Frequency tables

Count first

You asked 40 students: "What is your favourite sport?" You now have 40 answers in a long list. The first step is to count how many times each answer appears.

A frequency table lists each value and its frequency: how many times it appeared.

On paper, we count with tally marks. For each answer we make a stroke, and every fifth stroke crosses the four before it. A group of five is quick to read, and it stops us losing count.

Try it in Python

The program counts the 40 answers and prints the frequency table, with the share of each sport.

What to check

• The frequencies add up to the number of students asked: $18 + 10 + 6 + 6 = 40$.
• The percentages add up to $100\%$: $45 + 25 + 15 + 15 = 100$.

If they do not add up, you have lost or doubled an answer. This check catches mistakes.

Relative frequency

The relative frequency is the frequency divided by the total: $\dfrac{18}{40} = 0.45$. As a percentage it is $45\%$. It lets you compare two surveys of different sizes.

In class A, 18 of 40 students chose cricket. In class B, 21 of 60 chose cricket. Class B has more cricket fans, but a smaller share: $\dfrac{18}{40} = 45\%$ against $\dfrac{21}{60} = 35\%$. So cricket is more popular in class A.

You should find that the taller bar is not always the class with more fans. Make class B larger with the same number of fans, and its bar shrinks.

Grouping numerical data

If the data is numerical and has many different values, a row for each value makes the table too long. We put the values in groups, called class intervals. The marks out of 30 of 20 students could look like this:

The groups must not overlap, and every value must go in exactly one group. The graph of such a table is a histogram, in the last lesson of this course.

Practise with new numbers

Common mistakes

• Forgetting to check the total. The frequencies must add up to the number of cases.
• Comparing counts of surveys of different sizes. Use the relative frequency.
• Overlapping groups. A mark of $10$ must be in only one group.
• Worrying about $99\%$ or $101\%$. A small rounding difference is fine. A large one means a mistake.

Remember these

• A frequency table counts how many times each value appears.
• Relative frequency is the frequency divided by the total.
• The frequencies add up to the total. The percentages add up to 100.
• Group numerical data into class intervals.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A survey of 50 students has 20 who chose football. What percent is that?

2. What should the frequencies of a table add up to?

3. In a class of 25, 5 students chose chess. What is the relative frequency of chess?

4. In a table of marks, the frequencies are $3$, $8$ and $9$, and 21 students were asked. What went wrong?