Which of the three averages a given question actually wants

What to assume they know

• Dividing through by the class width to shrink the arithmetic further — the mean of grouped data by any of the three methods
• Finding the busiest class, then placing the mode inside it — the modal class and the mode formula
• Locating the middle class and interpolating across it — the median class and the interpolation formula
• Reading a frequency distribution for shape: where it peaks and whether it tails off to one side
• Percentage change, for judging how far apart two numbers are

What they should be able to do

• Compute all three measures for one grouped distribution and set them side by side
• Explain which observations each measure depends on, and which it ignores
• Given a stated purpose, choose the appropriate measure and justify the choice
• Predict whether the three will be close or far apart from the shape of the distribution
• Show, on a given table, that removing an extreme group moves the mean but not the mode
• Apply the empirical relation between the three, and state the accuracy it can and cannot be trusted to
• Recognise which distributions the chapter's formulas are not equipped for

Where it usually goes wrong

• "One of the three is the real average and the others are approximations." All three are exact answers to their own questions. Nothing in the chapter ranks them.
• "The mean is always the best because it uses all the data." Using all the data is precisely why one far-out group can move it, as Exercise 13.2 Q4 shows. The chapter says outright that the mean can stop representing the data in such cases.
• "Mode is always less than median is always less than mean." Exercise 13.3 Q6 has that order; Exercise 13.2 Q1 has the reverse; and Example 1's data has the mean below the median with the mode far off to one side. Three of the chapter's own datasets, three different orderings.
• "Use 3 × median = mode + 2 × mean to find whichever average is missing." On Example 1's data that rule is out by 11.5 on a quantity of about 180. It is a sanity check, not a formula, and the chapter calls it empirical for a reason.
• "The median ignores extreme values, so extreme values do not matter." They do not move the median much, but they are still part of the data and may be the most important part of it. Choosing the median is a decision to set them aside, and that decision should be conscious.
• "The mode tells you where most of the data is." It tells you where the single busiest class is. In Example 1's data the busiest class is 40–55 while eighteen of the thirty students score above 55.
• "These formulas work on any table." The mode formula needs equal class widths and the chapter declines unequal ones; the median formula is stated for equal widths too; and both need continuous classes. The mean alone is free of all three restrictions, which is why Example 3 could use unequal widths.

Questions to check understanding

• Compute mean, median and mode for one distribution and compare them — the form of Exercise 13.3 Q1
• Compute two of the three and interpret the pair in context — the form of Exercise 13.2 questions 1 and 4
• Given a described purpose, name the appropriate measure and justify it
• Given a distribution with a far-out group, say which measures move if it is removed
• Use the empirical relation to estimate a third measure from two, and comment on the reliability of the estimate
• Explain why extreme values affect the mean but not the median
• State the conditions under which the mode and median formulas may be applied

Examples worth working on the board

Values marked verified are worked out here on the chapter's printed data. The chapter computes only two of the twelve numbers below; the rest are derivations.

• The chapter's own flagship, three ways. The Example 1 marks, grouped in Table 13.3: classes 10–25, 25–40, 40–55, 55–70, 70–85, 85–100 with counts 2, 3, 7, 6, 6, 6 over 30 students. The chapter computes the mean, 62 (p. 174), and the mode, 52 (p. 185), and never computes the median. Verified: cumulative 2, 5, 12, 18, 24, 30; n ÷ 2 = 15; median class 55–70; median 55 + 15 × (15 − 12) ÷ 6 = 62.5. So the three are 52, 62 and 62.5 — the mode ten marks adrift of the other two, because the busiest class, 40–55, is not where the bulk of the distribution's weight sits.
• A balanced distribution, three ways — Exercise 13.3 Q1 (p. 198). Sixty-eight consumers, classes 65–85 to 185–205 in twenties, counts 4, 5, 13, 20, 14, 8, 4. This question asks for all three and for a comparison; Q6 below also asks for all three, but Q1 is the only place in the chapter that asks that they be set against each other. Verified: cumulative 4, 9, 22, 42, 56, 64, 68; n ÷ 2 = 34; median class 125–145; median 125 + 20 × (34 − 22) ÷ 20 = 137 exactly. Mode: same modal class, with neighbours 13 and 14, giving 125 + 20 × 7 ÷ 13 = 135.77. Mean, by step-deviation with a = 135 and h = 20: u values −3 to 3, products −12, −10, −13, 0, 14, 16, 12, total 7, giving 135 + 20 × 7 ÷ 68 = 137.06; cross-checked directly, the products total 9320 and 9320 ÷ 68 = 137.06. All three inside a range of 1.3 units, because the distribution is close to symmetric about its peak. This is the contrast that makes section 8 work.
• A right-tailed distribution — Exercise 13.3 Q6 (p. 200). One hundred surnames by letter count, classes 1–4 to 16–19 in threes, counts 6, 30, 40, 16, 4, 4. The question asks for all three. Verified: median class 7–10, median 7 + 3 × (50 − 36) ÷ 40 = 8.05; mode 7 + 3 × 10 ÷ 34 = 7.88; mean, from products 15, 165, 340, 184, 58, 70 totalling 832 over 100, is 8.32. Ordering: mode below median below mean, the classic signature of a tail stretching to the right.
• A left-tailed distribution — Exercise 13.2 Q1 (p. 186). Eighty patients by age, classes 5–15 to 55–65 in tens, counts 6, 11, 21, 23, 14, 5. The question asks for mode and mean and for an interpretation. Verified: mean, from products 60, 220, 630, 920, 700, 300 totalling 2830 over 80, is 35.375; mode, modal class 35–45 with neighbours 21 and 14, is 35 + 10 × 2 ÷ 11 = 36.82; median, cumulative 6, 17, 38, 61, 75, 80 and n ÷ 2 = 40, is 35 + 10 × (40 − 38) ÷ 23 = 35.87. Ordering: mean below median below mode — the exact reverse of the surnames. Two of the chapter's own exercises, two opposite orderings.
• Resistance made visible — Exercise 13.2 Q4 (p. 187). Thirty-five states by students per teacher, classes 15–20 to 50–55 in fives, counts 3, 8, 9, 10, 3, 0, 0, 2. Two classes are empty and two states sit far out at 50–55. Verified: with all thirty-five states the mean is 29.21; drop the two far-out states and the remaining thirty-three give products totalling 917.5, a mean of 27.80 — a shift of 1.4. The mode is 30.625 either way, because the modal class 30–35 and its two neighbours are untouched. Show both calculations side by side; this single comparison carries sections 3 and 6.
• The chapter's own illustration of the same point (p. 197). It sketches a distribution in which one class holds 2 observations while five others hold 20, 25, 20, 21 and 18, and argues the mean would misrepresent it. No class limits are given, so this stays a sketch.
• The thumb rule, tested — section 11, and entirely added here. The chapter states on p. 197 that three times the median is approximately the mode plus twice the mean. Verified on four of its own distributions:
  • Example 1's marks: three times 62.5 is 187.5, against 52 + 124 = 176. Out by 11.5, about 6%.
  • Exercise 13.3 Q1: three times 137 is 411, against 135.77 + 274.12 = 409.89. Out by 1.11, about 0.3%.
  • Exercise 13.3 Q6: three times 8.05 is 24.15, against 7.88 + 16.64 = 24.52. Out by 0.37, about 1.5%.
  • Exercise 13.2 Q1: three times 35.87 is 107.61, against 36.82 + 70.75 = 107.57. Out by 0.04, about 0.04%. The pattern is the argument: the rule is excellent when the three measures are already close together and poor when they are far apart — which is to say it is least reliable exactly where you would want to lean on it. Present it as a rough check, never as a way to compute a missing average.

Figures to have open

• One frequency chart carrying all three markers at once, reusable across sections 8, 9 and 10 with different data loaded. This is the topic's workhorse image. Standard schematic; the chapter contains no diagrams at all, so it must be built.
• A "what each measure looks at" triptych: the same chart three times, with the mean lighting every bar, the median lighting an ordered queue of positions, and the mode lighting three adjacent bars. Standard schematic, and the clearest statement of the whole thesis.
• A before-and-after pair for Exercise 13.2 Q4 with the two far-out states removed. Standard schematic.
• A four-row table of the thumb rule's performance, with the percentage error column sorted. Standard schematic.

Where this sits in the book

• NCERT Class 10 Mathematics, Chapter 13 "Statistics", §13.4, p. 197 — the whole discussion of which measure suits which requirement, the illustration with one sparse class among five busy ones, and the two Remarks giving the empirical relation and declining unequal class sizes for the median.
• Example 6 and Remarks 1 and 2, p. 185 — the mode-and-mean comparison and the warning that neither is reliably the larger.
• The Remark declining unequal class sizes for the mode, p. 186.
• §13.5 Summary, pp. 200–201, which lists the three mean formulas, the mode formula, cumulative frequency and the median formula — and nothing about choosing between them.
• A Note to the Reader, p. 201, requiring continuous classes before the mode and median formulas are applied.
• Exercise 13.2 questions 1 and 4, pp. 186–187; Exercise 13.3 questions 1 and 6, pp. 198–200.