Class 11 Statistics Notes · GSEB
Comparative Study of Mean, Median and Mode
Comparative Study of Mean, Median and Mode — choose the right average for the data, apply change-of-origin-and-scale rules, and use the empirical relation. GSEB Class 11 Statistics notes with a comparison table and solved sums.
Last updated: 5 Oct 2026
Notes
No Single Average Fits Every Problem
Words marked with a dotted underline carry the question they usually appear in — tap them to reveal it.
Each average carries its own applications and its own limitations — that is exactly why this chapter spends five topics building all of them. Among them, the mean satisfies most requisites of a good average: it is rigidly defined, uses every observation and — the property that matters most — it can be pushed around by algebra. Advanced statistical methods and cross-population comparisons all speak in means. That makes it the optimum average, not the automatic one.
Mean
For evenly spread data and every calculation that follows — GDP growth, class averages, combined groups.
Median
Agriculture, social sciences and qualitative data — education, skill, consumer satisfaction — anything not evenly distributed.
Mode
Business and consumer choice — restaurant dishes, readymade garments, footwear sizes: what sells the most.
Negative example:a hostel surveys 20 students' monthly expenses and reports the mean of ₹12,400 — because one student drives a car and spends ₹90,000. Nineteen students live near ₹8,500. The average was arithmetically right and practically useless; the median (₹8,500) would have told the truth.
Three Lenses on the Same Nine Numbers
Floors of nine buildings: 2, 10, 4, 1, 4, 8, 4, 2, 8. The same data, looked at three ways — each average is answering a different question.
x̄ = 43/9 = 4.78
Add them up and divide — nothing needs sorting. The balance point of the whole cloud sits at 4.78.
What each lens is really asking
Where Each Average Shines
x̄ = Σx / n — the sum spread evenly over every observation.
Rigidly defined, based on all values, and suitable for further algebraic operations — the optimum average.
Best for evenly spread data: marks, heights, daily sales of a uniform shop.
Weak spot: heavily affected by a few very large or very small values.
Mean vs Median vs Mode — Side by Side
| Aspect | Mean | Median | Mode |
|---|---|---|---|
| Best for | Evenly spread data and algebra | Skewed, open-ended or missing data | Business, commerce and consumer choice |
| Ordering needed | No — add and divide | Yes — arrange first | Grouping by repeats |
| Effect of extremes | Heavily affected | Barely affected | Not affected |
| Open-ended classes | Cannot be computed | The only average available | Not available — needs the modal class limits |
| Rigidly defined | Yes | No | No |
| Transform y = bx + a | ȳ = b·x̄ + a | b·M + a | b·Mo + a |
Choice of Average — The Routing Map
Five questions decide which average you are allowed to use. Tap a factor to follow its route.
y = bx + a applies to all three
Change of origin and scale hits the mean, the median and the mode identically — ȳ = b·x̄ + a, median(y) = b·M + a, mode(y) = b·Mo + a.
The five factors behind the routes
Some Results Worth Remembering
All values equal
Ages 15, 15, 15, 15, 15 → x̄ = M = Mo = G = x̄w = 15. Every average agrees.
Evenly distributed
evenly distributed ⇒ x̄ = M = Mo — symmetric data makes the three lenses land on one point.
Change of origin & scale
y = bx + a transforms all three the same way: ȳ = b·x̄ + a, b·M + a, b·Mo + a.
The chapter's headline identity
Mean, median and mode are not strangers — in moderately skewed data they lock into this single relation (Karl Pearson). Remember any two, get the third.
Transform Rules in Action
Mean
Mean of x is 9 and y = x + 4 → ȳ = 13
ȳ = x̄ + 4 = 9 + 4 = 13
The same shortcut exists for the other two — no re-sorting, no re-counting:
Median
Median of 10 observations is 14; double every observation → M = 28
M(y) = 2 × 14 = 28
Mode
Mode of salaries is ₹8,500; the company deducts 2% for a welfare fund → mode of the fund = ₹170
0.02 × 8500 = 170
Worked Example Gallery
Comparative Study — Solved Practicals
4 problemsTransform of the mean
(1) The mean of a variable x is 25. Find the mean of the variable obtained by first subtracting 3 from x and then dividing it by 2.
ȳ = (x̄ − 3) / 2 = (25 − 3) / 2
Mean of y = 11
- y is x with a change of origin (−3) and a scale (÷ 2).
- ȳ = (x̄ − 3) / 2.
- ȳ = (25 − 3) / 2 = 22 / 2 = 11.
Transform of the median
(2) The relation between price (p) of an item and its demand (d) is d = 50 − 2p. If the median of price is ₹ 11, find the mean of demand.
median(d) = 50 − 2(11) = 28
Median demand = 28 items
Even a negative b works — the rule does not care about the sign.
- d is a linear function of p with b = −2 and a = 50.
- Median(d) = 50 − 2 × median(p).
- 50 − 2(11) = 50 − 22 = 28.
Transform of the mode
(3) The mode of salaries of employees of a company is ₹ 8500. The company has decided to deduct 2% of each employee's salary for welfare fund. Find the mode for the amount of this fund.
mode(y) = 0.02 × 8500
Mode of the welfare fund = ₹170
- Fund = 0.02 × salary → b = 0.02, a = 0.
- Mode(fund) = 0.02 × mode(salary).
- 0.02 × 8500 = 170.
Three-lens comparison
Study the following data for a general comparison of mean, median and mode. x = No. of floors in the building, n = 9.
2, 10, 4, 1, 4, 8, 4, 2, 8
Find the mean, median and mode for the number of floors.
| Lens | Method | Answer |
|---|---|---|
| Mean | (2 + 10 + 4 + 1 + 4 + 8 + 4 + 2 + 8) / 9 = 43/9 | 4.78 |
| Median | 5th of 1, 2, 2, 4, 4, 4, 8, 8, 10 | 4 |
| Mode | value with frequency 3 | 4 |
x̄ = 4.78 · M = 4 · Mo = 4 — same data, three questions
- Mean: no ordering needed — 43/9 = 4.78.
- Median: order → 1, 2, 2, 4, 4, 4, 8, 8, 10; 5th observation = 4.
- Mode: 4 appears three times — more than any other value.
The mean sits slightly above the middle because 10 pulls it up.
Quick Reference
⭐ Mo = 3M − 2x̄ · evenly spread ⇒ x̄ = M = Mo · x̄ ≥ G · P₅₀ = M = Q₂ = D₅
Median of revenue = price × median of demand when price is constant
Mean, median and mode all obey the same change-of-origin-and-scale rule — y = bx + a turns ȳ into b·x̄ + a, M into b·M + a and Mo into b·Mo + a.
Mode is the average used in business, commerce and consumer choice.
The mean divides nothing — P₅₀ = M = Q₂ = D₅ is the set of cuts that split the data into two equal halves.
Key Takeaways
Key Takeaways
- No single average suits every problem — mean for evenly spread data and algebra, median for skewed/open-ended/missing data, mode for business and consumer choice.
- The mean is the optimum average because it satisfies most requisites of a good average — optimum, not automatic.
- Same nine floors seen three ways: x̄ = 4.78 (balance point), M = 4 (midpoint cut), Mo = 4 (tallest column).
- Five factors route you to an average: nature of data, nature of variable, purpose, classification, and further analysis — rates go to GM, weights to the weighted mean.
- y = bx + a applies to all three alike, and Mo = 3M − 2x̄ locks them into one relation.
- So what: whenever a number is quoted as “the average”, ask which lens produced it — the hostel example shows a mean can be correct arithmetic and still mislead everyone.