Averages and frequency tables
An average is one number that represents a set of data; the range shows how spread out it is. Frequency tables are a compact way to hold lots of data, and you need to find each average straight from the table.
Part 1 of 3: Learn it
In short
- Mean = total ÷ how many; median = middle value in order; mode = most common; range = biggest − smallest.
- Mean from a frequency table = Σ(value × frequency) ÷ Σ frequency.
- For grouped data, use the midpoints, so the mean is only an estimate.
Where this is in your specification
Spec points: DfE S3 and S4
| Board | Topic: Statistics and charts |
|---|---|
| DfE content | S1-S6 |
| AQA 8300 | Statistics |
| Edexcel 1MA1 | Statistics |
| OCR J560 | 14 Statistics |
| Eduqas C300 | HS1 to HS7 |
| Cambridge IGCSE 0580 | E9.1 to E9.7 |
| National 5 C847 75 | Statistical skills: standard deviation, semi-interquartile range, scattergraphs |
The four measures
| Measure | How to find it | Good to know |
|---|---|---|
| Mean | add the values, divide by how many | uses every value, but an outlier can drag it |
| Median | put in order, take the middle one | for an even count, take halfway between the middle two |
| Mode | the most common value | the only average for non-number data such as colours |
| Range | largest − smallest | measures spread, not an average |
Mean from a frequency table
- Add a column for value × frequency.
- Add up that column, and add up the frequencies.
- Divide the first total by the second.
For the median, find its position, (n + 1) ÷ 2, then count down the frequency column until you reach it.
Grouped data
When values are in classes such as 10 < t ≤ 20, you do not know the exact values. Use the midpoint of each class as its value, so the mean is an estimate. The modal class is the class with the highest frequency, and you can say which class contains the median.
Find the mean of 4, 7, 7, 9 and 13.
Show the answer
40 ÷ 5 = 8.
Part 2 of 3: See it worked
Worked examples
Example 1
Goals scored in 20 matches: 0 goals 4 times, 1 goal 7 times, 2 goals 5 times, 3 goals 3 times, 4 goals once. Find the mean, median and mode.
- Σ(goals × frequency) = 0 + 7 + 10 + 9 + 4 = 30; mean = 30 ÷ 20 = 1.5
- Median: between the 10th and 11th values. Running totals 4, 11, so both are 1
- Mode: 1 goal (frequency 7)
Answer: Mean 1.5, median 1, mode 1.
Example 2
Times (minutes): 0 < t ≤ 10: 3 people; 10 < t ≤ 20: 8; 20 < t ≤ 30: 6; 30 < t ≤ 40: 3. Estimate the mean.
- Midpoints: 5, 15, 25, 35
- Σ(midpoint × frequency) = 15 + 120 + 150 + 105 = 390
- Total frequency = 20; 390 ÷ 20 = 19.5
Answer: About 19.5 minutes.
Common mistakes
- Dividing by the number of rows rather than the total frequency.
- Finding the median without putting the data in order first.
- Giving the frequency as the mode instead of the value with that frequency.
- Calling a grouped mean exact rather than an estimate.
Find the median of 3, 8, 1, 6, 10, 4.
Show the answer
In order 1, 3, 4, 6, 8, 10, so (4 + 6) ÷ 2 = 5.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Find the mean of 4, 7, 7, 9 and 13.
40 ÷ 5 = 8.
Find the median of 3, 8, 1, 6, 10, 4.
In order 1, 3, 4, 6, 8, 10, so (4 + 6) ÷ 2 = 5.
Why is a mean from grouped data only an estimate?
The exact values are not known, so the midpoint of each class stands in for them.
Jobs that use this
- Data analyst-statistician (opens a new tab)
- Economist (opens a new tab)
- Data scientist (opens a new tab)
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses