The idea, step by step
Three different ideas of centre
The mean shares the total equally: add the values and divide by the number of observations. The median is the middle of the ordered list; for an even number of observations, average the two middle values. The mode is the most frequent value, not its frequency.
These summaries answer different questions. A mean need not be an observed value. There may be more than one mode, or no mode when no value repeats more frequently than the others.
Spread and frequency
The range is the highest value minus the lowest. It describes the span, not a centre. Range uses only two observations, so an unusual extreme can change it strongly. Include the data units in a mean, median or range.
A frequency table compresses repeated observations. To find its mean, multiply each value by its frequency, add those contributions and divide by the total frequency. Do not average the frequencies or treat each distinct value as one observation.
A common trap
The median is halfway between the minimum and maximum.
Instead: Order all observations and locate the middle positions. The midpoint of the extremes need not be the median.
See the thinking
An odd-sized dataset
Find the mean, median, mode and range of 3, 5, 5, 7, 10.
- The total is 30 and there are five values. Mean = 30÷5 = 6.
- The third ordered value is 5, so the median is 5. The value 5 occurs most often.
- Range = 10−3 = 7.
Even number of observations and two modes
Summarise 12, 8, 4, 4, 6, 8.
- Order: 4, 4, 6, 8, 8, 12. The sum is 42.
- Mean = 42÷6 = 7. Median = (6+8)÷2 = 7.
- Both 4 and 8 appear twice, so both are modes. Range = 12−4 = 8.
Check your understanding
Find the median of 2, 6, 8, 12.
Show answer
7.
The two middle values are 6 and 8; their average is (6+8)÷2=7.
Your turn
6 questions · 15 marks · Work at your own pace.
Read the questions online
- The data are 4, 6, 6, 8, 11.5 marks
- Find the median of 9, 3, 7, 5, 11, 13.2 marks
- What is the mode of 2, 4, 6, 8?2 marks
- Every score in a dataset increases by 3. What happens to the mean and range?2 marks
- Four scores have a mean of 9. Three scores are 6, 8 and 10. Find the fourth.2 marks
- Find the mean from this frequency table.2 marks
Score frequencies Score Frequency 2 1 4 2 6 1
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- The data are 4, 6, 6, 8, 11.5 marks
a) The sum is 35; 35÷5=7.
b) The third of five ordered values is 6.
c) 6 occurs twice.
d) 11−4=7.
a) 7.; b) 6.; c) 6.; d) 7. - Find the median of 9, 3, 7, 5, 11, 13.2 marks
Order: 3, 5, 7, 9, 11, 13.
The middle values are 7 and 9; (7+9)÷2=8.
8. - What is the mode of 2, 4, 6, 8?2 marks
Every value occurs once, so no value is more frequent.
There is no mode. - Every score in a dataset increases by 3. What happens to the mean and range?2 marks
Adding 3 to every value adds 3 to the equal share.
The new range is (maximum+3)−(minimum+3), so the added amounts cancel.
The mean increases by 3; the range stays the same. - Four scores have a mean of 9. Three scores are 6, 8 and 10. Find the fourth.2 marks
Required total = 4×9=36.
Known total = 6+8+10=24.
Missing score = 36−24=12.
12. - Find the mean from this frequency table.2 marks
Score frequencies Score Frequency 2 1 4 2 6 1 Total frequency is 1+2+1=4.
Weighted sum is 2×1+4×2+6×1=16.
Mean=16÷4=4.
4.
Curriculum reference & lesson scope
Australian Curriculum v9 / South Australian Mathematics Curriculum
Year 7 statistical variables, representations, summary statistics, outliers and an original small investigation; deeper sampling methods develop in Year 8.
Official curriculum source ↗