The idea, step by step
Describe the whole distribution
A roughly symmetric distribution has similar left and right sides. Positive skew has a longer tail towards large values; negative skew has a longer tail towards small values. Look for clusters, gaps and unusual observations as well as the overall shape.
Skew names the tail direction, not the side holding most observations. A long high tail can pull the mean above the median, so the median may better represent a typical value.
Centre needs spread
Mean is total divided by count. Median is the middle ordered value, or the average of the two middle values for an even count. Range is maximum minus minimum. Each result needs the variable’s units.
Two groups can have identical means but very different variability. Choose summaries that answer the question and describe both typical level and variation; one average cannot describe the entire distribution.
A common trap
A larger mean means every group member has a larger value.
Instead: Distributions may overlap; a group summary does not order every individual.
See the thinking
A skewed set
Find mean, median and range for waits 2, 3, 3, 4, 18 minutes.
- Total 30 divided by 5 gives mean 6.
- The middle ordered value is 3.
- Range is .
Equal centre, unequal spread
Compare A: 9, 10, 11 and B: 2, 10, 18.
- Both totals 30 give mean 10.
- Ranges are 2 and 16.
Check your understanding
Find median of 8, 2, 6, 4.
Show answer
5.
Order 2, 4, 6, 8 and average 4 and 6.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- Find mean, median and range for 4, 6, 6, 8, 11.2 marks
- A long tail extends to high values. Name the skew.2 marks
- Groups share median 15 but ranges 4 and 20. State one comparison.2 marks
- Why use median for strongly right-skewed income?2 marks
- Five observations have mean 12. Find their total.2 marks
- Find the mode of 2, 2, 3, 5, 5, 6.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Find mean, median and range for 4, 6, 6, 8, 11.2 marks
Total 35 divided by count 5; third value 6; .
Mean 7; median 6; range 7. - A long tail extends to high values. Name the skew.2 marks
The tail points towards larger values.
Positive/right skew. - Groups share median 15 but ranges 4 and 20. State one comparison.2 marks
The medians match; ranges differ.
The second group has wider overall spread. - Why use median for strongly right-skewed income?2 marks
The best summary also depends on the question.
It is less affected by a few very high incomes than the mean. - Five observations have mean 12. Find their total.2 marks
Mean multiplied by count.
60. - Find the mode of 2, 2, 3, 5, 5, 6.2 marks
Each occurs twice, the highest frequency.
Two modes: 2 and 5.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 General Mathematics: statistical investigation, univariate data, sampling, displays, summaries and interpretation.
Official curriculum source ↗