Measures of central tendency
How to find and choose the mean, median and mode, including from frequency tables and grouped data, and weighted means. Statistics is worth a good share of DSE Paper 1 marks.
What you will learn
- 30.1 Mean, median and mode/modal class: features and limits. One extreme value changes the mean a lot but not the median.
- 30.2 Mean, median and mode of ungrouped data.
- 30.3 Mean, median and modal class of grouped data (these are estimates).
- 30.4 Uses and abuses in daily life.
- NF 30.5 Effect of adding a constant to every item, or multiplying every item by a constant.
- 30.6–30.7 Weighted mean (report-card scores, university admission scores).
You need first: Geometry and data basics
Key terms
| English | 中文 |
|---|---|
| mean (average) | 平均數 |
| median | 中位數 |
| mode, modal class | 眾數，眾數組 |
| weighted mean, weight | 加權平均數，權 |
| ungrouped / grouped data | 不分組數據／分組數據 |
| class mark | 組中點 |
| extreme value | 極端值 |
Standard steps
Ungrouped data
- Mean = sum ÷ number of items.
- Median:
- Put the data in order.
- With n items, the median is the ((n + 1)/2)th item.
- If n is even, take the mean of the two middle items.
- Mode = the most frequent value (there can be more than one).
Frequency table or grouped data
- Mean = Σ(x × f) ÷ Σf, where x is the value (or class mark for grouped data).
- Class mark = (lower limit + upper limit) ÷ 2.
- Median: find the class containing the ((Σf + 1)/2)th item using cumulative frequency, or read it from a cumulative frequency curve.
Find a missing value
Total = mean × number of items. Subtract the known values.
Weighted mean
Weighted mean = Σ(value × weight) ÷ Σ(weight).
Effects of operations (NF)
| Every item | Mean, median, mode |
|---|---|
| + c | all increase by c |
| × k | all multiplied by k |
Worked examples
Example 1: Data: 3, 5, 5, 6, 8, 9, 13.
mean = 49 ÷ 7 = 7
median = 4th item = 6
mode = 5Example 2: The mean of 5 numbers is 12. Four of them are 10, 11, 15 and 9. Find the fifth.
total = 12 × 5 = 60
fifth = 60 − (10 + 11 + 15 + 9) = 15Example 3: Grouped data.
| Score | 0–9 | 10–19 | 20–29 | 30–39 |
|---|---|---|---|---|
| Class mark | 4.5 | 14.5 | 24.5 | 34.5 |
| Frequency | 4 | 7 | 6 | 3 |
mean ≈ (4.5×4 + 14.5×7 + 24.5×6 + 34.5×3) ÷ 20
= 370 ÷ 20
= 18.5Example 4: Coursework counts 40% and the exam 60%. A student scores 70 in coursework and 85 in the exam.
Weighted mean = 0.4 × 70 + 0.6 × 85 = 79.
Common mistakes
- Finding the median without ordering the data first.
- Taking the median as "(n + 1)/2" itself rather than the item in that position.
- In a frequency table, dividing by the number of rows instead of the total frequency.
- Using the class limits instead of the class mark.
- Reporting a grouped mean as exact.
How the DSE tests it
- A(1): a frequency table or stem-and-leaf diagram; find the mean or median, then find an unknown value from a given mean or median.
- Paper 2: which measure changes when a value is added or removed; NF effects of + c and × k.
- F6 statistics questions start from the same table.
Questions you might not ask in class
Which average should I use?
Use the median when there are extreme values (for example, salaries). Use the mean when every value matters. Use the mode for "most popular" (for example, shoe sizes).
Why is the grouped mean only an estimate?
We don't know the exact values in each class, so we pretend they are all at the middle.
What if every value appears once? Is there no mode?
Yes, that's allowed. The data has no mode.