Quantitative Aptitude · Measures of Central Tendency and Dispersion

Range, Quartile Deviation and Mean Deviation

Updated 1 October 2026

Range, quartile deviation and mean deviation measure how spread out data is. Range uses the largest and smallest values, quartile deviation uses Q3 and Q1, and mean deviation averages the absolute deviations from the mean or median. Divide each absolute measure by the matching average or total to get a unit-free coefficient.

Understand Range, Quartile Deviation and Mean Deviation

An average tells you where data is centred. It does not tell you how scattered the data is. Two classes can both average 50 marks, but one has everyone near 50 and the other has scores from 5 to 95. Dispersion measures this scatter.

Dispersion measures come in two kinds. Absolute measures carry the unit of the data (rupees, kg, marks). Relative measures, called coefficients, are pure numbers. Use a coefficient when you compare two series with different units or very different averages.

The range is the simplest measure: largest value minus smallest value. It uses only two values, so one extreme item changes it a lot. The quartile deviation (semi-interquartile range) uses the middle half of the data. It is half the gap between the third quartile Q3 and the first quartile Q1. Extreme values do not affect it.

The mean deviation uses every item. You find how far each value lies from the mean (or median), ignore the sign, and average these distances. Ignoring the sign matters, because plain deviations from the mean add up to zero. The mean deviation is smallest when taken about the median.

The coefficient of each measure divides the absolute measure by a matching base. For range and quartile deviation, the base is the sum of the two values used. For mean deviation, the base is the average (mean or median) you measured from.

Key formulas to remember

Range: Range = L − S; L = largest value, S = smallest value. For grouped data, L is the upper limit of the highest class and S is the lower limit of the lowest class.
Coefficient of range: (L − S) ÷ (L + S); A pure number. Use the same L and S as in the range.
Quartile deviation (QD): QD = (Q3 − Q1) ÷ 2; Also called semi-interquartile range.
Coefficient of quartile deviation: (Q3 − Q1) ÷ (Q3 + Q1); Do not divide by 2 here. The 2s cancel.
Mean deviation, ungrouped: MD about A = Σ|x − A| ÷ n; A is the mean or the median. |x − A| is the absolute deviation.
Mean deviation, grouped: MD about A = Σf|x − A| ÷ N; x is the class mid-point, f the frequency, N = Σf.
Coefficient of mean deviation: MD about A ÷ A; Divide by the same average (mean or median) that you measured from.
Minimum property: Σ|x − median| ≤ Σ|x − A| for any A; So mean deviation is least when measured about the median.

How to solve Range, Quartile Deviation and Mean Deviation questions

Use this order for any question on these three measures. It keeps the work short and avoids sign and base errors.

1Read what is asked: absolute measure or coefficient, and which average (mean or median) for mean deviation.
2Identify the data type: raw values, discrete frequency, or continuous classes. Convert classes to mid-points where needed.
3For range, pick L and S. For grouped data use class limits, not mid-points.
4For quartile deviation, find Q1 and Q3 using the quartile position and interpolation. Then apply (Q3 − Q1) ÷ 2.
5For mean deviation, find the mean or median first. Then write |x − A| for every item and ignore the signs.
6Multiply each |x − A| by f for grouped data, add them up, and divide by N (or n).
7For a coefficient, divide by the correct base: (L + S), (Q3 + Q1) or the average used.
8Check the unit and reasonableness. MD ≤ range and QD ≤ range ÷ 2 are weak upper bounds only. They rule out answers that are too large, but they do not confirm that an answer is right.

Quickest way: Option elimination and shortcuts for dispersion MCQs

When to use it: Use this in the objective paper when time is short. Most of these questions are one-line formula applications.

Check the type of answer first. A coefficient is a pure number with no unit, so eliminate options carrying units. For positive data, the coefficients of range and QD lie between 0 and 1 (they reach 1 only if S or Q1 is 0).
If the question gives Q1 and Q3, do the subtraction and addition mentally. Remember QD has a ÷ 2 but its coefficient does not.
For mean deviation about the mean, compute the mean first. Then add the absolute gaps. Items equal to the mean contribute zero, so skip them.
If the data is symmetric about the mean (such as 4, 6, 8, 10, 12), pair equal gaps and double them to save time.
For grouped data with equal class widths, you can use step deviations (x − A) ÷ h with a convenient assumed value A, but only to find the mean quickly. Then work out Σf|x − mean| using the true mean. If you scale the gaps by h to keep numbers small, measure them from the true mean and multiply by h at the end. Never take the absolute deviations about the assumed value A.
Sense check the answer: MD is never more than the range, so drop options that break this. It is only a weak upper bound, so it will not pick the right option by itself.
Skip a long grouped mean deviation question first time round if it needs many rows. Return after the quick ones, because a wrong answer costs 0.25 marks.

Common mistakes in Range, Quartile Deviation and Mean Deviation

Using mid-points instead of class limits to find the range of grouped data.
Mid-points are used for the mean, so students use them everywhere.
Fix: Range uses the upper limit of the highest class and the lower limit of the lowest class. Mid-points are only for mean deviation calculations.
Dividing by 2 in the coefficient of quartile deviation.
Students mix the formula for QD with the formula for its coefficient.
Fix: Coefficient = (Q3 − Q1) ÷ (Q3 + Q1). The 2 appears in both parts and cancels.
Keeping negative signs in |x − A| and getting a zero or tiny mean deviation.
Students drop the modulus and add the signed deviations. Positive and negative gaps cancel, and deviations from the mean add up to exactly zero.
Fix: Write every deviation as a positive number first. Then multiply by f for grouped data and add.
Dividing the coefficient of mean deviation by the wrong average.
Students always use the mean, even when MD was taken about the median.
Fix: Divide by the same average used for the deviations. MD about median ÷ median; MD about mean ÷ mean.
Forgetting to multiply by frequency, or dividing by the number of classes instead of N.
The grouped table looks like a short list, so it is treated as ungrouped.
Fix: Use Σf|x − A| ÷ N, where N = Σf. Always total the frequency column first.
Treating a large coefficient as a bigger spread without checking the base.
Students compare absolute measures of series with different units or averages.
Fix: Compare series only through coefficients when units or averages differ. A lower coefficient means less relative dispersion.

Worked examples

Example 1

The monthly sales (in ₹ thousand) of a shop in five months are 12, 18, 25, 30 and 45. The coefficient of range is: (A) 11/19 (B) 11/4 (C) 19/33 (D) 1/2

Show the solution

Largest value L = 45 and smallest value S = 12.
Range = L − S = 45 − 12 = 33.
L + S = 45 + 12 = 57.
Coefficient of range = 33 ÷ 57 = 11/19.
Check the options: 11/4 is above 1, so it cannot be a coefficient for positive data. 19/33 is the inverse. 1/2 does not equal 11/19.

Answer: (A) 11/19

Example 2

For a distribution, Q1 = 24 and Q3 = 56. The coefficient of quartile deviation is: (A) 0.4 (B) 0.2 (C) 0.67 (D) 2.5

Show the solution

Q3 − Q1 = 56 − 24 = 32.
Q3 + Q1 = 56 + 24 = 80.
Coefficient of QD = 32 ÷ 80 = 0.4.
For reference, QD = 32 ÷ 2 = 16, which is an absolute measure and is not an option.
0.2 comes from wrongly dividing by 160. 2.5 is the inverse.

Answer: (A) 0.4

Example 3

Class intervals 0–10, 10–20 and 20–30 have frequencies 2, 5 and 3 respectively. The mean deviation about the mean is: (A) 5.4 (B) 5.0 (C) 6.0 (D) 4.8

Show the solution

Mid-points x are 5, 15 and 25. N = 2 + 5 + 3 = 10.
Σfx = 2×5 + 5×15 + 3×25 = 10 + 75 + 75 = 160.
Mean = 160 ÷ 10 = 16.
Absolute deviations |x − 16| are 11, 1 and 9.
f|x − 16| = 2×11, 5×1, 3×9 = 22, 5, 27. Their sum is 54.
MD about mean = 54 ÷ 10 = 5.4.

Answer: (A) 5.4

Exam tips

Read whether the question wants the absolute measure or the coefficient. Many wrong options are the other one.
Memorise the pairing: range and QD coefficients use (sum of the two values) as the base; MD coefficient uses the average used.
In grouped mean deviation questions, build the table in the order x, f, fx, |x − mean|, f|x − mean|. Check Σf first.
Use sense checks as weak upper bounds only: MD ≤ range and QD ≤ range ÷ 2. They help you remove options that are too large, but they will not pick the right answer on their own.
Theory-style MCQs often ask which measure is least affected by extreme values (quartile deviation) or about which average MD is minimum (median).

Practice questions from Measures of Central Tendency and Dispersion

Range, Quartile Deviation and Mean Deviation: frequently asked questions

What is the difference between absolute and relative measures of dispersion?

An absolute measure has the unit of the data, such as ₹ or kg. A relative measure (coefficient) is a pure number found by dividing the absolute measure by an average or a sum of values. Use coefficients to compare series with different units or averages.

How do I calculate mean deviation for grouped data?

Find the mean or median using the class mid-points. Then find |x − A| for each mid-point and multiply it by its frequency. Add these products and divide by N, the total frequency.

What is the formula for the coefficient of quartile deviation?

It is (Q3 − Q1) ÷ (Q3 + Q1). It is not the quartile deviation divided by something else, so do not divide by 2 in the numerator separately.

Why is mean deviation least about the median?

The sum of absolute deviations is minimised at the median. So mean deviation about the median is never more than mean deviation about the mean or any other value.

Which dispersion measure is affected most by extreme values?

Range, because it depends only on the largest and smallest values. Quartile deviation ignores the lowest quarter and highest quarter of the data, so it is much more stable.