaning of Dispersion; Absolute and Relative Measures
Meaning of Dispersion; Absolute and Relative Measures
An average (mean, median or mode) tells us the central value of a set of data, but two very different sets can share the same average. Consider the daily earnings (in ₹) of two small shops over five days:
| Day 1 | Day 2 | Day 3 | Day 4 | Day 5 | Mean | |
|---|---|---|---|---|---|---|
| Shop A | 480 | 500 | 500 | 500 | 520 | 500 |
| Shop B | 100 | 300 | 500 | 700 | 900 | 500 |
Both shops average ₹500 a day, yet Shop A's takings barely move while Shop B's swing wildly. The average alone hides this. Dispersion (also called variation or scatter) measures how far the individual values are spread out from one another and from their central value — it is what distinguishes Shop A from Shop B.
Why dispersion matters
A small dispersion means the data is consistent, uniform and reliable (Shop A); a large dispersion means the data is erratic and less dependable (Shop B). In commerce and economics this decides which investment is safer, which supplier is steadier, and which batsman is more reliable — questions an average can never answer on its own.
This chapter studies four common measures of dispersion — Range, Quartile Deviation, Mean Deviation and Standard Deviation — and the Coefficient of Variation built from the last. These are the same measures of statistical dispersion set out in the standard national mathematics-and-statistics curriculum that the Maharashtra Std XI (FYJC) commerce course draws upon.
Absolute vs Relative measures
- An absolute measure of dispersion is expressed in the same units as the data (rupees, kilograms, marks). Range, Quartile Deviation, Mean Deviation and Standard Deviation are all absolute measures.
- A relative measure is a pure number (or percentage) with no units — it is an absolute measure divided by a suitable average. Because it is unit-free, a relative measure is what we use to compare two data sets measured in different units or with very different averages. The coefficients (coefficient of range, coefficient of quartile deviation, coefficient of mean deviation) and the coefficient of variation are the relative measures.
The reason both kinds are needed: to compare the spread of heights (in cm) with the spread of weights (in kg), an absolute measure is useless — 5 cm and 5 kg cannot be ranked — but their relative measures are both plain numbers and can be compared directly.
The extent to which individual values of a data set are spread out from their central value; small dispersion means consistent data, large dispersion means erratic data.
A measure of spread expressed in the same units as the original data (e.g. Range, Quartile Deviation, Mean Deviation, Standard Deviation).
A unit-free measure (a ratio or percentage) formed by dividing an absolute measure by an average; used to compare the variability of two different data sets (e.g. coefficient of range, coefficient of variation).