The marks (out of 40) obtained by 9 students of a college in an Economics test, arranged in ascending order, a

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Q. The marks (out of 40) obtained by 9 students of a college in an Economics test, arranged in ascending order, are: 12, 15, 18, 21, 24, 27, 30, 33, 36. Calculate the Quartile Deviation and its Coefficient.

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Concept understanding — Measures Of Dispersion

Measures of Dispersion

Imagine two friends, both scoring an average of 70 marks across five tests. One friend scores 68, 71, 69, 70, 72 — steady and predictable. The other scores 40, 95, 55, 80, 80 — wildly up and down. Their averages are identical, but their spread is completely different. That difference is what measures of dispersion capture.

The Core Idea

A measure of central tendency — mean, median, mode — tells you where the centre of the data lies. But it tells you nothing about how the data is scattered around that centre. Two datasets can have the same average yet be worlds apart in consistency, reliability, or risk.

Dispersion, also called variation or spread, quantifies how much the individual values differ from one another and from the central value. Low dispersion means the data points cluster tightly; high dispersion means they are spread far apart.

Why It Matters in Commerce and Humanities

In commerce, dispersion is the language of risk. A stock with high price volatility (high dispersion) is riskier than one with stable returns (low dispersion), even if their average returns are the same. A factory producing bolts with low variation in diameter is more reliable than one with high variation — quality control depends on minimising dispersion.

In humanities, dispersion helps compare consistency across groups. If you measure literacy rates across districts, a low dispersion means uniform access to education; high dispersion reveals inequality. If two teachers have the same average student performance, the one with lower dispersion is likely more effective — their students are uniformly good, not a mix of brilliant and failing.

What the NCERT Textbook Emphasises

The NCERT Class 11 Economics textbook (Statistics for Economics) introduces dispersion as the second step after central tendency. It states clearly: "Measures of dispersion are needed to supplement the information given by measures of central tendency." The textbook lists the main measures — range, quartile deviation, mean deviation, standard deviation — but for a prose subject, you need only understand their purpose, not their formulas.

Important

A measure of central tendency without a measure of dispersion is incomplete. Always ask: Average of what? And how spread out?

Key Points to Remember

Dispersion tells you about consistency, reliability, and homogeneity. Low dispersion = more uniform data; high dispersion = more varied data.
It is always a non-negative number. Zero dispersion means all values are identical — every student scored exactly 70, every bolt is exactly 5 cm.
Different measures capture different aspects of spread. Some are based on extremes (range), some on middle values (quartile deviation), some on every value (mean deviation, standard deviation).
Dispersion is relative. A spread of 10 units means something different when the average is 100 versus when the average is 10. That is why relative measures (coefficient of variation) exist — but again, you only need the concept.

A Common Misunderstanding

Students often think that a high average is always good. But a high average with high dispersion means some values are very high and some very low — the average may be misleading. A low average with low dispersion might actually be more desirable in many contexts (e.g., consistent but modest profits versus erratic high profits with occasional losses).

Note

Dispersion does not tell you why the data varies — only how much it varies. Explaining the causes is a separate step, often involving other statistical tools.

In Summary

Measures of dispersion answer the question: How far apart are the data points? They are the necessary companion to averages, giving you the full picture of any dataset. Without them, you are only seeing half the story.

"Measures of Dispersion: Definition, Formula & Real-World Examples" is a heavily searched topic for the NCERT Class 11 Mathematics chapter on Statistics (and the parallel Statistics for Economics syllabus), commonly paired with searches like "range mean deviation standard deviation important questions class 11." This topic is tested regularly in CBSE Class 11 exams and also appears in competitive exams like CUET that draw on the NCERT statistics syllabus.

Step 1 — Locate Q₁. With $n = 9$ , $Q_1$ lies at the $\frac{n+1}{4} = \frac{10}{4} = 2.5^{th}$ item, i.e., halfway between the 2nd item (15) and the 3rd item (18):

Q_1 = 15 + 0.5(18-15) = 15 + 1.5 = 16.5

Step 2 — Locate Q₃. $Q_3$ lies at the $\frac{3(n+1)}{4} = \frac{30}{4} = 7.5^{th}$ item, i.e., halfway between the 7th item (30) and the 8th item (33):

Q_3 = 30 + 0.5(33-30) = 30 + 1.5 = 31.5

Step 3 — Quartile Deviation.

QD = \frac{Q_3-Q_1}{2} = \frac{31.5-16.5}{2} = \frac{15}{2} = 7.5

Step 4 — Coefficient of QD.

\text{Coefficient of QD} = \frac{Q_3-Q_1}{Q_3+Q_1} = \frac{15}{48} = 0.3125

✓Final answer

QD = 7.5 marks; Coefficient of QD \approx 0.3125.

Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

CBSE 2025Set MARCH1 markMCQ
$Q_1=30$
(a) $10$
(b) $20$
(c) $0.25$
(d) $40$
›Reveal solutionSolution

Apply the coefficient of quartile deviation $\dfrac{Q_3-Q_1}{Q_3+Q_1}=0.25$ .

The coefficient of quartile deviation is a relative (unit-free) measure of dispersion defined by

$\text{Coefficient of Q.D.}=\frac{Q_3-Q_1}{Q_3+Q_1}.$

Substituting $Q_1=30$ and $Q_3=50$ :

$=\frac{50-30}{50+30}=\frac{20}{80}=0.25.$

(Note: the quartile deviation itself is $\dfrac{Q_3-Q_1}{2}=10$ , but the question asks for the coefficient.) This is from the measures-of-dispersion section of the TN HSC Class-11 Business Statistics syllabus.

✓Final answer
$0.25$
CBSE 2022Set MARCH1 markMCQ
$= 45$
(a) $0.0056$
(b) $11.25$
(c) $45$
(d) $180$
›Reveal solutionSolution

Mean deviation $=$ coefficient of M.D. $\times$ median $=0.25\times45=11.25$ .

The coefficient of mean deviation about the median is defined as

$\text{Coefficient of M.D.} = \frac{\text{Mean deviation about median}}{\text{Median}}$

Rearranging for the mean deviation:

$\text{M.D.} = \text{Coefficient} \times \text{Median} = 0.25 \times 45 = 11.25$

✓Final answer
$11.25$
CBSE 2020Set MARCH1 markMCQ
$Q. If median = 45 and its co-efficient is 0.25, then the mean deviation about median is :$
(a) 45
(b) 11.25
(c) 180
(d) 0.0056
›Reveal solutionSolution

Since the coefficient of mean deviation about the median is $\dfrac{\text{M.D.}}{\text{median}}$ , we have M.D. $=0.25\times 45 = 11.25$ .

The coefficient of mean deviation about the median is defined as:

$\text{Coefficient of M.D.} = \frac{\text{Mean deviation about median}}{\text{Median}}.$

Given median $=45$ and coefficient $=0.25$ :

$0.25 = \frac{\text{M.D.}}{45}.$

So:

$\text{M.D.} = 0.25 \times 45 = 11.25.$

✓Final answer
$11.25$

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Q.The marks (out of 40) obtained by 9 students of a college in an Economics test, arranged in ascending order, are: 12, 15, 18, 21, 24, 27, 30, 33, 36. Calculate the Quartile Deviation and its Coefficient.

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