Grouping loses the raw values, so we stand the class mark in for them

What to assume they know

• The mean of ungrouped data as total-of-values divided by number-of-values
• Reading a frequency table: that a frequency f against a value x means x was observed f times, so it contributes f copies to the total
• Sigma notation for a sum, and that Σf is the number of observations while Σfx is the sum of them
• Forming a grouped frequency distribution from a list of raw values, from Class IX
• Upper and lower limits of a class, and the arithmetic average of two numbers

What they should be able to do

• Compute the mean of an ungrouped frequency distribution as Σf·x ÷ Σf, laying the products out in a column
• Regroup a set of raw values into classes of a stated width, applying the convention that a value sitting exactly on a boundary is counted in the class above
• Compute the class mark of a class as the average of its two limits
• Explain why a grouped table cannot yield the exact mean, and name what has been discarded
• Compute the mean of a grouped distribution using class marks, and compare it with the ungrouped mean of the same data
• State the assumption the class mark encodes, and say when it is nearly true and when it fails
• Identify, for a given class, whether its observations sit above or below its mid-point, and predict the direction in which the grouped mean will err

Where it usually goes wrong

• "One of 59.3 and 62 is a mistake." Neither is. They are answers to two different questions, because after grouping the marks 10 and 20 no longer exist in the table — only "two students somewhere in 10–25" does.
• "Grouping always pushes the mean up." It pushed it up here because most classes happened to be bottom-heavy. Class 25–40 in this very data leans the other way. The direction is a fact about the particular data, not about grouping.
• "The class mark is the average of the observations in the class." It is the average of the class's two limits. Whether it equals the average of the observations is exactly the thing being assumed, and in Example 1 it does not.
• "A mark of 40 could go in either 25–40 or 40–55." Not once the convention is fixed. Every boundary value goes to the class above, or the same student would be counted twice and the frequencies would not total 30.
• "Wider classes are simpler, so use them." Wider classes discard more position information, so the stand-in has further to stretch. Width buys tidiness with accuracy.
• "Σf·x ÷ Σf is a new formula for grouped data." It is the same formula as for ungrouped data, run on a table whose x column has been replaced by stand-ins. Nothing about the mean changed; the data did.

Questions to check understanding

• Compute the mean of a small ungrouped frequency table using an f·x column
• Given raw values and a stated class width, build the grouped table, applying the boundary convention correctly
• Write down the class marks of a given set of classes
• Compute a grouped mean by the direct method and state the assumption used
• Explain in words why the grouped and ungrouped means of one data set differ
• Given a class and the actual observations in it, say whether the class mark over- or under-states that class's contribution
• Short-answer: what does the mid-point assumption assume, and when is it safe?

Examples worth working on the board

Values marked verified are worked out here on the chapter's printed data.

• Example 1, the raw table (p. 172). Thirty students, marks out of 100. Marks 10, 20, 36, 40, 50, 56, 60, 70, 72, 80, 88, 92, 95 with frequencies 1, 1, 3, 4, 3, 2, 4, 4, 1, 1, 2, 3, 1 respectively. Table 13.1 lays out the f·x column. Verified: the frequencies total 30; the products total 1779; the mean is 1779 ÷ 30 = 59.3 exactly.
• The regrouping (Table 13.2, p. 173). The same thirty marks, cut into six classes of width 15: 10–25, 25–40, 40–55, 55–70, 70–85, 85–100, with counts 2, 3, 7, 6, 6, 6. Verified by rebuilding it from Table 13.1: 10 and 20 give 2; 36 gives 3; 40 and 50 give 7; 56 and 60 give 6; 70, 72 and 80 give 6; 88, 92 and 95 give 6. The four students on 40 land in 40–55, not in 25–40 — this is the boundary convention doing visible work.
• The class marks (Table 13.3, p. 174). 17.5, 32.5, 47.5, 62.5, 77.5, 92.5. Verified: each is the average of its two limits, e.g. (10 + 25) ÷ 2 = 17.5, and consecutive marks differ by 15, the class width.
• The grouped mean. Verified: the f·x products are 35, 97.5, 332.5, 375, 465, 555, totalling 1860; 1860 ÷ 30 = 62 exactly.
• The gap, class by class — this is the section-9 payload, and it is not in the book. Compare the true total each class contributes against what the class mark claims. Verified from Table 13.1: class 10–25 truly contributes 10 + 20 = 30 but is credited 35, so it is over-credited by 5; 25–40 truly contributes 3 × 36 = 108 against a credited 97.5, under-credited by 10.5; 40–55 truly contributes 4 × 40 + 3 × 50 = 310 against 332.5, over by 22.5; 55–70 truly 2 × 56 + 4 × 60 = 352 against 375, over by 23; 70–85 truly 4 × 70 + 72 + 80 = 432 against 465, over by 33; 85–100 truly 2 × 88 + 3 × 92 + 95 = 547 against 555, over by 8. The five over-credits total 91.5, the single under-credit is 10.5, so the net is 81 — which is exactly 1860 − 1779, and over thirty students comes to 81 ÷ 30 = 2.7, precisely the gap between 59.3 and 62. Every marked value here is worked out here; the book computes only the two means.
• A class where the assumption holds. 25–40 contains three students all on 36, whose own mean is 36 against a class mark of 32.5 — the one class the mid-point under-states. Useful as the counter-instance that stops students believing grouping always inflates.

Figures to have open

• A bin diagram: the thirteen distinct marks as labelled tokens dropping into six class boxes, with the count showing on each box. Standard schematic; the chapter has no picture of this and the explanation needs one.
• A number line from 10 to 100 marked at every class boundary, with each class mark shown as a tick at the centre of its class and the real observations shown as dots at their true positions inside the class. This one image carries section 9 and is the most important thing to build — the visible offset between the dots and the tick is the 2.7.
• Two side-by-side tables, Table 13.1 and Table 13.3, sharing a totals row. Redraw rather than reproduce.
• No photograph is needed, and nothing needs to come from the printed page — this chapter prints no diagrams at all (see Notes).

Where this sits in the book

• NCERT Class 10 Mathematics, Chapter 13 "Statistics", §13.1 Introduction (p. 171) and §13.2 Mean of Grouped Data (pp. 171–174).
• Example 1 and Table 13.1, p. 172; the mean 59.3, p. 173.
• Table 13.2 and the class mark definition, p. 173.
• Table 13.3, the mean 62, and the paragraph naming the mid-point assumption as the source of the difference, p. 174.