Running totals, and converting between the two cumulative tables

What to assume they know

• Grouping loses the raw values, so we stand the class mark in for them — grouped tables, class limits and frequencies
• The median of ungrouped data: the middle value once the data is ordered, and the average of the two middle values when the count is even
• Ordering a list of values and counting positions within it
• Adding a running column, and subtracting to reverse it
• That the frequencies of all classes total the number of observations

What they should be able to do

• Build a cumulative frequency column from a frequency column, upward and downward
• State what a given cumulative entry counts, in words, for a named boundary
• Show that the two cumulative forms sum to n at every shared boundary, and use that to convert one into the other
• Recover a class-frequency column by differencing a cumulative column
• Construct the class intervals implied by a table given only in cumulative form
• Identify which limits — upper or lower — each cumulative form is indexed by
• Explain why the two cumulative tables cover different boundary sets

Where it usually goes wrong

• "Cumulative frequency is just another column to fill in." It changes what the table can answer. Frequencies answer "how many here?"; running totals answer "how many so far?", and only the second lets you locate a position.
• "The two cumulative tables are two different distributions." They are one distribution counted from opposite ends. Either can be produced from the other by subtracting from n, without going back to the frequencies at all.
• "Less than 20 means the class 10–20." It means every class below 20 — here 0–10 and 10–20 together, so 8 and not 3. This is the single commonest error in building the column.
• "The last cumulative entry can be anything." It must equal n. If it does not, the column is wrong, and that is a free self-check on every question.
• "Both cumulative tables are indexed by the same numbers." The upward one is indexed by upper limits, the downward one by lower limits. They differ by one class width all the way along.
• "A table given cumulatively is ready to use." It is not — the mode and median formulas need class frequencies, so the column must be differenced first. Example 7 and Exercise 13.3 Q3 both open with that step.
• "An open first class is a misprint." Example 7's lowest class genuinely has no printed lower limit. Sometimes context supplies one, as the age-18 floor does in Exercise 13.3 Q3, and sometimes nothing does.

Questions to check understanding

• Build a cumulative frequency column, upward or downward, from a frequency table
• Convert an upward cumulative table into a downward one without recomputing from the frequencies
• Recover the class frequencies from a cumulative table by differencing
• Construct the class intervals implied by a table stated only as "less than" or "below" values
• State what a named cumulative entry counts, in the context of the data
• Use the last cumulative entry as a check on the arithmetic
• Handle a first class whose lower limit must be read out of the question's wording rather than the table

Examples worth working on the board

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

• The ungrouped warm-up (Tables 13.9–13.11, pp. 188–189). One hundred students, marks out of 50: the marks 20, 25, 28, 29, 33, 38, 42, 43 with frequencies 6, 20, 24, 28, 15, 4, 2, 1. Verified: the frequencies total 100; the running totals are 6, 26, 50, 78, 93, 97, 99, 100.
• The "why" the chapter leaves as a question (p. 190). It states that the 50th observation is 28 and the 51st is 29, and asks the reader why without answering. Supply the reasoning: the running total reaches 26 at the mark 25, so positions 1 to 26 are accounted for; the 24 students on 28 therefore occupy positions 27 through 50, and 28 is where the total lands exactly. The next block, the 28 students on 29, begins at position 51. Verified: 26 + 24 = 50 and 50 + 28 = 78, so positions 51 to 78 all carry 29. Hence the median is (28 + 29) ÷ 2 = 28.5. This is section 11 and it is the best thing in the ungrouped case — a running total does not just count, it tells you which positions belong to which value.
• The grouped distribution (Table 13.12, p. 190). Fifty-three students, marks out of 100, classes 0–10 through 90–100 in tens, frequencies 5, 3, 4, 3, 3, 4, 7, 9, 7, 8. Verified: they total 53.
• The upward accumulation (Table 13.13, p. 191), indexed by upper limits. Verified: 5, 8, 12, 15, 18, 22, 29, 38, 45, 53 at the boundaries 10, 20, 30, 40, 50, 60, 70, 80, 90, 100.
• The downward accumulation (Table 13.14, p. 192), indexed by lower limits. Verified: 53, 48, 45, 41, 38, 35, 31, 24, 15, 8 at the boundaries 0, 10, 20, 30, 40, 50, 60, 70, 80, 90.
• The identity, tested exhaustively — section 7, and not in the book. Verified at all nine shared boundaries: at 10, 5 + 48 = 53; at 20, 8 + 45 = 53; at 30, 12 + 41 = 53; at 40, 15 + 38 = 53; at 50, 18 + 35 = 53; at 60, 22 + 31 = 53; at 70, 29 + 24 = 53; at 80, 38 + 15 = 53; at 90, 45 + 8 = 53. Nine boundaries, nine identical totals, no exception.
• Why the boundary sets differ — section 8. Verified: the upward table is indexed by the ten upper limits, 10 to 100; the downward one by the ten lower limits, 0 to 90. Each has ten rows, but they overlap in only nine values. The two unmatched ends are the trivial ones — everybody scored under 100, and everybody scored at least 0, so both read 53.
• Differencing, and Example 7 (p. 194). A survey of 51 girls' heights given only as running totals: under 140 cm, 4; under 145, 11; under 150, 29; under 155, 40; under 160, 46; under 165, 51. Verified by differencing: the classes are below 140, then 140–145, 145–150, 150–155, 155–160, 160–165, with frequencies 4, then 11 − 4 = 7, 29 − 11 = 18, 40 − 29 = 11, 46 − 40 = 6 and 51 − 46 = 5; these total 51, which is the check that the differencing was done right. Note the first class has no stated lower limit — it is open below, and the chapter simply labels it by its upper end.
• Exercise 13.3 Q3, the same operation with a wrinkle (pp. 198–199). One hundred policy holders by age, given cumulatively: under 20, 2; under 25, 6; under 30, 24; under 35, 45; under 40, 78; under 45, 89; under 50, 92; under 55, 98; under 60, 100. The question states that policies are issued only from age 18 and below 60. Verified by differencing: 2, 4, 18, 21, 33, 11, 3, 6, 2, totalling 100. The wrinkle is the first class: it is not "everything below 20" but 18–20, because the stated eligibility floor supplies the lower limit the table omits. That is a reading-comprehension trap worth a beat of the explanation.
• Exercise 13.3 Q1, a plain accumulation (p. 198). Sixty-eight consumers by monthly electricity use, classes 65–85 through 185–205 in twenties, counts 4, 5, 13, 20, 14, 8, 4. Verified: running totals 4, 9, 22, 42, 56, 64, 68.

Figures to have open

• An ordered queue of observations with a walking counter and a running tally above it. This is what a cumulative frequency is, and the chapter shows only the table. Standard schematic, must be built.
• A paired-bar sweep across the nine shared boundaries, each pair summing to a fixed 53 line. Carries section 7. Standard schematic.
• One axis marked with all eleven class edges, 0 to 100, with the upward table's ten index points above it and the downward table's ten below, showing the one-class offset. Standard schematic.
• Not required, but flagged: an ogive would be the natural picture for this topic and there is none to reproduce — the chapter draws no graph at all. If the teacher wants one, it must be built from the Table 13.15 data and presented as an addition made here, not as the chapter's.

Where this sits in the book

• NCERT Class 10 Mathematics, Chapter 13 "Statistics", §13.4 Median of Grouped Data, pp. 188–192.
• The ungrouped median recalled, and Tables 13.9 to 13.11, pp. 188–189.
• The 50th and 51st observations, the median 28.5, and the Remark naming the Cumulative Frequency Table, p. 190.
• Table 13.12 and the opening questions, p. 190; Table 13.13 and the upward form, p. 191; Table 13.14 and the downward form, and Table 13.15 combining frequency with cumulative frequency, p. 192.
• Example 7 and Table 13.16, p. 194 — cited here for the differencing step; the median it goes on to compute belongs to Locating the middle class and interpolating across it.
• Exercise 13.3 questions 1 and 3, pp. 198–199.
• §13.1 Introduction, p. 171, and A Note to the Reader, p. 201 — the two places ogives are named.