Chapter at a glance
Chapter 10 extends the idea of an average to the weighted average (weighted mean), used for averages of averages, mixtures and custom-weight scores. The second half is about stacked bar charts and 100% stacked bar charts: how to read them and what you can and cannot infer from them.
- 10.1 Combining things: average of averages, mixtures, custom weights
- 10.2 Visualising and interpreting data: stacked columns, an alternative to the pie chart
10.1.1 Average of averages
Badminton example. 8 seniors average 165.5 cm and 3 juniors average 149.33 cm. The tempting answer (165.5 + 149.33) ÷ 2 = 157.415 cm is wrong, because it treats both groups equally even though there are many more seniors. The correct average is (165.5 × 8 + 149.33 × 3) ÷ 11 = 1772 ÷ 11 ≈ 161.09 cm.
Combining two collections
If collection 1 has n values with average a and collection 2 has m values with average b, the combined average is
(an + bm) ÷ (n + m)
For three collections with averages a, b, c and sizes p, q, r: (ap + bq + cr) ÷ (p + q + r).
When does the simple “average of averages” work? Only when all collections have the same size. Jaspreet cycled 5 days in each of 3 weeks (weekly averages 12.8, 15.8, 18), so (12.8 + 15.8 + 18) ÷ 3 = 15.53 min is correct.
10.1.2 Mixtures
- Equal quantities: two equal glasses of lemonade with 10% and 20% jaggery give exactly 15%, midway.
- Unequal quantities: 500 mL at 10% plus 200 mL at 20% gives (50 + 40) ÷ 700 ≈ 13%, closer to 10% because more of the mixture comes from the 10% bowl.
- Finding an unknown (brass): A = 200 kg at 70% copper, B = 120 kg at 50%, C = y kg at 45%, mixture 55%. Then 0.55(320 + y) = 140 + 60 + 0.45y, so 0.1y = 24 and y = 240 kg.
Weighted mean
For values x₁, x₂, …, xₙ with weights w₁, w₂, …, wₙ:
x̄ = (w₁x₁ + w₂x₂ + … + wₙxₙ) ÷ (w₁ + w₂ + … + wₙ)
Each value counts according to how big or how important it is. “Weights” are often abstract, not actual masses.
History: Brahmagupta wrote this formula in the Brāhmasphuṭasiddhānta (c. 628 CE) to find the mean depth of an irregular pit. Śrīdharācārya (c. 750 CE) used weighted means for the purity of gold alloys, measured in varṇa (16 varṇa = pure gold = 24 karat). Europe used it widely only by the 1700s.
10.1.3 Custom weights
Rehmat’s marks: internals 60%, project 64%, final 73%, combined in the ratio 3 : 2 : 5. Plain mean = 65.6%, but the weighted mean is (60×3 + 64×2 + 73×5) ÷ 10 = 673 ÷ 10 = 67.3%. It is as if the data were 60, 60, 60, 64, 64, 73, 73, 73, 73, 73.
- Restaurant rating with food : service : ambience = 4 : 3 : 2 and ratings 5, 3, 4: (20 + 9 + 8) ÷ 9 ≈ 4.11.
- Product ratings: a shop may count recent reviews twice as much as old ones.
When to use a weighted mean
Whenever parts of the data differ in amount, time or importance: combining groups of different sizes, mixtures, marks with weightage, ratings.
Useful properties (End-of-chapter Ex. 6 and 15): doubling every weight (or multiplying all by the same number) does not change the weighted mean, because the factor cancels. Adding the same constant to every weight generally does change it, pulling it towards the plain average.
10.2 Visualising and interpreting data
Cluster column chart
Bars grouped side by side. Choose the cluster to suit the question: cluster by category to compare families on housing, or by family to see which category one family spends most on.
Stacked bar / column chart
The bars in a cluster are joined end to end. The full bar shows the total, and each piece shows a component.
- Best for comparing totals while still seeing the parts.
- Harder: reading individual parts, since only the first piece starts at 0.
100% stacked bar chart
Every bar has the same length (100%), split in proportion to its parts, like a pie chart stretched into a strip. Comparing lengths is often easier than comparing angles in pie charts.
What a 100% stacked bar can and cannot tell you
- Can: compare shares within one bar. In Fatima’s garden, 50% of blooms came in summer and 30% in monsoon, so she had more in summer.
- Cannot: compare actual amounts across bars. 50% of Fatima’s blooms may be fewer than 40% of Naveen’s if his total is larger.
- Cannot: tell whether the totals are equal.
From a stacked bar chart you can make the 100% version, but not the other way round.
Worked example (electricity, 2005): lighting 360, cooling 300, kitchen 120, others 220 units; total 1000. Shares: 36%, 30%, 12%, 22%.
Time-use chart: the average Indian’s day (National Time Use Survey 2024) is both a stacked bar and a 100% stacked bar, because every day totals 24 hours. Remember, an average can hide big differences between age groups, regions and genders.
Practice questions with answers
- Section A (30 students) averages 72 and Section B (25 students) averages 76. Combined average? (72×30 + 76×25) ÷ 55 = 4060 ÷ 55 ≈ 73.8
- A stork averaged 44.5 km a day over 20 days, then flew 55 km on day 21. New average? (890 + 55) ÷ 21 = 45 km
- Śrīdharācārya: 9 units at 12 varṇa, 5 at 10 varṇa, 17 at 11 varṇa. Purity? 345 ÷ 31 ≈ 11.13 varṇa
- Shop: ₹8000 of books at 30% profit and ₹1000 of covers at 50% profit. Overall profit %? (2400 + 500) ÷ 9000 ≈ 32.2%
- 600 mL of 5% salt solution is mixed with 300 mL of 8% sugar solution. Concentrations? Salt 30 ÷ 900 ≈ 3.33%, sugar 24 ÷ 900 ≈ 2.67%
- T20: the run rate after 19 overs is 6 and the team scores 12 in over 20. New run rate? (114 + 12) ÷ 20 = 6.3
- Pool of 30 hastas: sections 4, 5, 6, 7, 8 hastas long at depths 9, 7, 7, 3, 2. Mean depth? 150 ÷ 30 = 5 hastas
- Vaishnavi holds 5 shares at ₹150 and buys some at ₹30; her average becomes ₹70. How many did she buy? 750 + 30x = 70(5 + x), so x = 10 shares
Chapter summary
- Weighted mean x̄ = Σwᵢxᵢ ÷ Σwᵢ.
- Use it for averages of averages, adding or removing values, mixture concentrations, ratings and weighted marks.
- Stacked bar charts compare totals along with their components.
- 100% stacked bar charts, like pie charts, compare proportions, not absolute values.
- A stacked chart can be converted to a 100% stacked chart, but not vice versa.
Source: NCERT, Ganita Manjari, Grade 9 (2026-27). NCERT chapter PDF. Spotted a mistake? Email edura.class9.yt@gmail.com. Last updated 11 October 2026.