The idea, step by step
The mean shares the total equally
The arithmetic mean is the total divided by the number of observations. For a frequency table, multiply each value by its frequency before adding, and divide by the sum of frequencies. A mean need not be an observed value. State its unit and remember that it can be pulled towards unusually large or small observations.
The median and mode answer different questions
Order the data before finding the median. For odd n it is the middle observation; for even n it is the mean of the two middle observations. The mode is a most frequent value; there may be several or no unique mode. Median is often more resistant to outliers than mean. A mode can describe categories, whereas an arithmetic mean of category codes is not meaningful.
A common trap
Finding the middle of the original, unsorted list.
Instead: The median uses ordered values. Sort or use cumulative frequencies first.
See the thinking
An unusual value
Find mean, median and mode for 2,3,3,4,18. Which centre is more typical?
- The sum is 30, so mean 30/5=6.
- The middle ordered value is 3;3 occurs most often.
- The unusually large 18 pulls the mean upwards.
Use frequencies
Scores 1,2,4 have frequencies 2,3,1 respectively. Find the mean and median.
- Total score=1×2+2×3+4×1=12; total frequency 6.
- Mean 12/6=2.
- Ordered scores are 1,1,2,2,2,4, so the middle pair is 2,2.
Check your understanding
Find the median of 2,9,4,6.
Show answer
5.
Order 2,4,6,9 and average the middle pair: (4+6)/2=5.
Your turn
6 questions · 14 marks · Work at your own pace.
Read the questions online
- Find mean, median and mode for 4,5,5,6,10.2 marks
- The mean of 6 values is 8. Find their total.2 marks
- Four scores are 3,7,8,x and their mean is 6. Find x.2 marks
- Find the median and mode of 1,2,2,5,5,9.2 marks
- A set has mean 12. Adding a new observation of 12 changes what? Explain the mean.3 marks
- A small business has salaries 40,42,43,45 and 180 thousand dollars. Calculate mean and median and recommend one summary of a typical worker's pay.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Find mean, median and mode for 4,5,5,6,10.2 marks
Sum 30 over 5 gives 6; the middle value is 5 and 5 is most frequent.
Mean 6; median 5; mode 5. - The mean of 6 values is 8. Find their total.2 marks
Total=mean×number=8×6=48.
48. - Four scores are 3,7,8,x and their mean is 6. Find x.2 marks
Total must be 24; known total 18, so x=6.
6. - Find the median and mode of 1,2,2,5,5,9.2 marks
The middle pair is 2,5.
Both 2 and 5 occur twice.
Median 3.5; modes 2 and 5. - A set has mean 12. Adding a new observation of 12 changes what? Explain the mean.3 marks
The total increases by 12 at the same time the count increases by 1.
If old total 12n, new mean(12n+12)/(n+1)=12.
The mean stays 12. - A small business has salaries 40,42,43,45 and 180 thousand dollars. Calculate mean and median and recommend one summary of a typical worker's pay.3 marks
Sum 350 divided by 5 is 70.
The middle value is 43; the 180 salary pulls the mean above most salaries.
Mean 70; median 43 thousand dollars. Median is more typical of this group.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 Mathematics: counting, binomial expansion, data, samples, summary measures and introductory normal distributions. These lessons provide the Counting and Statistics strand of the approved Methods preparation route.
Official curriculum source ↗