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Chapter 10 · Period 1 of 16 · 50 min

Why Averages of Averages Can Mislead

Combining Things · Problem solving

This opening unit establishes the central problem of the chapter's first strand — that simple averaging of averages goes wrong when group sizes differ — so it is worth spending time on the prediction step before revealing the calculation. A common error is believing that both methods always give the same answer; prompt students to check whether the group sizes matter. Example 2 on p.9 (Jaspreet's cycling record) shows a case where the two methods do agree and can be read independently.

0-10Open by recalling what a plain average means: add all values, divide by count. Pose the question: if seniors in a badminton group average one height and juniors average another, is the whole group's average simply the midpoint of those two? Let students commit to a prediction before any calculation is shown.
10-30Work through the two methods described in Section 10.1 using the badminton scenario introduced there. Show that adding the two averages and halving gives 157.415 cm, then sum all eleven heights and divide by 11 to get 161.09 cm. Ask: why do the answers differ? Students articulate that the two groups are not the same size, so treating them equally misweights the result.
30-45Introduce Shreyas's fix — multiply each group's average by its group size, add the products, divide by the total count — and verify it yields 161.09 cm. Students check by re-deriving: 165.5 × 8 = 1324, 149.33 × 3 = 447.99, total ÷ 11 ≈ 161.09. Generalise to two groups of sizes n and m with averages a and b: combined average = (an + bm)/(n + m).
45-50Quick consolidation: each student writes one sentence explaining in their own words why Method 1 fails for the badminton group. Two or three responses are shared aloud. No new content introduced.

Homework

Exercise Set 10.3 Q2, p.15: A white stork averages 44.5 km per day over 20 days and flies 55 km on the 21st day; guess, then calculate, its average over 21 days.

Assessment

Competency (C-6.1). Extends the mean to the weighted mean by showing why simply averaging unequal-group averages goes wrong and deriving and applying the correct combined-average formula.

Learning outcome. Students can explain why averaging the averages of two unequal-sized groups gives a different result from computing the overall average directly, and can apply the formula (an + bm)/(n + m) to find the correct combined average.

A school has two sections in Grade IX. Section A has 40 students with an average mark of 68, and Section B has 25 students with an average mark of 74. A student claims the overall average is (68 + 74)/2 = 71. Show why this claim is incorrect and find the correct overall average mark for Grade IX. Round your answer to two decimal places if needed.

70.31

Chapter 10 · 16 periods of 50 min

Also saved as 10 periods or 13 periods.

  • 2When Averaging Averages Works — and the General Rule
    Average of Averages
  • 3Mixing Quantities — Concentration as a Weighted Average
    Mixtures
  • 4Working the Weighted Mean Backwards and the Historical Setting
    Mixtures
  • 5Custom Weights — When Importance Differs, Not Just Size
    Custom Weights
  • 6Weighted Mean in Varied Contexts — Practice and Consolidation
    Custom Weights
  • 7Clustered Bar Charts — Choosing the Right Grouping
    Visualising and Interpreting Data
  • 8Stacking the Bars — Reading and Constructing Stacked Column Charts
    Stacking Columns
  • 9From Stacked Bars to 100% Bars — Proportions, Not Totals
    An Alternative to Pie Chart
  • 10What 100% Bars Can and Cannot Tell You — Fatima, Naveen and the Inference Test
    An Alternative to Pie Chart
  • 11Choosing and Reading Charts — Multi-Chart Practice
    An Alternative to Pie Chart
  • 12What Percentage Charts Cannot Do — Inference Limits Across Contexts
    An Alternative to Pie Chart
  • 13Multi-Step Weighted Mean Problems — Langurs, Shares and Water
    Mixtures (Revisit)
  • 14Algebraic Reasoning About Weighted Means — Cricket, Shuttlecocks and a Pool
    Average of Averages (Revisit)
  • 15Estimation, Judgment and Chart Choice — End-of-Chapter Consolidation
    An Alternative to Pie Chart (Revisit)
  • 16Whole-Chapter Synthesis — Weighted Means, Chart Inference and the Shape of Combined Data
    Synthesis
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