Averages from a Frequency Table

Why a frequency table changes things

You already know the mean, median and mode for a short list of numbers. But data often arrives pre-counted in a frequency table, where one row can stand for dozens of values. Writing them all out would be slow, so we use shortcuts that work straight from the table.

Here is the data we will use: the number of goals scored by a team in 20 matches.

Goals (x)	Frequency (f)
0	4
1	6
2	7
3	2
4	1

First, check the total frequency: 4 + 6 + 7 + 2 + 1 = 20. That matches the 20 matches, so the table is complete.

The mode

The mode is the value that appears most often — the row with the highest frequency.

The biggest frequency is 7, in the "2 goals" row. So the mode is 2 goals.

Watch out: the mode is the value, not the frequency. The answer is 2 (goals), not 7 (matches). And the highest value (4) is not automatically the mode.

The mean using an fx column

To find the mean you need the total of all the goals, then divide by how many matches there were. Multiplying each value by its frequency does the adding-up quickly. This new column is called fx (value × frequency).

Goals (x)	Frequency (f)	f × x
0	4	0
1	6	6
2	7	14
3	2	6
4	1	4
Total	20	30

Now apply the formula:

Mean = total of (f × x) ÷ total frequency

Mean = 30 ÷ 20 = 1.5 goals per match.

The most common mistake: dividing by 5 (the number of rows) instead of 20 (the total frequency). Always divide by the total frequency. An average of 1.5 goals makes sense for this team; an "average" of 6 would not.

The median

The median is the middle value when the data is in order. A frequency table is already in order, so you just count to the middle position.

1. There are 20 values, so the median is between the 10th and 11th values. (For n values the middle position is the (n+1)/2 th; here (20+1)/2 = 10.5, so we average the 10th and 11th.)
2. Count through the frequencies:
3. 0 goals fills positions 1–4.
4. 1 goal fills positions 5–10.
5. 2 goals fills positions 11–17.
6. The 10th value is a 1; the 11th value is a 2.
7. Median = (1 + 2) ÷ 2 = 1.5 goals.

A running total like this (4, then 10, then 17, …) is called the cumulative frequency, and it is the key to locating the median quickly.

Quick summary

Average	How to find it	Answer here
Mode	Value with the highest frequency	2
Median	Value at the middle position	1.5
Mean	Total of (f × x) ÷ total frequency	1.5

Activity: roll and record

1. Roll a die 30 times and record results in a frequency table (values 1–6).
2. Find the mode by spotting the highest frequency.
3. Build an fx column and calculate the mean. (For a fair die the mean should be near 3.5.)
4. Use cumulative counting to find the median.
5. Compare your mean to the theoretical 3.5 and discuss any difference.

Why this matters

Frequency tables are how real data is usually stored — survey results, test scores, sports stats. Being able to summarise them with one number lets you compare groups fast. To revisit the core ideas of each average, see mean, median, mode and range, and to picture the data, try bar charts and pictograms.