Class 11 Statistics Notes · GSEB
Combined and Weighted Mean
Combined and Weighted Mean — merge group averages with the combined mean and weight unequal observations correctly with the weighted mean. GSEB Class 11 Statistics notes with formulas, reverse problems and solved sums.
Last updated: 5 Oct 2026
Notes
Combined Mean and Weighted Mean — Definitions and Formulas
OCR note — the printed formulas have stray minus signs
Words marked with a dotted underline carry the question they usually appear in — tap them to reveal it.
Why the Weighted Mean Exists
This is disadvantage #6 of the arithmetic mean turned into a fix: using the arithmetic mean is not appropriate if the importance of all observations is not the same. The weighted mean simply gives each observation the voice it deserves.
✗ Wrong way — plain mean over unequal importance
A stationery shop owner averaged his three margins to price a Diwali discount: notebooks 5%, chart paper 10%, staplers 40% → plain mean = (5 + 10 + 40)/3 = 18.3%. But notebooks are 70% of his sales — he offered 10% off “from an 18% margin” and his real blended margin was only 9.5%. The week ended ₹18,000 down.
✓ Right way — weight by share of sales
Weights 70 : 20 : 10 → x̄w = (70×5 + 20×10 + 10×40) / 100 = 950 / 100 = 9.5%. Capping the discount at 4% kept the festival week profitable. Same three numbers — only the weights changed the decision.
When to reach for which
GroupMergeBars — Watch Groups Merge
Illustration 10's three production groups sit on one number line, sized by their nᵢ. When the means of two or more groups are known, the groups merge into a single block at x̄c — and the thicker the arrow, the harder that group pulls the combined mean toward itself. Click a group block.
Blue blocks = the three groups (height ∝ nᵢ) · emerald block = the merged 12-unit group at x̄c = 298.33.
WeightBalance — Marks × Weights on a See-Saw
The chapter's balance metaphor, now with weights: each mark sits on the beam at its true value, the block height is its weight, and the fulcrum rests at x̄w — the weighted mean — where the moments on both sides are equal.
Block height = weight · mark position = actual score · emerald fulcrum = x̄w = Σwx / Σw = 175 / 7 = 25.
Weights need not sum to 100
The Ratio Trick — Let a Cancel
When only a ratio of group sizes is given (women : men = 1 : 2), invent a multiplier: n₁ = a, n₂ = 2a. The a appears in every term of the combined mean and cancels completely:
x̄c = (a·x̄₁ + 2a·x̄₂) / (a + 2a) = (x̄₁ + 2x̄₂) / 3
With x̄₁ = 34 and x̄₂ = 37: x̄c = (34 + 2×37) / 3 = 108 / 3 = 36 years.
Never leave a in your answer — it is scaffolding, not part of the data. For a plain ratio a : b, write n₁ = a, n₂ = b and the same cancellation happens.
Three groups in ratio 1 : 2 : 1 would collapse the same way — multiply each mean by its ratio part, add, divide by 1 + 2 + 1 = 4.
Reverse Problems — Find the Missing Mean
The whole reverse game is one line of algebra: x̄c × total = n₁x̄₁ + n₂x̄₂. Everything else is arithmetic. In cricket: total runs = overall rate × total overs.
Solved Example
Problem
Solution
9.75 runs per over
General form: unknown mean = (x̄c × total − known sum) / n₂ — the same shape appears for wage bills, batch averages and fee collections.
Worked Example Gallery
One solved example per sum type — expand any row for the full working and answer.
Combined & Weighted — one example per sum type
4 problemsCombined mean — three groups
A factory owner knows that the mean monthly production from January to March is 350 items, from April to August it is 254 items and from September to December it is 315 items. Find the mean monthly production for that year.
Σ(nᵢx̄ᵢ) = 3 × 350 + 5 × 254 + 4 × 315 = 1050 + 1270 + 1260 = 3580
Σnᵢ = 3 + 5 + 4 = 12
x̄c = Σ(nᵢx̄ᵢ) / Σnᵢ = 3580 / 12
x̄c = 298.33 items
Combined mean — ratio groups (a cancels)
The ratio of women and men employees in an office is 1:2. If the means of ages of women and men are 34 years and 37 years respectively, find the mean age of all the employees of the office.
Let n₁ = a, n₂ = 2a — the a is scaffolding, not data
x̄c = (a × 34 + 2a × 37) / (a + 2a) = (34 + 74) / 3 = 108 / 3
x̄c = 36 years
Never leave a in the answer — write the ratio as n₁ = a, n₂ = 2a, cancel, then substitute.
Reverse combined mean (cricket run rate)
A team has to score with a run rate of 8.25 in 20 overs to win the match. The run rate at the end of 12 overs is 7.25. What should be the least run rate to be maintained in the remaining overs to win the match?
x̄c × total = n₁x̄₁ + n₂x̄₂ → 8.25 × 20 = 12 × 7.25 + 8 × x̄₂
165 = 87 + 8x̄₂
8x̄₂ = 165 − 87 = 78 → x̄₂ = 78 / 8
x̄₂ = 9.75 runs per over
Weighted mean (theory · practical · oral components)
A student gets 35 marks in theory paper, 15 marks in practical examination and 5 marks in oral examination of a subject. The school gives weights 4, 2 and 1 respectively to these types of examinations. Find the weighted mean of marks for this student.
| Component | x | w | wx |
|---|---|---|---|
| Theory | 35 | 4 | 140 |
| Practical | 15 | 2 | 30 |
| Oral | 5 | 1 | 5 |
| Σ | 7 | 175 |
x̄w = Σwx / Σw = 175 / 7
x̄w = 25 marks
Weights sum to 7, not 100 — the denominator is always Σw.
Quick Reference
- Combined: x̄c = (n₁x̄₁ + n₂x̄₂) / (n₁ + n₂)
- Weighted: x̄w = Σwx / Σw (Σw need not be 100)
- Ratio a : b → n₁ = a, n₂ = b — a cancels, never leaves the answer.
- Reverse: unknown mean = (x̄c × total − known sum) / n₂
- Printed formula OCR quirk: stray minus signs mean + — all working uses addition.
Key Takeaways
Key Takeaways
- Combined mean merges group means weighted by group sizes: x̄c = Σ(nᵢx̄ᵢ) / Σnᵢ — bigger groups pull x̄c toward themselves.
- Weighted mean gives every observation its due voice: x̄w = Σwx / Σw, used whenever importance is not equal.
- Ratio problems: invent n₁ = a, n₂ = b — the a cancels; weights need not sum to 100 (Σw is the denominator).
- Reverse problems collapse to one line: x̄c × total = n₁x̄₁ + n₂x̄₂ — cricket run rates, wage bills, batch averages all use it.
- So what? Your semester GPA is a weighted mean (credits are weights), an IPL chase is a reverse combined mean, and the stationery shop failed because it averaged margins that were never equally important.