Chapter 9.2 — Exercise 9.1 — Presentation of Data

Reconstructing Frequencies From Cumulative Totals

Some tables hand over running totals instead of individual counts directly. Given cumulative figures of 5, 11, 19, 31, 40, 45 across marks 5 through 10, each individual frequency is simply the difference between one running total and the one before it:

Each frequency here is a subtraction: 11 − 5 = 6, 19 − 11 = 8, 31 − 19 = 12, and so on down the row — the running total only ever grows, so every step's increase is exactly that row's individual frequency. This reconstruction skill genuinely turns out to run in both directions across this exercise: sometimes cumulative totals need converting down into frequencies, and — as the mean/median exercise ahead shows — frequencies often need building back up into cumulative totals for an entirely different purpose. Three coin tosses repeated 30 times work the same way in reverse: instead of a running total, the raw outcomes (0, 1, 2, or 3 heads each trial) get tallied directly into frequencies of 3, 10, 10, and 7 — no subtraction needed at all, since nothing was pre-summed before the counting began.

Categories Instead of Numbers

Not every frequency table organizes numeric values — blood groups, survey answers, and vehicle types are all categories rather than measurements, and the same tallying process applies just as directly. The blood groups of 36 students:

Once tallied, reading off the most and least common categories is immediate: O, with 15 students, is the most common group, and AB, with only 2, is the rarest. No calculation beyond counting is needed — the table itself already sorts this information into a directly comparable form, which is really the entire point of building a frequency table in the first place. A related problem asks the same question about a bar graph rather than a table — reading vehicle counts for cycles, autos, bikes, and cars straight off bar heights (40, 30, 45, 25) and rebuilding them into a frequency table — a reminder that a frequency table and a bar graph are really two different presentations of the identical underlying counts, convertible back and forth in either direction without losing any information.

Grouping Responses From a Live Poll

A television poll on smoking prohibition collected 65 text-message responses across three options — A (complete prohibition), B (prohibition in public places only), C (not necessary):

All 65 messages received were valid, appropriate responses — no stray, blank, unreadable, or ambiguous entries needed discarding — and option B, with 36 of the 65 responses, is the clear majority opinion: most respondents wanted prohibition limited to public places rather than a complete ban or no restriction at all. This is a genuinely different kind of "most common" than the blood-group example above: there, every category was roughly comparable in size; here, B alone accounts for more than half the total, a much starker majority than a simple "most frequent" label conveys on its own. Reading a bar graph's scale correctly matters just about as much as reading the table itself — a related problem gives axis scales of "1 cm = 1 class" and "1 cm = 10 students," and getting the frequency values exactly right depends entirely on multiplying each bar's measured height by the correct per-centimetre scale factor before recording it carefully in the table.

Deciding How Many Classes a Continuous Range Needs

Electricity bills for 25 houses ranged from ₹170 to ₹724 — a range of 724 − 170 = 554. Choosing a class size of 75, the number of classes needed is 554 ÷ 75 ≈ 7.3, rounded up to 8 classes, since a leftover partial class still genuinely needs a full row to hold its values:

Notice the table starts at 150, below the actual minimum of 170, and ends at 750, above the actual maximum of 724 — rounding the range's endpoints outward to tidy, convenient multiples of the class size is standard practice, since it keeps every class the same fixed width rather than leaving an oddly-sized final class to absorb whatever remainder is left over. Rounding the number of classes up rather than down is deliberate too: 7.3 rounds up to 8 specifically because rounding down to 7 would leave the highest bill, ₹724, with no class able to actually contain it — every single value in the original data set has to land inside some row of the finished table, with none left over unaccounted for.

Continuous Measurements Need Exclusive Classes

The lifespans of 40 car batteries, in years, were measured to one decimal place — values like 2.6, 3.7, 4.1 — genuinely continuous data rather than whole-number counts. Grouped into exclusive classes of size 0.5, starting from 2.0–2.5:

This table puts the boundary-value convention from the introduction to direct, practical use: a battery lasting exactly 3.0 years belongs in the 3.0–3.5 class, not 2.5–3.0, because by convention a boundary value is always assigned to the class where it sits as the lower limit. Reading the completed table, the 3.0–3.5 class holds far more batteries (14 of 40) than any other — a first, genuinely informal hint of where this data set's center sits, well before the mean, median, and mode get formally computed in the next exercise. A separate problem asks for a frequency table built from 30 students' test marks out of 75, with equal class intervals starting from 0–10 — a case where the class size and starting point are both simply handed over directly rather than needing to be computed from a range first, which is by far the more common situation in practice once a convenient starting point and interval size have already been agreed on beforehand.

From Presenting Data to Summarizing It

Every single table across this exercise organizes data without reducing it down to a single number. Exercise 9.2 takes the next step, computing the mean, median, and mode directly from frequency tables built using exactly this same process — the tally and grouping work carried out carefully here becomes the direct input to every single calculation carried out there.