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JEE Mains Maths · Statistics

Mean Deviation and Median

The mean deviation of a data set about its mean, its median or its mode, often after finding unknown values from the mean and variance.

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Why this matters

Twelve PYQs, ten of them multiple choice, and three from 2026. Six ask for the mean deviation about the mean — four after finding unknown values from the mean and variance, two for runs of consecutive integers; six measure it about the median, or in one case the mode. Two ideas cover the page.

Concept 1 of 2: Mean deviation about the mean

The mean deviation averages the distances ∣xi−xˉ∣|x_i-\bar x|. Unlike the variance, it has no shortcut through ∑x\sum x and ∑x2\sum x^2: you need the actual values. So when values are unknown, find them first from the mean and variance, then list the distances. For consecutive integers the distances are the same wherever the run starts.

Definition

  • MD(xˉ)=1n∑∣xi−xˉ∣\text{MD}(\bar x)=\frac{1}{n}\sum|x_i-\bar x|.
  • Unknown values: the mean gives their sum, the variance their sum of squares.
  • nn consecutive integers: MD=n2−14n\text{MD}=\frac{n^2-1}{4n} for odd nn, n4\frac{n}{4} for even nn.
  • Shifting the data leaves the MD unchanged; scaling by kk multiplies it by ∣k∣|k|.

Mean deviation about the mean

MD(xˉ)=1n∑i=1n∣xi−xˉ∣\text{MD}(\bar x)=\frac{1}{n}\sum_{i=1}^{n}\left|x_i-\bar x\right|

Worked example

The mean and variance of 2,4,6,a,b2,4,6,a,b are 5 and 4, with a<ba<b. Find the mean deviation about the mean.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2024 · 4 April 2024 · Q73Moderate

Example 1 · Statistics · Mean Deviation and Median

Let a,b∈Ra,b \in R. Let the mean and the variance of 6 observations −3,4,7,−6- 3,4,7, - 6, a, b be 2 and 23, respectively. The mean deviation about the mean of these 6 observations is:

The mean deviation is not the SD

1n∑∣xi−xˉ∣\frac{1}{n}\sum|x_i-\bar x| and 1n∑(xi−xˉ)2\sqrt{\frac{1}{n}\sum(x_i-\bar x)^2} are different numbers, and the MD is never larger than the SD. Do not square the distances or take a root.
Drill 5 more on mean deviation about the mean

Concept 2 of 2: Mean deviation about the median or the mode

Sort the data first. The median is the middle value, or the average of the two middle values when nn is even. Then add the distances from it. The total distance ∑∣xi−A∣\sum|x_i-A| is smallest when AA is the median, so the MD about the median is never more than the MD about the mean. The mode, the most frequent value, is used the same way.

Definition

  • Median: the middle of the sorted list; for even nn, the mean of the two middle values.
  • MD(M)=1n∑∣xi−M∣\text{MD}(M)=\frac{1}{n}\sum|x_i-M|.
  • ∑∣xi−A∣\sum|x_i-A| is least at A=MA=M.
  • Mode: the most frequent value; the same formula with the mode as centre.

Mean deviation about the median

MD(M)=1n∑i=1n∣xi−M∣\text{MD}(M)=\frac{1}{n}\sum_{i=1}^{n}\left|x_i-M\right|

Worked example

Find the mean deviation about the median of 12,3,18,7,10,512,3,18,7,10,5.
Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

JEE Mains · 2026 · 28 Jan 2026 Shift 1 · Q65Moderate

Example 2 · Statistics · Mean Deviation and Median

The mean and variance of 10 observations are 9 and 34.2 , respectively. If 8 of these observations are 2,3,5,10,11,13,15,212,3,5,10,11,13,15,21, then the mean deviation about the median of all the 10 observations is

Sort before you pick the median

The median is the middle of the sorted list, not of the list as printed. With unknown values, first decide where they fall in the order; each case can give a different answer.
Drill 5 more on mean deviation about the median or the mode

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

  • Mean deviation about the mean

    Mean deviation about the mean

    MD(xˉ)=1n∑i=1n∣xi−xˉ∣\text{MD}(\bar x)=\frac{1}{n}\sum_{i=1}^{n}\left|x_i-\bar x\right|
  • Mean deviation about the median or the mode

    Mean deviation about the median

    MD(M)=1n∑i=1n∣xi−M∣\text{MD}(M)=\frac{1}{n}\sum_{i=1}^{n}\left|x_i-M\right|

Watch out for (2)

Test yourself on Statistics

20 past JEE Mains questions from this chapter, timed at 48 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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