Understand the idea.
Summaries answer different questions
The mean uses the total of all observations; the median uses their ordered middle; the mode uses frequency and may have several values. The range measures the maximum-to-minimum span, rather than the typical distance from the centre.
Inspect the pattern as well as the summaries. Clusters concentrate observations, gaps contain none, and a long tail toward large values indicates right skew. Two sets can have equal means and ranges while differing in their frequencies and shape.
Investigate unusual values
An unusually distant observation may be a recording error or a genuine feature of the situation. Check its source before changing it. Correct a confirmed error and record the correction; do not remove a valid observation just to improve the appearance of the data.
An extreme value can strongly affect the mean and range because they use its magnitude. The median is often less affected because it depends on ordered position. Explain which centre describes the situation usefully, and disclose any comparison made with an observation omitted.
| Data | Mean | Median | Range |
|---|---|---|---|
| 2, 3, 3, 4, 18 | 6 | 3 | 16 |
| 2, 3, 3, 4 | 3 | 3 | 2 |
See the method in action.
A long upper tail
For 2, 3, 3, 4, 18 calculate mean, median and range.
Worked solution
- 1
Calculate the mean.
The mean includes the magnitude of every observation.
- 2
Read the median.
The third of five ordered observations is the middle value.
- 3
Find the range.
Subtract the smallest value from the largest.
- 4
Explain the difference between the centres.
Four observations cluster from to , while the isolated pulls the mean upward. The median is nearer this cluster, but the unusual value still needs investigation rather than automatic deletion.
Mean 6; median 3; range 16.
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 |
Worked solution
- 1
Count all observations.
Frequencies count repetitions; there are ten observations, not four.
- 2
Calculate the frequency-weighted total.
Each value contributes once for every occurrence.
- 3
Divide the total by the count.
The mean is the total value of all ten observations divided by ten.
- 4
Check the size of the result.
A mean must lie within the observed minimum and maximum, so this necessary size check is satisfied.
2.4.
Try it yourself.
Check your understanding
For 4, 4, 5, 5, 22, which centre is more resistant to 22?
Show answer & reasoning
The median.
The ordered middle value is , while the mean is . The large value raises the mean, so the median is the more resistant centre here.
Keep practising
6 worksheet questions · 17 marks
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 ↗