The direct method, and where it becomes unwieldy

What to assume they know

• Grouping loses the raw values, so we stand the class mark in for them — the class mark as the stand-in for a whole class, and why a grouped mean is approximate
• Reading a grouped frequency table and writing down class marks
• Multiplying a two- or three-digit number by a two- or three-digit number, and totalling a long column
• Division producing a non-terminating decimal, and rounding a result to two decimal places
• That a class mark is unchanged if both limits are shifted by the same amount

What they should be able to do

• State the direct-method formula and identify each symbol in a given table
• Lay out a grouped-mean calculation as a table with an f·x column and a totals row
• Compute the mean of a grouped distribution by the direct method and round the result sensibly
• Judge, before starting, whether the direct method or a reduced-arithmetic method suits a given table, and give the reason in terms of the size of the class marks and frequencies
• Explain why the direct method requires no proof of validity while the assumed mean and step-deviation methods do
• Show that the mean is unaffected by converting inclusive classes into continuous ones, and say why

Where it usually goes wrong

• "The direct method is the crude one and the others are better." All three return the same number. The later two are labour-saving devices, not improvements in accuracy — a point the chapter states flatly after Example 2.
• "Choose the method by the size of the answer." Choose it by the size of the intermediate products. Q7's answer is about 0.1 and its arithmetic is still fiddly; Q1's answer is 8.1 and its arithmetic is trivial.
• "Σf·x is the total of the data." It is the total of the reconstructed data. In Example 1 the real total was 1779 and the reconstructed total 1860.
• "You may simplify the frequencies too." You may not. Frequencies are the observed counts; changing one changes the data. Only the class marks are relabelled, and only because the relabelling is later undone.
• "Continuity correction changes every calculation." It changes the class limits, and therefore the mode and median, which are computed from limits. The class mark is the average of the two limits and both move by the same half unit in opposite directions, so the mean does not budge.
• "Round every intermediate value to two places." Example 2's mean is 39.714…, rounded once at the end. Rounding the products first will move the last digit.

Questions to check understanding

• Compute a grouped mean by the direct method, showing the f·x column and both totals
• "Which method did you use, and why?" — the justification is examinable, not just the number
• Given a table with a missing frequency and a stated mean, form and solve the resulting linear equation
• Convert inclusive classes to continuous ones and state which of mean, median and mode are affected
• Compare two distributions by their means and draw a conclusion in context
• Round a mean correctly and state the units, especially for money and ppm

Examples worth working on the board

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

• Example 2 (pp. 178–179). The percentage of women among primary-school teachers in rural areas, by state and union territory. Classes 15–25, 25–35, 35–45, 45–55, 55–65, 65–75, 75–85 with counts 6, 11, 7, 4, 4, 2, 1. The chapter credits the figures to the seventh of NCERT's All India School Education Surveys. Verified: class marks 20, 30, 40, 50, 60, 70, 80; the counts total 35; the products are 120, 330, 280, 200, 240, 140, 80, totalling 1390; the mean is 1390 ÷ 35 = 39.714…, reported to two places as 39.71. Note: the total 35 is a coincidence worth flagging — it is the number of states and union territories, not a class count.
• Where the products get big. In Example 2 the largest single product is 330. Set that beside Exercise 13.1 Q5 (p. 182), where 135 boxes sit against a class mark of 57 and the product is 7695 — the same method, an order of magnitude more pen-work. Verified: Q5 class marks 51, 54, 57, 60, 63 against counts 15, 110, 135, 115, 25; the counts total 400 and the mean is 57.1875.
• Exercise 13.1, the easy end (p. 181). Q1: plants per house, classes 0–2 up to 12–14, counts 1, 2, 1, 5, 6, 2, 3 over 20 houses. Verified: class marks 1, 3, 5, 7, 9, 11, 13; products 1, 6, 5, 35, 54, 22, 39, totalling 162; mean 8.1 plants per house. Every number here fits in the head — this is the case the direct method is for, and Q1 explicitly asks which method was used and why.
• Exercise 13.1, the awkward end. Q2 (p. 181): daily wages in ₹, classes 500–520 up to 580–600, counts 12, 14, 8, 6, 10 over 50 workers. Verified: class marks 510, 530, 550, 570, 590; products 6120, 7420, 4400, 3420, 5900, totalling 27260; mean ₹545.20. Three-digit marks against two-digit counts is exactly the situation the chapter says to escape from.
• Exercise 13.1 Q7 (p. 182): the concentration of sulphur dioxide in ppm across 30 localities, classes 0.00–0.04 up to 0.20–0.24, counts 4, 9, 9, 2, 4, 2. Verified: class marks 0.02, 0.06, 0.10, 0.14, 0.18, 0.22; products 0.08, 0.54, 0.90, 0.28, 0.72, 0.44, totalling 2.96; mean 2.96 ÷ 30 = 0.0987 ppm to three places. Small numbers, but decimal ones — a different kind of awkwardness, and the one the step-deviation method is best at removing.
• The continuity point (section 10). Exercise 13.1 Q5's classes are printed 50–52, 53–55, 56–58, 59–61, 62–64, with gaps between them. Converting them to continuous classes gives 49.5–52.5, 52.5–55.5 and so on. Verified: the class mark is unmoved — (50 + 52) ÷ 2 and (49.5 + 52.5) ÷ 2 are both 51 — because the correction lowers the lower limit and raises the upper limit by the same half unit. So the mean of Q5 is the same either way, while the mode and median of such a table are not. The chapter's closing note on p. 201 demands continuity for the mode and median formulas and says nothing about the mean; this is why.
• Exercise 13.1 Q3 as the method run backwards (p. 181). Daily pocket money, classes 11–13 up to 23–25, counts 7, 6, 9, 13, f, 5, 4, with the mean given as ₹18 and f unknown. Verified: the class marks are 12, 14, 16, 18, 20, 22, 24; taking deviations about 18 the known classes contribute −42, −24, −18, 0, +20, +24, a net −40, against 44 + f children, so −40 + 2f = 0 and f = 20.

Figures to have open

• A single class drawn on a number line twice — once as 50–52 and once as 49.5–52.5 — with arrows showing each limit sliding half a unit outward and the mid-point tick fixed at 51. This carries section 10 and is the one image the topic cannot do without. Standard schematic.
• A "same shape, four sizes" panel: four calculation skeletons side by side for Exercise 13.1 Q1, Q7, Q2 and Q5, with the product column's digit-count highlighted. Standard schematic.
• Example 2's table, redrawn, with the product column appearing last. Do not reproduce the printed table's styling.
• Nothing needs to come from the printed page; this chapter contains no diagrams at all.

Where this sits in the book

• NCERT Class 10 Mathematics, Chapter 13 "Statistics", §13.2, p. 174 — the direct method named, and the paragraph on when the products become tedious.
• Example 2 and Tables 13.6 and 13.7, pp. 178–179, including the Remark on choosing between the three methods, p. 179.
• Exercise 13.1, questions 1–9, pp. 181–183.
• Exercise 13.3 Q4's hint on converting to continuous classes, p. 199, and A Note to the Reader, p. 201 — both cited here for the contrast that the mean is exempt.