EconomicsISC Class 11 · Measures of Dispersion

Range, Quartile Deviation and Mean Deviation (ISC Class 11 Economics)

Updated 17 October 2026 · Fact-checked

Dispersion measures how widely the values of a series are scattered around an average. Range = Largest - Smallest, with coefficient of range = (L - S)/(L + S). Quartile deviation QD = (Q3 - Q1)/2, with coefficient (Q3 - Q1)/(Q3 + Q1). Mean deviation is the average of the absolute deviations from the mean (or median), MD = Sum |d| / n, with coefficient = MD / mean.

Understand Dispersion: Range, Quartile Deviation and Mean Deviation

Meaning and purpose. An average alone does not show how items are spread. Dispersion tells the degree of variation, the reliability of the average and helps compare the consistency of series. *Absolute measures* have the units of the data; *relative measures (coefficients)* are pure numbers used to compare series.

Range. Range = L - S (largest minus smallest). Coefficient of range = (L - S)/(L + S). *For grouped data:* L = upper limit of the highest class; S = lower limit of the lowest class. Simple but based on two values and affected by extremes.

Quartile deviation (semi-interquartile range).

*Individual series:* Q1 = ((n + 1)/4)th item, Q3 = (3(n + 1)/4)th item. QD = (Q3 - Q1)/2. Coefficient of QD = (Q3 - Q1)/(Q3 + Q1).
*Continuous series:* Q1 = L + [(N/4 - cf)/f] x h; Q3 = L + [(3N/4 - cf)/f] x h, using the quartile classes.

It ignores the extreme 25% at each end.

Mean deviation. Average of the absolute deviations from the mean (or median): MD = Sum |X - mean| / n (individual) or Sum f|X - mean| / N (grouped, using mid-points). Coefficient of MD = MD / mean (or / median).

Which to use? Range for a quick idea; QD when there are extreme values or open-ended classes; MD when all items matter but the algebra is simple; SD (next topic) for advanced use.

Properties. The absolute measures (range, QD, MD) are unaffected by a change of origin and change in proportion to a change of scale. The coefficients are pure numbers; they are unchanged by a change of scale but do change with a change of origin.

Dispersion: Range, Quartile Deviation and Mean Deviation: key facts

Range: Range = L - S | Coefficient = (L - S) / (L + S); Simplest.
Quartile deviation: QD = (Q3 - Q1) / 2 | Coefficient = (Q3 - Q1) / (Q3 + Q1); Uses the middle half.
Mean deviation: MD = Sum |X - mean| / n | Coefficient = MD / mean; Absolute deviations.

How to answer Dispersion: Range, Quartile Deviation and Mean Deviation questions

Arrange the data, locate the needed items, and compute step by step.

1Arrange the data in order (or compute cumulative frequencies).
2Find L and S, or Q1 and Q3, or the mean and deviations.
3Substitute in the formula.
4Find the coefficient to compare series.

Dispersion: Range, Quartile Deviation and Mean Deviation in real life

The idea, met in an everyday situation. Answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Consistency

Suppose two bowlers have the same average of 3 wickets a match, but one takes 2, 3, 4 wickets and the other 0, 3, 6.

Which is more consistent, using the range?

Try it first, then show the solutionSolution

First range = 4 - 2 = 2; second range = 6 - 0 = 6.
The smaller range means more consistent.

Answer: The first bowler is more consistent (range 2 against 6).

Memory hook: Less spread, more consistent.

Tricks to remember Dispersion: Range, Quartile Deviation and Mean Deviation

Short memory aids. Say each one aloud once, then use it on a question the same day.

L minus S
Range uses largest and smallest only.
Use it when: Range.
Middle half
QD uses Q3 and Q1: half their difference.
Use it when: QD.
Absolute deviations
Ignore the signs in MD.
Use it when: MD.

Dispersion: Range, Quartile Deviation and Mean Deviation: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Range · 1 mark · MCQBoard level

The range of 12, 25, 18, 40, 33 is:
(a) 28 (b) 40 (c) 12 (d) 52

Try it first, then show the solutionSolution

40 - 12 = 28.

Answer: (a) 28

Example 2: Range and coefficient · 3 marks · Short answerBoard level

Find the range and the coefficient of range of 12, 25, 18, 40, 33. For classes 0-10 to 40-50, find the range. (3 marks)

Try it first, then show the solutionSolution

Range = 40 - 12 = 28; coefficient = 28 / 52 = 0.538.
For the classes: range = 50 - 0 = 50.

Answer: 28 and 0.538; the grouped range is 50.

Example 3: QD of an individual series · 4 marks · Short answerHarder board level

Find the quartile deviation and its coefficient for 5, 7, 9, 12, 14, 18, 21, 25, 30. (4 marks)

Try it first, then show the solutionSolution

n = 9; Q1 = 2.5th item = (7 + 9)/2 = 8.
Q3 = 7.5th item = (21 + 25)/2 = 23.
QD = (23 - 8)/2 = 7.5.
Coefficient = (23 - 8)/(23 + 8) = 15/31 = 0.484.

Answer: QD 7.5; coefficient 0.484.

Example 4: QD of a continuous series · 6 marks · Long answerHarder board level

Classes 0-10, 10-20, 20-30, 30-40, 40-50 with frequencies 4, 8, 15, 9, 4. Find Q1, Q3, the QD and its coefficient. (6 marks)

Try it first, then show the solutionSolution

cf: 4, 12, 27, 36, 40; N = 40.
N/4 = 10: Q1 class 10-20: Q1 = 10 + (10 - 4)/8 x 10 = 17.5.
3N/4 = 30: Q3 class 30-40: Q3 = 30 + (30 - 27)/9 x 10 = 33.33.
QD = (33.33 - 17.5)/2 = 7.92.
Coefficient = 15.83/50.83 = 0.311.

Answer: Q1 = 17.5, Q3 = 33.33, QD = 7.92, coefficient 0.311.

Example 5: Mean deviation and comparison · 5 marks · Long answerHOTS / competency

Find the mean deviation about the mean and its coefficient for 4, 8, 10, 12, 16. A second series 20, 22, 25, 28, 30 has a mean deviation of 3.2 about its mean 25; compare the consistency of the two series. (5 marks)

Try it first, then show the solutionSolution

Mean = 50/5 = 10; |d| = 6, 2, 0, 2, 6; Sum = 16; MD = 3.2.
Coefficient = 3.2 / 10 = 0.32.
Second series: 3.2 / 25 = 0.128.
The first series has a coefficient 0.32, the second 0.128 although both MDs are 3.2: the second is more consistent in relative terms.

Answer: MD 3.2, coefficient 0.32; the second series (0.128) is more consistent.

Where students lose marks in Dispersion: Range, Quartile Deviation and Mean Deviation

The slips examiners see most often, and what to write instead.

Mistake: Including signs in mean deviation
Forgetting absolute values.
Wrong working
Sum d = 0
Right working
Sum |d| = 16
Do this: Ignore signs.
Usually costs: 2 marks.
Mistake: Q1 position as n/4
Formula slip.
Wrong working
9/4
Right working
(9 + 1)/4
Do this: Use (n + 1)/4.
Usually costs: 2 marks.
Mistake: Coefficient of range with (L - S)/2
Wrong denominator.
Wrong working
28/2
Right working
28/52
Do this: Use L + S.
Usually costs: 1 mark.

Exam tips

Show the ordered list and the positions of Q1 and Q3.
Always give the coefficient when comparing series.
State the base: mean or median for MD.

Practice questions on Dispersion: Range, Quartile Deviation and Mean Deviation

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The quartile deviation is half of the:
(a) range (b) interquartile range (c) mean deviation (d) variance
Hint
(Q3 - Q1)/2.
Answer
(b) interquartile range
Q2 · 2 marks · Short answerBoard level
Find the range and its coefficient for the daily sales (Rs): 520, 480, 610, 570, 390, 450. (2 marks)
Hint
L = 610, S = 390.
Answer
Range = 610 - 390 = 220; coefficient = 220 / 1,000 = 0.22.
Q3 · 5 marks · Long answerHarder board level
Find Q1, Q3, the quartile deviation and the coefficient for the marks 22, 28, 30, 34, 36, 40, 45, 47, 50, 55, 60. (5 marks)
Hint
n = 11.
Answer
Q1 = 3rd item = 30; Q3 = 9th item = 50; QD = (50 - 30)/2 = 10; coefficient = 20/80 = 0.25.
Q4 · 5 marks · Long answerHarder board level
Find the mean deviation about the mean of the frequency distribution x = 10, 20, 30, 40, 50 with f = 2, 4, 8, 4, 2, and its coefficient. (5 marks)
Hint
Mean = 30.
Answer
N = 20; Sum fx = 20 + 80 + 240 + 160 + 100 = 600; mean = 30. |d| = 20, 10, 0, 10, 20; f|d| = 40, 40, 0, 40, 40; Sum = 160; MD = 8; coefficient = 8 / 30 = 0.267.
Q5 · 5 marks · Long answerHOTS / competency
Two shops record weekly sales (Rs thousand): Shop P: 40, 44, 50, 56, 60 (mean 50). Shop Q: 20, 40, 50, 60, 80 (mean 50). Compute the range and mean deviation for each, and decide which shop has the steadier sales and whether range alone would mislead. (5 marks)
Hint
Both means are 50.
Answer
P: range 20; |d| = 10, 6, 0, 6, 10; MD = 32/5 = 6.4. Q: range 60; |d| = 30, 10, 0, 10, 30; MD = 80/5 = 16. Shop P has steadier sales. Here the range and MD agree; but if one outlier is present, the range may mislead, so MD gives a fuller picture.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Range?Show answer
Largest minus smallest.
2.QD?Show answer
(Q3 - Q1)/2.
3.MD is from?Show answer
Mean (or median), ignoring signs.
4.Relative measure?Show answer
Coefficient of dispersion.
5.Q1 position?Show answer
(n + 1)/4.

Dispersion: Range, Quartile Deviation and Mean Deviation: frequently asked questions

Why use coefficients?

To compare series with different units or averages.

Is QD affected by extremes?

No.

Can MD be negative?

No.

What does zero dispersion mean?

All values equal.