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Fundamentals of Business Mathematics and Statistics · Measures of Central Tendency and Dispersion

Range, Quartile Deviation and Mean Deviation Explained

Updated 10 October 2026 · Fact-checked

Measures of dispersion show how spread out data is. Range is Largest − Smallest. Quartile deviation is (Q3 − Q1) ÷ 2. Mean deviation is the average of absolute deviations from the mean or median. Dividing each by the matching average gives a unit-free coefficient for comparing different series.

Understand Range, Quartile Deviation and Mean Deviation

An average tells you the centre of a data set. It does not tell you how far the values are spread around that centre. Two shops can both have average daily sales of ₹10,000, but one may sell steadily while the other swings wildly. Dispersion measures this spread.

Measures of dispersion are of two kinds. Absolute measures are in the same unit as the data, such as rupees or kilograms. Relative measures (coefficients) are pure numbers, usually ratios, so you can compare series with different units or very different sizes.

Range is the simplest measure. It is the gap between the largest and smallest values. It uses only two values, so one extreme value can change it a lot. Quartile deviation (semi-inter-quartile range) ignores the extreme quarters of the data and uses only Q1 and Q3, so it is not affected by outliers.

Mean deviation uses every value. You find how far each value is from an average (mean or median), ignore the sign, and take the average of these distances. Mean deviation is smallest when taken about the median, but exam questions will tell you which average to use.

Key formulas to remember

Range
Range = L − S
L = largest value, S = smallest value. For grouped data, use upper limit of the highest class minus lower limit of the lowest class.
Coefficient of range
(L − S) ÷ (L + S)
A pure number. Always uses L + S in the denominator, not L − S.
Quartile deviation
QD = (Q3 − Q1) ÷ 2
Also called semi-inter-quartile range.
Coefficient of quartile deviation
(Q3 − Q1) ÷ (Q3 + Q1)
Denominator is Q3 + Q1, not divided by 2 separately; the 2s cancel.
Mean deviation (ungrouped)
MD = Σ|x − A| ÷ n
A is the mean or median, as asked. Take absolute values.
Mean deviation (frequency data)
MD = Σf|x − A| ÷ Σf
x is the value or class mid-point.
Coefficient of mean deviation
MD ÷ A
Divide by the same average used to find MD (mean or median).

How to solve Range, Quartile Deviation and Mean Deviation questions

Use this method for any question on range, quartile deviation or mean deviation.

  1. 1Read what is asked: absolute measure or coefficient, and which average (mean or median) for mean deviation.
  2. 2Arrange the data in ascending order if it is ungrouped.
  3. 3For range, pick L and S. For quartile deviation, find Q1 and Q3 using the position formulas.
  4. 4For mean deviation, find the required average A first.
  5. 5Write |x − A| for every item. Ignore signs. For frequency data, multiply each by f.
  6. 6Add the deviations and divide by n (or Σf) to get MD.
  7. 7For a coefficient, divide by the right base: L + S, Q3 + Q1, or the average A.
  8. 8Check the answer is positive and that the coefficient has no unit.

Quickest way: Shortcut for MCQs

When to use it: Use when the question gives a small data set or directly gives L, S, Q1 and Q3 and the options are far apart.

  1. For range and quartile questions, plug straight into the formula. Do not build a table.
  2. For mean deviation about the median in a small odd-sized data set, the median is the middle value, so its own deviation is zero.
  3. Add deviations mentally and divide by n. Check the result against the options.
  4. If the mean is a whole number, use it directly; otherwise check whether the median is simpler.
  5. Eliminate options that are negative. For positive data, the coefficients of range and quartile deviation lie between 0 and 1, so eliminate options outside this range.

Common mistakes in Range, Quartile Deviation and Mean Deviation

  • Using L − S in the denominator of the coefficient of range.

    Students mix up the range with its coefficient.

    Fix: Remember: range uses minus, coefficient divides by plus. Coefficient = (L − S) ÷ (L + S).

  • Forgetting to divide by 2 in quartile deviation.

    Students stop at Q3 − Q1, which is the inter-quartile range.

    Fix: QD = (Q3 − Q1) ÷ 2. The coefficient has no extra 2 because it cancels.

  • Keeping negative signs in mean deviation.

    Deviations from the average are naturally positive and negative and they sum to zero around the mean.

    Fix: Take the absolute value of every deviation before adding.

  • Using the mean when the question says median, or the reverse.

    Students default to the mean out of habit.

    Fix: Underline the word mean or median in the question and use that average for both deviations and the coefficient.

  • Forgetting the frequency in grouped data.

    Students copy the ungrouped method.

    Fix: Use Σf|x − A| ÷ Σf and use class mid-points for x.

  • Wrong quartile position for ungrouped data.

    Students mix the position formulas of Q1 and Q3.

    Fix: Q1 is the ((n + 1) ÷ 4)th item and Q3 is the (3(n + 1) ÷ 4)th item in ordered data.

Worked examples

Example 1

The marks of 7 students are 12, 18, 20, 25, 30, 35, 40. Find the quartile deviation and its coefficient.

Show the solution
  1. Data is already ordered and n = 7.
  2. Q1 is the (7 + 1) ÷ 4 = 2nd item = 18.
  3. Q3 is the 3 × (7 + 1) ÷ 4 = 6th item = 35.
  4. QD = (35 − 18) ÷ 2 = 17 ÷ 2 = 8.5.
  5. Coefficient = (35 − 18) ÷ (35 + 18) = 17 ÷ 53 ≈ 0.32.

Answer: Quartile deviation = 8.5; coefficient ≈ 0.32.

Example 2

Find the mean deviation about the median and its coefficient for the values 4, 6, 8, 10, 12.

Show the solution
  1. Data is ordered and n = 5. The median is the 3rd item = 8.
  2. Deviations |x − 8| are 4, 2, 0, 2, 4.
  3. Sum of deviations = 12.
  4. MD = 12 ÷ 5 = 2.4.
  5. Coefficient of MD = 2.4 ÷ 8 = 0.3.

Answer: Mean deviation about the median = 2.4; coefficient = 0.3.

Exam tips

  • Questions often give L and S and ask only for the coefficient of range. Use the plus sign in the denominator.
  • Check whether the question asks for the absolute measure or the coefficient before you start.
  • For mean deviation, note whether it is about the mean or median. Options are often built from the wrong choice.
  • Remember that the range is affected by extreme values, and quartile deviation is not. Theory MCQs test this.
  • There is no negative marking, so attempt every question after eliminating clearly wrong options.

Practice questions from Measures of Central Tendency and Dispersion

Range, Quartile Deviation and Mean Deviation in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Range, Quartile Deviation and Mean Deviation: frequently asked questions

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

An absolute measure is in the unit of the data, such as rupees. A relative measure is a ratio with no unit. Use relative measures to compare series with different units or sizes.

What is the formula for the coefficient of quartile deviation?

It is (Q3 − Q1) ÷ (Q3 + Q1). It is a pure number used to compare the spread of the middle half of different data sets.

How do I calculate mean deviation about the median?

Find the median, take the absolute difference of each value from it, then add them and divide by n. For frequency data, multiply each deviation by its frequency and divide by Σf.

Why is quartile deviation better than range?

Range depends only on the two extreme values, so one outlier distorts it. Quartile deviation uses Q1 and Q3, so it is not affected by extreme values, though it ignores the spread of the outer quarters.