Q.Two salespersons, A and B, recorded the following daily sales (in ₹'000) over 5 days.
A: 20, 22, 24, 26, 28
B: 15, 20, 25, 30, 35
Find the coefficient of variation for each, and state which salesperson has more consistent sales.
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.
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).
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.
Find the mean and standard deviation for each salesperson, then compute CV=(σ/xˉ)×100 and compare — the smaller CV is more consistent.
CV=(σ/xˉ)×100; smaller CV = more consistent.
A: xˉ=24, σ=8≈2.83, CVA≈11.79%.
B: xˉ=25, σ=50≈7.07, CVB≈28.28%.
CVA≈11.79%, CVB≈28.28% — Salesperson A has more consistent sales.
Salesperson A: 20, 22, 24, 26, 28.
Mean: xˉA=520+22+24+26+28=5120=24.
Deviations from mean: −4,−2,0,2,4. Squared: 16,4,0,4,16. Sum =40.
σA=40/5=8≈2.83.
CVA=242.83×100≈11.79%.
Salesperson B: 15, 20, 25, 30, 35.
Mean: xˉB=515+20+25+30+35=5125=25.
Deviations from mean: −10,−5,0,5,10. Squared: 100,25,0,25,100. Sum =250.
σB=250/5=50≈7.07.
CVB=257.07×100≈28.28%.
Comparison. CVA(≈11.79%)<CVB(≈28.28%), so Salesperson A's sales are more consistent (less variable relative to their own average), even though B's average sales are slightly higher.
Independent check. A's raw spread (28−20=8) is much tighter around a similar-sized mean than B's raw spread (35−15=20) — the range comparison alone already points toward A being more consistent, matching the CV-based conclusion reached through the full calculation.
CVA≈11.79%; CVB≈28.28%. Since a lower CV means greater consistency, Salesperson A has more consistent sales.
Comparing the two raw standard deviations (2.83 vs 7.07) directly, without dividing by each series' own mean first, can mislead when the two means are noticeably different — always convert to CV before declaring one series 'more consistent' than another, especially when the means themselves are not close.
- CA Foundation 2025Set may-20251 markMCQQ.Which of the following is not a method of dispersion ? (A) Standard deviation (B) Mean deviation (C) Range (D) Concurrent deviation method
›Reveal solutionSolution
The concurrent deviation method measures correlation, not dispersion — so (D) is not a measure of dispersion.
Step 1 — List genuine measures of dispersion
Dispersion measures how spread out data are around a central value. Standard measures include:
- Range (A)
- Mean deviation (B)
- Standard deviation (A) — wait, that is option (A); also quartile deviation.
So options (A) Standard deviation, (B) Mean deviation and (C) Range are all valid dispersion measures.
Step 2 — Identify the intruder
The concurrent deviation method is a way of finding the coefficient of correlation between two variables (based on the direction of change of paired items). It measures relationship, not spread.
Watch out“Deviation” appears in both “mean deviation” (dispersion) and “concurrent deviation” (correlation) — do not let the shared word fool you. Concurrent deviation is a correlation tool.
TipGroup them: dispersion = range, quartile deviation, mean deviation, standard deviation. Anything about relationship between two series (concurrent deviation, rank correlation) is correlation.
✓Final answer(D) Concurrent deviation method
- CA Foundation 2022Set dec-20221 markMCQQ.________ is based on all the observations and ________ is based on the central fifty percent of the observations. (A) Mean deviation, Range (B) Mean deviation, quartile deviation (C) Range, standard deviation (D) Quartile deviation, standard deviation
›Reveal solutionSolution
Mean deviation uses all observations; quartile deviation uses the central 50%.
Step 1 — First blank: based on all observations
Mean deviation = average of |x − central value| over every item, so it is based on all observations. (Standard deviation also is, but it is not offered with the correct second blank.)
Step 2 — Second blank: central 50% of observations
Quartile Deviation=2Q3−Q1
Q1 to Q3 spans exactly the middle 50% of the data, so QD is based on the central fifty percent.
Watch outRange is based only on the two extreme values, not on all observations — that rules out options (A) and (C).
TipPositional measures (quartiles, deciles, percentiles) ignore the tails; deviation measures like mean deviation and standard deviation use every value.
✓Final answer(B) Mean deviation, quartile deviation
NoteThis answer is verified — independently worked and reviewed by our experienced subject lecturers before we published it.
- CA Foundation 2022Set dec-20221 markMCQQ.Which one of the following is not a method of measures of dispersion? (A) Standard deviation (B) Mean deviation (C) Range (D) Concurrent deviation method
›Reveal solutionSolution
The concurrent deviation method measures correlation, not dispersion.
Step 1 — Recall the measures of dispersion
Absolute measures: Range, Quartile Deviation (semi-interquartile range), Mean Deviation and Standard Deviation. Their relative counterparts are the coefficients of the same.
Step 2 — Identify the odd one out
- Standard deviation — a measure of dispersion.
- Mean deviation — a measure of dispersion.
- Range — a measure of dispersion.
- Concurrent deviation method — a shortcut method to find the coefficient of correlation based on the direction of change of the two series.
Watch outDo not confuse 'deviation' in the concurrent deviation method with 'mean deviation'. The shared word tempts you, but the concurrent method is a correlation tool.
TipIf a term ends in '...correlation method' or refers to matching signs/directions of two series, it belongs to correlation, not dispersion.
✓Final answer(D) Concurrent deviation method
NoteThis answer is verified — independently worked and reviewed by our experienced subject lecturers before we published it.
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