How to Find Averages from a Frequency Table: KS3 Maths

A frequency table lists each data value and how many times it appears. To find the mean, multiply each value by its frequency, add the products, and divide by the total frequency. The mode is the value with the highest frequency. The median is the middle value once data are ordered.

What is a frequency table?

A frequency table organises raw data so you can see, at a glance, how often each value occurs. Instead of listing 30 individual scores, the table shows each distinct score once alongside its frequency — how many times it was recorded.

Score (x) | Frequency (f)
2 | 3
3 | 7
4 | 10
5 | 6
6 | 4
Total | 30

This table will be used in all worked examples below.

How do you find the mode from a frequency table?

The mode is the value with the highest frequency. Scan the frequency column and find the largest number.

From the table above: the highest frequency is 10, which belongs to score 4. The mode is 4.

If two values share the highest frequency, the data is bimodal — there are two modes.

How do you find the median from a frequency table?

The median is the middle value when all data are arranged in order. With a frequency table:

Find the total number of values: here, n = 30.

Identify the position of the median: for an even number of values, the median is halfway between the (n/2)th and (n/2 + 1)th values — i.e. the 15th and 16th values.

Use a running total (cumulative frequency) to locate those positions.

Score (x) | Frequency (f) | Cumulative frequency
2 | 3 | 3
3 | 7 | 10
4 | 10 | 20
5 | 6 | 26
6 | 4 | 30

The 15th and 16th values both fall in the score-4 row (cumulative frequency goes from 10 to 20). Both are 4, so the median = 4.

How do you find the mean from a frequency table?

Add a column for f × x (frequency × value), sum that column, then divide by the total frequency.

Formula: Mean = Σ(f × x) ÷ Σf

Score (x) | Frequency (f) | f × x
2 | 3 | 6
3 | 7 | 21
4 | 10 | 40
5 | 6 | 30
6 | 4 | 24
Totals | 30 | 121

Mean = 121 ÷ 30 = 4.03 (to 2 d.p.)

What is the range from a frequency table?

The range is the difference between the largest and smallest values in the table.

From the table: largest value = 6, smallest = 2. Range = 6 − 2 = 4.

What mistakes are most common in frequency table questions?

Dividing by the number of rows instead of Σf. Always divide by the total of the frequency column, not the number of distinct values.

Reading the frequency as the value. The score (x) and the frequency (f) are in separate columns. A common error is adding the frequency column instead of computing f × x.

Finding the wrong median position. For n values, the median is at position (n + 1)/2 when n is odd. When n is even, average the two middle positions. Use cumulative frequency to locate them.

Frequently asked questions

Why do we multiply value by frequency to find the mean?

Each value in the table appears more than once. A score of 3 recorded 7 times contributes 3 × 7 = 21 to the total. Summing f × x for all rows gives the same result as writing out every individual data value and adding them up — it is just a much faster method.

What if the frequency table uses class intervals (grouped data)?

With grouped data (e.g. 0–9, 10–19, …), you use the midpoint of each interval as the value of x. This gives an estimate of the mean, not the exact value, because you do not know where within the interval each data point falls.

Can the mean be a value not in the table?

Yes, and this is perfectly normal. The mean of 4.03 does not appear as a score in the table above. The mean is a calculated measure of centre — it does not have to be a value that was actually recorded.

How do I decide which average to report?

Use the mean for symmetric data without extreme values. Use the median when data are skewed or contain outliers — it is not affected by unusually large or small values. Use the mode when the most frequent value is what matters most (e.g. the most popular shoe size a shop should stock).

For Socratic statistics practice including frequency tables, see aitutors.me.