# Mean Deviation About Mean and Median

> Mean deviation in JEE Mathematics: minimum about the median, MD of first n naturals (n^2-1)/4n, coefficient of MD and comparison with standard deviation.

- Canonical URL: https://prepelephant.com/topics/jee/mathematics/mean-deviation
- Exam / course: JEE · Subject: Mathematics
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- First published: 2026-10-02
- Last updated: 2026-10-02
- How to cite: "Mean Deviation About Mean and Median", PrepElephant, https://prepelephant.com/topics/jee/mathematics/mean-deviation

## Direct answer

Mean deviation is the average of absolute deviations from a chosen central value: MD about a equals (1/n) Σ |xi - a| for raw data, or Σ fi |xi - a| / Σ fi for a frequency distribution. Its defining property is that the median minimises it — MD about the median is the smallest such average over all choices of a, a fact that distinguishes it from the standard deviation, which the mean minimises in the squared sense. Two results JEE expects on tap: the MD of the first n natural numbers about their mean is (n^2 - 1)/4n, and for any data MD ≤ SD, with the normal distribution displaying the tidy ratios QD = 2σ/3, MD = 4σ/5, SD = σ. The coefficient of mean deviation, MD divided by the same central value, renders the measure unit-free for comparisons.

## What you must remember

- **Definition:** MD about a = Σ|xi - a|/n (or with frequencies, Σ fi |xi - a|/Σfi); absolute values are non-negotiable — signs must not cancel.
- **Median minimality:** the MD about the median is the least possible; about the mean it is larger (or equal for symmetric data), a one-mark fact asked repeatedly.
- **First n naturals:** MD about the mean = (n^2 - 1)/4n; for n = 5 this is 24/20 = 1.2, verifiable by hand on {1, 2, 3, 4, 5}.
- **Relation to SD:** for any data, MD ≤ SD; equality-type questions exploit data concentrated at two values where the gap narrows.
- **Normal-curve ratios:** for a normal distribution, QD : MD : SD = 10 : 12 : 15, i.e., QD = 2σ/3 and MD = 4σ/5 ≈ 0.7979σ.
- **Coefficient of MD:** MD about mean ÷ mean, or MD about median ÷ median — a unit-free relative measure for comparing scatter across data sets.
- **Grouped data shortcut:** for a modest frequency table, take the median class, estimate the median, then compute Σ fi |xi - median| directly rather than building a full deviation column first.

## Watching the median win

Take the data 3, 5, 7, 15. The median of the four values is (5 + 7)/2 = 6, and the mean is 30/4 = 7.5. MD about the median: |3 - 6| + |5 - 6| + |7 - 6| + |15 - 6| = 3 + 1 + 1 + 9 = 14, so MD = 14/4 = 3.5. MD about the mean: |3 - 7.5| + |5 - 7.5| + |7 - 7.5| + |15 - 7.5| = 4.5 + 2.5 + 0.5 + 7.5 = 15, so MD = 15/4 = 3.75. The median's total beats the mean's by exactly one unit here — not an accident but the visible edge of the minimisation theorem: each step of a away from the median flips the sign-contribution of the observations on one side. The outlier 15 drags the mean upward, and the mean then pays for its own displacement; the median, resistant to the outlier, stays cheaper. This is the standard exam demonstration — a small data set, both central values computed, and the comparison asked as "verify that MD about the median is least".

## How the exam frames it

JEE Main asks for the MD of a small data set about the mean or median, the (n^2 - 1)/4n result for naturals, or the coefficient of MD — three to four marks of careful arithmetic, usually as a numerical value. Frequency-table versions with ten to twelve classes appear too, and the combined-formula trap lands there: students average deviations of class marks without weighting by frequencies. Advanced seldom features MD alone; it embeds the concept in comparisons (which measure of dispersion is least affected by extreme values — answer MD; which is least — range; which enters the normal machinery — SD) and in the normal ratios. The predictable losses: dropping absolute values midway and summing to zero; computing MD about the mean when the question asks about the median on symmetric-looking data; and misstating the minimiser (the mean minimises the sum of squared deviations; the median minimises the sum of absolute ones — swapping the two is the classic single-correct trap). Statistics sits explicitly in the Main syllabus; MD itself is the least examined of the dispersion measures, which is exactly why it is safe marks when it appears.

## Frequently asked questions

### About which central value is the mean deviation minimum?

The median — Σ|xi - a| is smallest when a is the median, making MD about the median the least of all such averages.

### What is the mean deviation of the first n natural numbers about their mean?

(n^2 - 1)/4n; for the data 1 to 5 it equals 24/20 = 1.2.

### How is the coefficient of mean deviation defined?

As MD divided by the central value used — MD about mean over the mean, or MD about median over the median — producing a unit-free measure of relative scatter.

### How does mean deviation compare with standard deviation?

MD is always less than or equal to SD for the same data; squaring punishes large deviations more, so SD exceeds MD whenever spread is uneven.

### Which measure of dispersion is least affected by extreme values?

The mean deviation (and the quartile deviation), because absolute — not squared — deviations, and medians rather than means, keep outliers from dominating.