Grouped Frequency Tables

Sort a list of values into classes, place boundary values in the right class, and read totals, the modal class and percentages from a grouped table.

In the exam for Cambridge IGCSE Maths 0580, Cambridge O Level Maths 4024 · 2 real past-paper questions below

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Why Group the Data?

With lots of different values, a list is hard to read. So you sort the values into groups called classes, and count how many are in each.

Score	Frequency
1 to 10	3
11 to 20	5
21 to 30	8
31 to 40	4

The frequency is how many values are in the class.

Key idea

Grouping makes the table short, but you lose the exact values. You only know which class each one is in.

Building the Table From a List

The 2-mark question: "Complete the grouped frequency table." These are 20 quiz scores out of 40.

12, 25, 7, 33, 18, 21, 29, 4, 16, 38, 23, 27, 11, 35, 19, 26, 9, 31, 24, 22

Worked example

1
$1 to 10 : 7, 4, 9$ — Go along the list once and pick out every score from 1 to 10. There are 3, so this frequency is 3.
2
$11 to 20 : 12, 18, 16, 11, 19$ — Same again for 11 to 20. 11 is the smallest score that fits and 19 the biggest. There are 5.
3
$21 to 30 : 8 scores$ — 25, 21, 29, 23, 27, 26, 24 and 22 all lie from 21 to 30. Count them: 8.
4
$31 to 40 : 4 scores$ — The scores left are 33, 38, 35 and 31, all from 31 to 40. That makes 4.
5
$3 + 5 + 8 + 4 = 20$ — Add the frequencies. The total is 20, the same as the number of scores, so none was missed or counted twice.

Rule

Go through the list once, in order, and tick off each value as you tally it.

Common mistake

Missing a value or counting one twice. Always check that the frequencies add up to the number of values.

Continuous Data: Reading the Classes

The 1-mark question: "Which class does this value go in?" Heights are measured, not counted, so the classes use inequalities.

Height (cm)	Frequency
$150 < h \leq 155$	6
$155 < h \leq 160$	11
$160 < h \leq 165$	14
$165 < h \leq 170$	9

Worked example

1
$155 \to 150 < h \leq 155$ — Read 150 < h ≤ 155 as “more than 150, up to and including 155”. The ≤ lets 155 in, so it goes here.
2
$155.5 \to 155 < h \leq 160$ — 155.5 is more than 155, so it is too big for the first class. It is not more than 160, so it fits the second.
3
$160 \to 155 < h \leq 160$ — 160 is the top end of this class, and ≤ includes it. It is not in 160 < h ≤ 165, because that class needs more than 160.

Rule

In $150 < h \leq 155$ the top end, 155, is in the class. The bottom end, 150, is not. So each value fits in exactly one class.

Key idea

Some tables put the ≤ at the bottom, like $10 \leq t < 20$ . Then the bottom end is in and the top end is not, so 20 goes in $20 \leq t < 30$ . Always check which end has the ≤.

Common mistake

Classes like 150 to 155 and 155 to 160 overlap: 155 could go in both. Use inequalities so there are no gaps and no overlaps.

Reading Off the Table

The 1 or 2-mark questions: “Write down the modal class” and “How many students are taller than 160 cm?” Use the 40 heights from the last card: 6, 11, 14 and 9.

Worked example

1
$modal class: 160 < h \leq 165$ — The modal class is the class with the biggest frequency. 14 is bigger than 6, 11 and 9.
2
$160 < h \leq 165 and 165 < h \leq 170$ — Taller than 160 cm means these two classes. The < sign leaves 160 itself out, so every height in them is more than 160.
3
$14 + 9 = 23$ — Add their frequencies: 23 students are taller than 160 cm.
4
$\frac{23}{40}$ — For a fraction or a probability (one student picked at random), put that count over the total, 40.
5
$\frac{23}{40} \times 100 = 57.5%$ — For a percentage, multiply the fraction by 100.

Common mistake

"How many are taller than 162 cm?" You cannot say exactly: 162 is inside a class, and you do not know the exact heights.

The Class With the Median

The 1-mark question: “Which class contains the median?” The median is the middle value when all the values are in order. You do not need the exact values, only which class the middle one falls in.

Time (min)	Frequency
$0 < t \leq 10$	5
$10 < t \leq 20$	9
$20 < t \leq 30$	8
$30 < t \leq 40$	15

Worked example

1
$5 + 9 + 8 + 15 = 37$ — Add the frequencies to find how many people there are.
2
$\frac{37 + 1}{2} = 19$ — With 37 values in order, the middle one is the 19th: 18 values below it and 18 above it.
3
$0 < t \leq 10 : 5$ — Count up from the first class. 5 people so far, so the 19th is not in 0 < t ≤ 10.
4
$5 + 9 = 14$ — Add the next class. 14 people so far, still short of 19.
5
$14 + 8 = 22$ — Add the next class. 22 people so far, so the 19th person is in this class.
6
$median class: 20 < t \leq 30$ — Write the class, not the 19 and not the frequency 8.

Common mistake

Choosing 30 < t ≤ 40 because it has the biggest frequency, 15. That is the modal class. The median class comes from counting up to the middle.

Key idea

If the total is even, the method is the same. For 40 values, $\frac{40 + 1}{2} = 20.5$ , so count up to the 20th and 21st values. Write the running totals down: they show your method and let you check your count.

Which Method? Read the Question

Check whether the data is a list, a table, or a diagram.

You see	You do
Complete the table	tally each value once, check the total
Which class?	the top end of a class is included
Modal class	the class with the biggest frequency
Class with the median	find the middle position, then count up the running totals
How many more than	add the classes above
Estimate the mean	Estimating the Mean for Grouped Data
Draw a diagram	Histograms (Equal Class Intervals)

Key idea

The modal class is a class, like 160 to 165 cm, not a single number and not the frequency 14.

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