Chapter 7.3 — Exercise 7.2 — Grouped Data

Two Ways to Write a Class Interval

Class intervals come in two styles, and telling them apart matters for exactly where a boundary value gets counted:

• Inclusive classes (like 1–10, 11–20, 21–30) leave gaps between consecutive classes — 10 and 11 are both accounted for, but nothing falls exactly on a shared edge.
• Exclusive classes (like 0–10, 10–20, 20–30) share a common edge between consecutive classes — by convention, a value exactly on that shared edge belongs to the class where it's the lower limit, not the upper one.

Every inclusive class interval has a corresponding pair of class boundaries — true limits shifted by half a unit — which convert it into the exclusive form for calculation purposes: 1–10 becomes 0.5–10.5, 11–20 becomes 10.5–20.5, and so on. This half-unit adjustment closes the gaps between inclusive classes without changing which original values fall in which class.

Why the Half-Unit Adjustment Is Exactly Half, Not Any Other Amount

The shift from inclusive limits to class boundaries always uses exactly half a unit, and it's worth seeing why. Between the inclusive classes 1–10 and 11–20, there's a genuine one-unit gap (the values between 10 and 11, of which there are none for whole numbers, but the gap exists in principle). Splitting that one-unit gap exactly in half — giving 0.5 to the end of the first class and 0.5 to the start of the second — is what turns two classes with a gap between them into two classes that meet edge-to-edge with nothing missing and nothing double-counted. Any other split would either leave a small gap unaccounted for somewhere, or cause the two adjacent classes to overlap one another entirely instead of meeting cleanly.

Choosing How Many Classes to Use

For 45 ages ranging from 5 to 63 (range = 58), grouped into 6 classes: 58/6 ≈ 9.66, rounded up to a class length of 10 — giving intervals 5–14, 15–24, 25–34, 35–44, 45–54, 55–64. Rounding up, not down, matters here: rounding down could leave the largest value just outside the last class entirely.

The same idea runs the other direction too — given a fixed class length instead of a fixed class count, divide the range by the length instead: a range of 25 with class length 4 gives 25/4 = 6.25, rounded up to 7 classes.

What Gets Lost Once Data Is Grouped

Grouping data trades away some information in exchange for a manageable summary — once 45 individual ages become "9 people aged 5–14," the exact age of any one of those 9 people is no longer recoverable from the table alone. This trade-off is deliberate and usually worthwhile: a table of 6 rows is far easier to read and reason about than a list of 45 separate numbers, even though some precision is genuinely lost in the process. Choosing the number of classes is really choosing how much of that trade-off to accept — too few classes hides real structure in the data (everything blurs into one or two giant bars), while too many classes recreates almost the same problem the grouping was meant to solve in the first place — somewhere between five and ten classes is the range that usually works well for data sets of the size typically seen in this exercise.

Working Backward From Class Marks Alone

Sometimes only the midpoints of each class (the class marks) are given, without the class intervals themselves. If consecutive class marks are 10, 22, 34, 46, 58, 70, the gap between them (h) is 12, so each class extends half that gap on either side of its mark: class mark 10 → interval (10−6)–(10+6) = 4–16; class mark 22 → 16–28; and so on, giving 4–16, 16–28, 28–40, 40–52, 52–64, 64–76.

Cumulative Frequency — Running Totals in Either Direction

A less-than cumulative frequency answers "how many observations fall below this upper boundary?" by adding up frequencies from the first class onward. A greater-than (or more-than) cumulative frequency does the reverse, adding from the last class backward. For classes 0–10, 10–20, 20–30 with frequencies 2, 5, 3: the less-than cumulative frequencies are 2, then 2+5=7, then 7+3=10 (the total); the greater-than cumulative frequencies are 10, then 10−2=8, then 8−5=3.

Converting a cumulative frequency table back into ordinary frequencies just reverses the running total: each class's frequency is the difference between its cumulative value and the one before it. Given a less-than cumulative frequency of 3, 8, 19, 25, 30 across five classes, the individual frequencies are 3, 8−3=5, 19−8=11, 25−19=6, 30−25=5 — subtracting each cumulative value from the next one undoes the accumulation.

Reading Which Type of Cumulative Frequency You've Been Given

If a cumulative frequency table increases from the first row to the last, it's a less-than distribution (more observations accumulate as the upper boundary rises). If it decreases from first to last, it's a greater-than distribution (fewer observations remain as the lower boundary rises). Recognising which type you're looking at — before trying to recover the original frequencies — determines whether to subtract each value from the next, or the next from each value. Getting this backwards produces a column of negative frequencies, which is itself a reliable warning sign that the direction of subtraction genuinely needs to be reversed before going any further.

Reading a Real Example All the Way Through

It helps to walk one complete example from raw list to finished table. Suppose 30 classrooms report their student counts, ranging from 15 to 39 (range 24), and the instruction asks for exclusive class intervals of width 4. That gives 24/4 = 6 classes: 15–19, 19–23, 23–27, 27–31, 31–35, 35–39. Every one of the 30 numbers gets sorted into exactly one class (remembering that, for exclusive classes, a value sitting exactly on a shared boundary belongs to the class where it's the lower limit) — tallying them produces the frequency for each row, and the six frequencies together must add back up to 30, the original total. That final check — do the frequencies sum to the original count of observations? — is a simple, reliable way to catch a tallying mistake before moving on to any further calculation done with the finished table.

Where This Leads

Class intervals, boundaries, and cumulative frequency are exactly the raw material Exercise 7.3 turns into histograms, frequency polygons, and ogive curves. For the mean, median and mode calculations this exercise's tables feed into, revisit Exercise 7.1 — the deviation method there works just as well once class marks stand in for individual observations.