The idea, step by step
Summaries answer different questions
The mean is total value divided by the number of observations. The median is the middle ordered value, averaging the middle pair for an even sample. The mode is the most frequent value and may not be unique. The range describes the gap from minimum to maximum, not a typical distance from the centre.
Look at shape too. A cluster is a concentration of observations; a gap has none. A long tail toward large values makes a right-skewed distribution. Two sets can share a mean and range but have different patterns.
Investigate unusual values
An outlier lies unusually far from the main pattern. It could be a recording error or a genuine important observation. Check the source before changing data. Correct a confirmed error, but do not delete a real result merely to obtain a nicer mean.
The mean and range can change greatly when an extreme value is added; the median is often less affected. Report both centres when useful and explain why they differ. If results are compared with and without an unusual value, describe that decision openly.
| Data | Mean | Median | Range |
|---|---|---|---|
| 2, 3, 3, 4, 18 | 6 | 3 | 16 |
| 2, 3, 3, 4 | 3 | 3 | 2 |
A common trap
The mode is the largest value, and the mean is always the best centre.
Instead: Mode refers to frequency. Choose a useful centre after considering context, shape and unusual values.
See the thinking
A long upper tail
For 2, 3, 3, 4, 18 calculate mean, median and range.
- Total , so mean .
- Median is 3; range .
- The isolated 18 pulls the mean upward; the median describes the main cluster more closely.
A frequency-weighted mean
Values 1, 2, 3, 4 have frequencies 2, 3, 4, 1. Find the mean.
| Value | Frequency |
|---|---|
| 1 | 2 |
| 2 | 3 |
| 3 | 4 |
| 4 | 1 |
- Total frequency .
- Total value .
- Mean .
Check your understanding
For 4, 4, 5, 5, 22, which centre is more resistant to 22?
Show answer
The median.
Median is 5; mean is 8 because the large value pulls it upward.
Your turn
6 questions · 17 marks · Work at your own pace.
Read the questions online
- Find mean, median, mode and range of 2, 4, 4, 6, 9.4 marks
- Compare mean and median for 1, 2, 2, 3, 17.3 marks
- Is an inconvenient but valid long wait a reason to delete a value?2 marks
- Find the mean: values 0, 1, 2, 3 with frequencies 1, 3, 4, 2.3 marks
- Most values lie near 10, with a few near 40. Describe the likely tail direction.2 marks
- Must two sets with mean 12 and range 4 be identical? Give a counterexample.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Find mean, median, mode and range of 2, 4, 4, 6, 9.4 marks
Total , giving mean 5.
Middle and most frequent are 4; .
Mean 5; median 4; mode 4; range 7. - Compare mean and median for 1, 2, 2, 3, 17.3 marks
Total , mean .
17 pulls the mean above the main cluster; the middle remains 2.
Mean 5; median 2. - Is an inconvenient but valid long wait a reason to delete a value?2 marks
The value may reveal a real problem.
Deleting valid data merely to improve results would misrepresent the experience.
No. Investigate and explain it. - Find the mean: values 0, 1, 2, 3 with frequencies 1, 3, 4, 2.3 marks
Total frequency .
Total value ; .
1.7. - Most values lie near 10, with a few near 40. Describe the likely tail direction.2 marks
Most observations cluster low.
The few large values extend the upper end.
Right-skewed: a tail toward larger values. - Must two sets with mean 12 and range 4 be identical? Give a counterexample.3 marks
Both totals are 60 for five values, giving mean 12.
Both run from 10 to 14, but frequencies differ.
No: 10, 11, 12, 13, 14 and 10, 10, 12, 14, 14.
Curriculum reference & lesson scope
Australian Curriculum v9 / SA Curriculum Mathematics
Year 9 sampling, comparative distributions, centre/spread/outliers, displays, critique and statistical investigations.
Official curriculum source ↗