Why one number is never enough to describe a data set

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What to assume they know

The arithmetic mean as fair-share — the arithmetic mean
Outliers, and why the median survives them — the median, and what an outlier does to each
Dot plots: seeing spread and clustering at a glance — reading a dot plot and comparing two on one scale
Subtracting two three-digit numbers, and reading a decimal to two places
The difference between a zero entry and a missing entry in a table

What they should be able to do

State the four things a data set can be described by, in the chapter's own list: the ends, the total, the gap between the ends, and the two measures of central tendency
Explain why a comparison of two averages says nothing about any individual pair drawn from the two groups
Read a set of dot plots with printed means and medians and infer the shape of each group
Distinguish a recorded zero from a missing value, and choose the denominator accordingly
Explain how a median of zero can sit alongside a large total
Decide which values belong in a data set before summarising it
Compare two groups whose centres are close but whose spreads differ
Justify why a claim about spread cannot be made from a mean alone
Explain why the mean and the median always fall between the minimum and the maximum, and use that as a check on a computed answer

Where it usually goes wrong

"The average tells you what the group is like." It tells you one thing about the group. The class on p.109 has an average of about 144 and contains a child of 128 and a child of 158; nobody looking only at 144 would know that.
"If group A's average is higher, every member of A is bigger." The chapter attaches the correction to the claim in the same breath, and it is the single most examinable idea in this topic. Show two overlapping dot plots whose means differ by two centimetres.
"Two groups with the same average are the same." The two one-minute groups have centres about a second apart and visibly different spreads. Same centre, different data.
"A blank cell is a zero." They change different parts of the fraction. A zero adds nothing to the total and one to the count; a blank adds nothing to either. The chapter separates them on p.111 with a worked case.
"A median of zero means the team scored nothing." More than half the batters scored nothing; the rest scored 388 between them. Median is a statement about position, not about the total, and the cricket example exists to make that impossible to forget.
"All the values you have belong in the summary." Sita's tree gives no mangoes for seven months of the year. Including those months answers a different question from the one anybody wanted to ask. Deciding the scope of a data set is part of the analysis, not something that happens before it.
"The extremes are just the two least interesting values." The chapter's own sub-heading pairs the ends with the essence for a reason: on p.110 the ends are what tell you the boys are the more varied group, and no measure of centre could have told you that.

Questions to check understanding

Given raw heights or weights, compute the mean, the median, the two extremes and the gap between them, and write a two-sentence description using all four
Given two groups' means, say what does and does not follow about individuals
Count how many values in a set lie above the set's own mean
Given a table with a zero and a blank, compute the mean correctly and justify the divisor
Given two dot plots with similar centres, describe the difference in spread
Decide whether a proposed mean or median is possible for a stated data set, using the rule that both must lie between the minimum and the maximum — the chapter's own item at Part II, §5.2, p.112
"The mean of a class is 154.2 cm. What must be true of the other section?" — the chapter's own multiple-choice item at Part II, §5.4, p.130, whose answer is that nothing follows at all
Decide which months, days or cases belong in a data set before summarising it, and defend the choice

Examples worth working on the board

The Class 5 classroom (Part II, §5.2, p.109). Checked against the printed page. Heights in centimetres. Boys (17 of them): 147, 135, 130, 154, 128, 135, 134, 158, 155, 146, 146, 142, 140, 141, 144, 145, 150. Girls (11 of them): 143, 136, 150, 144, 154, 140, 145, 148, 156, 150, 150. The chapter prints, beside three dot plots on a shared line labelled 125–155 (it runs on past 155, which is where the 156 and 158 sit): whole class mean 144.4 and median 145; boys' mean 142.94 and median 144; girls' mean 146.9 and median 148. Check the whole-class mean before speaking it — see Notes; the two group means and all three medians check out, the whole-class mean is a tenth low.
The three printed inferences (Part II, §5.2, p.110). In paraphrase: the boys' heights run from 128 to 158 and the girls' from 136 to 156, so the class contains both its tallest and its shortest boy; the boys' average is nevertheless below both the class average and the girls' average, and the chapter adds at once that this does not make every girl taller than every boy; and in both groups the mean falls below the median, which the chapter reads as a small pull from the low side. All three are worth showing exactly as three separate claims — the second one is the important one and it comes with its own caveat attached.
Two open questions on the same page (Part II, §5.2, p.110). How many students, and how many boys, stand above the class average? Both are asked and neither is answered in print. They are answerable from the raw heights above.
How long is a minute? (Part II, §5.2, p.110). Checked against the printed page. Two dot plots on a shared 40–70 second line with their summaries printed beside them: Group A mean 58.21 and median 60; Group B mean 59.28 and median 59.5. The seconds themselves are drawn, not typeset. Use the shapes as the input: Group A reaches further to the left and trails a thin low tail, which is why its mean sits nearly two seconds below its median; Group B is packed tighter round the middle with a tall stack at the centre and its two summaries almost coincide. Do not state how many children were in either group — the dots are artwork and a miscount would be invented data.
Zero Median Runs Scored! (Part II, §5.2, p.111). Inputs: a team's innings closes at 407 for the loss of 10 wickets, so eleven batters batted; the extras come to 19; the chapter computes the per-player average as 388 divided by 11 and prints 35.27. The puzzle it poses is how the median can be 0 at the same time. The page also carries a mock newspaper cutting — checked against p.111 — which prints the whole eleven-man scorecard from a July 2025 Test: 19, 0, 0, 22, 158, 0, 184, 5, 0, 0, 0, plus 19 extras. Six zeros, a total of 388 off the bat, 407 all in. Those eleven values are the strongest single input in this topic, because the median and the mean can both be computed live and the answer is genuinely surprising.
Zero vs. No value (Part II, §5.2, p.111). Input: a player's series reads 57, 13, 0, 84, —, 51, 27. The 0 in match three is a duck; the dash in match five is an absence. The chapter states the consequence — the divisor is 6, not 7 — and prints the sum without evaluating it. Leave it unevaluated too; the point is the choice of divisor.
Sita's mango tree (Part II, §5.2, p.111). Inputs: mangoes per month from January to December — 0, 0, 8, 24, 41, 16, 5, 0, 0, 0, 0, 0. The chapter's own reading is that a mean or median over all twelve months would not be the useful number, because only the growing months are meaningful.
A Mean Foot (Part II, §5.2, p.112). Inputs: in early-1500s Europe the rod was defined as sixteen feet; sixteen adult men were lined up heel to toe, and the resulting line was taken to be one rod; that rod was then divided into sixteen equal parts to fix one foot. The chapter's point is that this is an arithmetic mean taken three centuries before anyone used the word. The page carries a period woodcut credited to Jacob Kobel; redraw rather than reproduce.
Daily water usage, and the rule it is there to establish (Part II, §5.2, "Figure it Out" item 4, p.112). Inputs, in litres, for nine days: 5.6, 8, 3.09, 12.9, 6.5, 12.1, 11.3, 20.5, 7.4. The item asks two things and prints no answer to either. (a) Could the mean or the median fall between 25 and 30? Worked here: the total is 87.39, so the mean is 9.71; sorted, the middle value is 8; the minimum is 3.09 and the maximum 20.5. Both centres sit far below 25, so no. (b) Could either be smaller than the minimum or larger than the maximum? Also no — and this one is structural rather than a fact about these nine numbers. No value exceeds the maximum, so a set of n values totals at most n times the maximum, and dividing that total by n leaves the mean at or below the maximum; the same argument at the low end puts it at or above the minimum. The median is either one of the values or the halfway point between two of them, and every value already lies between the ends. Put (b) as a rule that holds for every data set, not as an observation about this one — it is the most reusable claim in §5.2 and it is the check a student can run on their own arithmetic. The same block's item 3 also asks whether there are quicker ways to find a mean, and prints no method.
Sumo wrestlers and ballet dancers (Part II, §5.2, p.113). Inputs, in kilograms. Wrestlers: 295.2, 250.7, 234.1, 221.0, 200.9. Dancers: 40.3, 37.6, 38.8, 45.5, 44.1, 48.2. The question asks roughly how many times heavier one is than the other, which forces the student to pick a summary before dividing — and to notice that the two groups have different sizes.
The second Class 5 section (Part II, §5.2, p.113). Checked against the printed page. Three more dot plots on the same 125–155 line, from another section of the same school, with whole-class mean 141.21 and median 142.5; boys' mean 142.05 and median 143; girls' mean 140.14 and median 140. Put this beside the first section: here the boys are the taller group, which is exactly the reversal §5.4 will build on.

Figures to have open

Three dot plots on one shared line labelled 125–155 and running on past 158, each able to carry a solid mean rule and a dashed median rule. Standard schematic, matching the book's own convention on p.109.
Two overlapping distributions with different means, with one member of the lower-mean group visibly above most of the higher-mean group. This figure is the one that kills the "every A beats every B" error and it must be built for that purpose.
A cricket scorecard strip that can show a 0 and a dash in adjacent cells with the divisor written below.
A twelve-month strip for the mango data with the growing months distinguishable from the empty ones.
Sixteen figures standing heel to toe along a line, and the line then divided into sixteen. Redraw; the book's page carries a period woodcut credited to Jacob Kobel, which should not be reproduced.
No figure needs to be taken from the textbook.

Where this sits in the book

NCERT Ganita Prakash, Class 7, Part II, printed Chapter 5, "Connecting the Dots...", §5.2 "Representative Values", pp.109–113. The named sub-headings used are "Of Ends and the Essence" (pp.109–112) and, inside it, "How Tall is Your Class?" (p.109), "How long is a minute?" (p.110), "Zero Median Runs Scored!" (p.111), "Zero vs. No value" (p.111) and "A Mean Foot" (p.112), then the second "Figure it Out" (pp.112–113). None carries a number.
The SUMMARY's list of the ways data can be described: Part II, p.134.
The two-section multiple-choice item used as an assessment model: Part II, §5.4, p.130.
Source file gegp205.pdf / gegp205.txt, printed pp.97–135; PDF page 1 is printed page 97.