A grouped table holds four things you must read correctly: boundaries, class size, midpoints and cumulative frequency. Get the table right once and every later calculation stays right.
This lesson opens dispersion of grouped data. The next lesson, estimating mean from grouped data, uses the midpoints you find here.
What does each column mean?
| Item | How to find it | Example for 6 to 10 |
|---|---|---|
| Lower boundary | Halfway between this lower limit and the previous upper limit | 5.5 |
| Upper boundary | Halfway between this upper limit and the next lower limit | 10.5 |
| Class size | Upper boundary − lower boundary | 5 |
| Midpoint | (lower limit + upper limit) ÷ 2 | 8 |
| Cumulative frequency | Running total of the frequencies | See below |
Worked example: waiting times
A clinic records the waiting times of 28 visitors, in minutes.
| Time (min) | Frequency | Upper boundary | Cumulative frequency | Midpoint |
|---|---|---|---|---|
| 1 to 5 | 6 | 5.5 | 6 | 3 |
| 6 to 10 | 11 | 10.5 | 17 | 8 |
| 11 to 15 | 8 | 15.5 | 25 | 13 |
| 16 to 20 | 3 | 20.5 | 28 | 18 |
Check: the last cumulative frequency is 28, which equals the total. If it does not match, an addition slipped.
The modal class is 6 to 10, because 11 is the largest frequency. The class size is 5, from 10.5 − 5.5.
The limit-versus-boundary slip
A common slip is to plot the cumulative frequency against the printed upper limit, such as 5, instead of the upper boundary 5.5. The point then sits half a unit too far left.
The 6 visitors in the first class all waited less than 5.5 minutes, so 5.5 is where that class truly ends. Build the boundary column first, and the ogive points come straight from it.
A quick way to check every table
Do three checks before using any table. Does the last cumulative frequency equal the sum of the frequencies? Do the classes have equal size? Does each midpoint sit halfway inside its class?
These checks take ten seconds and catch the common errors. Record any check you fail in the mistake log and paper-error review, so you can see which one you skip.
Check yourself
The masses of 30 parcels are grouped as 1 to 10 (3), 11 to 20 (7), 21 to 30 (12), 31 to 40 (6) and 41 to 50 (2). Find the class size, the modal class and the cumulative frequencies.
Answer
The boundaries of the first class are 0.5 and 10.5, so the class size is 10.
The highest frequency is 12, so the modal class is 21 to 30.
The cumulative frequencies are 3, 10, 22, 28, 30. The last value equals the total of 30, so the table checks out.
What to study next
Continue with estimating mean from grouped data. You will need this table again for constructing and interpreting an ogive.
For a teacher to check your table habits as you build one, see online one-to-one Mathematics tuition.