Statistics
Averages and range
In this note
- 9.3Find the mean, median, mode and range of a list or a frequency table.
- 9.3Choose the best average for a purpose, and work backwards from a mean.
Before you start: ordering numbers and division. Next: Grouped data.
Formula bank
Averages of a list
An average is one number that represents the data. Data: 3, 7, 4, 9, 7, 2, 8. In order: 2, 3, 4, 7, 7, 8, 9. Mean \(=\tfrac{40}{7}=5.71\); median = 7 (the 4th of 7); mode = 7; range = 9 − 2 = 7.
General methodHow to find each average
- Mean: add everything, divide by how many there are.
- Median: put the values in ORDER first, then take the middle one. Even number of values? Halfway between the two middle ones.
- Mode: the value that appears most.
- Range: largest − smallest (one number, not '2 to 9').
Example Find the median and mean of 12, 15, 9, 20, 14, 10.
Six values: the median is halfway between the 3rd and 4th.
- in order: 9, 10, 12, 14, 15, 20
- median \(=\tfrac{12+14}{2}=13\)
- mean \(=\tfrac{80}{6}=13.33\)
Answer median 13, mean 13.33
Example The mean of five numbers is 8. Four of them are 6, 9, 5 and 10. Find the fifth.
Total = mean × how many.
- total \(=5\times8=40\)
- \(40-(6+9+5+10)=10\)
Answer 10
Which average? The mean uses every value but is pulled by one very big or small value (an outlier). The median is not affected by outliers. The mode is the only one for non-number data (favourite colour).
Try it yourself
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Answer 11
Show answer
Answer 5.71
Averages from a frequency table
| Goals x | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|
| Frequency f | 4 | 7 | 5 | 3 | 1 |
| f × x | 0 | 7 | 10 | 9 | 4 |
General methodHow to find averages from a frequency table
- Mean: add a row f × x; mean \(=\sum fx\div\sum f\) (divide by the number of matches, not by 5).
- Median: find the position \(\tfrac{n+1}{2}\), then count along the frequencies to it.
- Mode: the value with the highest frequency (not the frequency itself).
Example Find the mean, median and mode of the goals table.
20 matches.
- mean \(=\tfrac{30}{20}=1.5\)
- median: 10th and 11th values; first 4 are 0, next 7 are 1, so both are 1
- mode: 1 (frequency 7)
Answer mean 1.5, median 1, mode 1
Try it yourself
Show answer
Answer 6.85
Ways they test it
Type 1 Average of a list
"Find the median." (1-2 marks)
Spot it a list of values.
- Order first.
- Middle position.
Type 2 Frequency table mean
"Calculate the mean number of goals." (3 marks)
Spot it a frequency table.
- \(\sum fx\div\sum f\).
For example goals table
- \(30\div20\)
Answer 1.5
Type 3 Missing value
"The mean is 8. Find the missing number." (2 marks)
Spot it a mean and a gap.
- Total = mean × n.
- Subtract the known values.
Type 4 Choose and explain
"Which average best represents…? Give a reason." (1 mark)
Spot it 'which average'.
- Outlier? median.
- Non-number data? mode.
Mistakes to avoid
- ✗ Median without ordering✓ order the data first
- ✗ Table mean ÷ number of rows✓ ÷ \(\sum f\)Divide by the number of data items.
- ✗ Mode = the highest frequency (7)✓ the VALUE with that frequency (1)
- ✗ Range written as '2 to 9'✓ one number: 7
Summary
- Mean = total ÷ count; median = middle in order; mode = most common; range = biggest − smallest.
- Frequency table: mean \(=\sum fx\div\sum f\).
- Working back: total = mean × count.
If you see …
- an outliermedian is better than mean.
- a frequency tableadd an f × x row.
Practice
Practice: easy, medium, hard
Easy
Show answer
Answer 7
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Answer 7
Show answer
Answer 13
Medium
Guide me
What is it asking? a mean
What do we know? \(\sum fx=30\)
Plan \(\div20\)
Show working
- 1.5
Answer 1.5
Guide me
What is it asking? a value
What do we know? total 40
Plan 40 − 30
Show working
- 10
Answer 10
Guide me
What is it asking? a median
What do we know? 10th and 11th values
Plan count along: 4 zeros, then 7 ones
Show working
- 1
Answer 1
Hard
Guide me
What is it asking? a mean
What do we know? \(\sum fx=137\)
Plan \(\div20\)
Show working
- 6.85
Answer 6.85
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(\sum fx=137\) | M1 | ||
| \(\div20\) | M1 | ||
| 6.85 | A1 | ||
| Total: 3 marks | |||
Guide me
What is it asking? a mean
What do we know? total 80
Plan \(\div6\)
Show working
- 13.33
Answer 13.33
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 80/6 | M1 | ||
| 13.33 | A1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? a median
What do we know? 10th and 11th values
Plan 2 + 5 = 7, so both are in the 8 sevens
Show working
- 7
Answer 7
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 7 | B1 | ||
| Total: 1 mark | |||
Can I do this?
- I can find the mean, median, mode and range of a list.
- I can find averages from a frequency table.
- I can work back from a mean and choose the best average.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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