Averages from Frequency Tables

A frequency table is a short way to write a list in which values repeat. The first column gives the value that was recorded. The frequency says how many times that value was recorded.

Here, two pupils went swimming 0 times, three pupils went 1 time and one pupil went 2 times. Written out in full, the list is 0, 0, 1, 1, 1, 2

Adding the frequencies gives the number of people: 2+3+1=6 pupils. The table has only 3 rows, but it shows 6 pieces of data. The two pupils who went 0 times still count.

The same information is sometimes drawn as a bar chart or a vertical line chart. The height of each bar or line is the frequency.

In the exam you are usually asked to find the mode, the median, the range or the total from a frequency table, for 1 or 2 marks each, or to work out the mean, for 3 marks. Harder questions give a table with a missing frequency, or ask you to explain the mistake in someone's working.

The mode is the value with the highest frequency. The question may call it the modal value. Give the value from the first column, not the frequency.

The range is the largest value minus the smallest value. Use only values that were recorded: a row with frequency 0 means nobody had that value, so leave it out.

To find the total of all the data, multiply each value by its frequency, then add the results. For example, a value of 3 with frequency 5 adds 3\times5=15 to the total, because five people each had 3

Add a third column to the table for value × frequency. It helps you to include every row, and a row with frequency 1 is easy to miss.

To find the mean, divide the total of the data by the total frequency:

\text{mean}=\frac{\text{total of the data}}{\text{total frequency}}

Divide by the number of people, not by the number of rows. A mean must lie between the smallest and the largest value, so if your mean is bigger than every value in the table, something has gone wrong.

The median is the middle value when all the data is in order. A frequency table already has its values in order, so you do not need to write out the list.

If there are n values, the median is the value in position

\frac{n+1}{2}

For 21 values that is the 11th value. For 16 values it is 8.5, which means halfway between the 8th and 9th values.

To find which value is in that position, keep a running total of the frequencies as you go down the table. The position, such as the 11th, is not the median: the median is the value found in that position.

Some questions leave one frequency out and tell you a total instead.

• If you are told the total frequency, add the frequencies you know and subtract from it.
• If you are told the total of the data, work out value × frequency for the rows you know, subtract from the total, then divide by the value in the missing row.

A frequency counts people or things, so it must be a whole number. If you get a decimal, check your working.

If you know a mean, you can work back to the total:

\text{total}=\text{mean}\times\text{number of values}

For example, 10 pupils with a mean score of 4 scored 4\times10=40 altogether. Another 15 pupils with a mean of 6 scored 6\times15=90

To find the mean of all 25 pupils, add the totals and add the numbers of pupils: (40+90)\div25=5.2. Taking the mean of the two means, (4+6)\div2=5, is wrong, because the groups are different sizes.

If more data is added to a table, change both the total of the data and the total frequency before you divide again. If a wrong value is corrected, the total frequency stays the same, but the total of the data changes.

• Adding one column instead of multiplying. To find the total, multiply each value by its frequency, then add the products.
• Working out 0\times4 as 4. Zero times any frequency is 0.
• Dividing by the number of rows. For the mean, divide by the total frequency, the number of people.
• Giving the highest frequency as the mode. The mode is the value next to the highest frequency.
• Finding the middle of the table, not the middle of the data. Use (n+1)\div2 and a running total of the frequencies.
• Leaving out a row. Check that you have one product for every row, including rows with frequency 1