Grouped frequency distributions
A class of 40 students scores marks from 8 to 98 on a test. Listing every distinct mark separately would produce a table almost as long as the class register — useless for spotting any pattern. The fix is to group the data into class intervals, such as –, –, and so on, and record how many observations fall in each. This arrangement is a grouped frequency distribution, and it is the standard first step before drawing a frequency polygon, histogram, or ogive. Raw data almost never arrives already grouped correctly. Sometimes the classes are written with gaps between them (discontinuous form), which must be closed before the data is graph-ready. This article builds a grouped frequency table from raw values, then shows the exact adjustment that converts a discontinuous table into a continuous one — the two skills the Class IX Statistics chapter asks for.
A grouped frequency distribution sorts data into class intervals with recorded frequencies, and any discontinuous (gapped) intervals must be converted to continuous ones by subtracting/adding half the gap to get class boundaries.
The core idea
A class interval is a span of values such as – into which raw observations are grouped. Written as , is the lower limit and is the upper limit. The class size (class width) is , and good practice keeps every class the same size in a table. The frequency of a class is simply the count of observations landing inside it, and the completed list of classes with their frequencies is the frequency table. Class intervals come in two forms, and telling them apart is the crux of this topic.
Continuous (exclusive) form — e.g. –, –, – — has no gap: the upper limit of one class is exactly the lower limit of the next. By convention a value equal to a shared boundary (say, exactly ) is placed in the next class (–), never the one it closes. This is the only form that can feed directly into a frequency polygon or histogram.
Discontinuous (inclusive) form — e.g. –, –, – — leaves a visible gap: to . A value like would fall between two classes and belong to neither, which makes this form unusable for graphing until it is repaired. The repair is the adjustment factor, half the size of the gap between one class's upper limit and the next class's lower limit:
where: the lower limit and upper limit are read straight off the table as written, and the result is a single number applied uniformly across the whole table. Subtracting the adjustment from every lower limit and adding it to every upper limit produces the class boundaries (true class limits) — a gap-free continuous table with the frequencies completely unchanged.
The goal of this derivation is to turn a discontinuous frequency table — one with visible gaps such as to between successive classes — into a continuous, gap-free table without disturbing a single frequency, by finding one adjustment number and applying it uniformly to every limit in the table.
Step 1. Identify the gap. Look at any two consecutive classes, say – and –. The upper limit of the first class is ; the lower limit of the second is . The gap between them is .
Step 2. Halve the gap to get the adjustment factor. . Halving matters because the missing half-unit belongs equally to both neighbouring classes — extending one class's upper limit up by and the next class's lower limit down by makes them meet exactly in the middle of the old gap.
Step 3. Apply the adjustment to every class. Subtract from every lower limit and add to every upper limit: – becomes –, – becomes –, and so on. The new upper limit of one class () now equals the new lower limit of the next (), so the table has become continuous.
Step 4. Confirm the frequencies are untouched. The adjustment only moves the boundary labels; no observation has been recounted or reassigned, so every original frequency carries over unchanged into the new, continuous table — which is now ready for a frequency polygon.
Worked example
The marks (out of 40) scored by 30 students, grouped in discontinuous form, are given below. Convert the table to continuous form and state the class size of the converted table.
| Marks (discontinuous) | Frequency |
|---|---|
| 1 – 10 | 3 |
| 11 – 20 | 8 |
| 21 – 30 | 12 |
| 31 – 40 | 7 |
Step 1. Find the gap and adjustment. Consecutive classes – and – have a gap of , so the adjustment factor is .
Step 2. Apply the adjustment to every class: subtract from each lower limit and add to each upper limit.
| Original class | Class boundaries (continuous) | Frequency |
|---|---|---|
| 1 – 10 | 0.5 – 10.5 | 3 |
| 11 – 20 | 10.5 – 20.5 | 8 |
| 21 – 30 | 20.5 – 30.5 | 12 |
| 31 – 40 | 30.5 – 40.5 | 7 |
Step 3. Check the table is now continuous. Each class's upper boundary meets the next class's lower boundary exactly (), and the total frequency is still , matching the original 30 students.
Step 4. State the class size of the new table. Each converted class spans , , and so on — the class size is unchanged at , since the adjustment shifted both limits of each class by the same . Answer: the continuous class boundaries are –, –, –, –, with frequencies and class size .
Common traps
- Applying the adjustment factor to only the lower limits (or only the upper limits) instead of both — this keeps the gap or creates an overlap.
- Confusing class limits (values as originally written, ) with class boundaries (adjusted, gap-free values, ); the two coincide only when the data was already continuous.
- Placing a value exactly equal to a class's upper limit into that same class instead of the next one, in a continuous table (e.g. a mark of exactly belongs to "–", not "–").
- Recomputing or shifting frequencies during the conversion — the adjustment changes only the boundary labels, never the counts.
Ask the tutor: not sure whether a given class-interval table is already continuous, or how to find the adjustment factor for your own data set? Talk it through with the AI tutor.
Why do we even need continuous class intervals if the discontinuous ones already show all the data
Discontinuous intervals leave real gaps on the number line — between and , for instance, no value has a class to belong to. That is fine for a simple table, but a frequency polygon or histogram needs bars or points that sit edge to edge along the whole range with no blank space between them. Converting to class boundaries closes those gaps without changing any frequency, so the same data becomes ready for graphing.
Do I always subtract and add the same adjustment number to every class in the table
Yes, as long as every class was written with the same size gap between them, which is the usual case in exam problems. Compute the adjustment once from any pair of consecutive classes, then apply that single number consistently: subtract it from every lower limit and add it to every upper limit across the entire table. Only if the gaps differ from pair to pair would you need to recompute the adjustment separately for each boundary.
Does converting to class boundaries change the class size
No. Because the same adjustment is subtracted from a class's lower limit and added to the next class's upper limit, each individual class gains the adjustment on one side and loses none of its original width — every class stretches by the full gap amount split across its two boundaries, so the width stays exactly the same before and after conversion. In the worked example every class kept its width of .