Measures of dispersion
Range, inter-quartile range, box-and-whisker diagrams and standard deviation, and (NF) standard scores, the normal distribution and the effects of operations on data. Statistics is worth a good share of DSE Paper 1 marks.
What you will learn
- 17.1 What dispersion means.
- 17.2 Range and inter-quartile range (IQR).
- 17.3 Box-and-whisker diagrams (boxplots); compare distributions with them.
- 17.4 Standard deviation for grouped and ungrouped data; variance = (standard deviation)². The population formula σ = √[Σ(x − μ)² / N].
- 17.5 Compare dispersions using a suitable measure.
- NF 17.6 Standard scores; the normal distribution in real-life problems (percentages are given, no table).
- NF 17.7 Effect on dispersion of adding a constant to every item, or multiplying every item by a constant.
You need first: Measures of central tendency, Geometry and data basics
Key terms
| English | 中文 |
|---|---|
| dispersion | 離差 |
| range | 分佈域 |
| quartile (lower, upper) | 四分位數（下，上） |
| inter-quartile range | 四分位數間距 |
| box-and-whisker diagram | 框線圖 |
| standard deviation | 標準差 |
| variance | 方差 |
| standard score | 標準分 |
| normal distribution | 正態分佈 |
Standard steps
Range and IQR
- Range = largest − smallest.
- Order the data. Find the median Q₂.
- Lower quartile Q₁ = median of the lower half; upper quartile Q₃ = median of the upper half. (This is the usual Hong Kong textbook method; check the student's book.)
- IQR = Q₃ − Q₁.
- From a cumulative frequency curve, read Q₁ at ¼ and Q₃ at ¾ of the total.
Box-and-whisker diagram
Five numbers: minimum, Q₁, median, Q₃, maximum. The box is Q₁ to Q₃; the whiskers go to the minimum and maximum.
Standard deviation (use the calculator)
- Put the calculator in statistics (SD) mode.
- Enter the data (with frequencies for a table; class marks for grouped data).
- Read σx (population standard deviation), not sx. The DSE uses σ.
- Write down the value; variance = σ².
Standard score (NF)
z = (x − mean) ÷ standard deviation. Higher z means better relative performance.
- Find the mean and standard deviation of the group.
- Work out z = (x − mean) ÷ standard deviation.
- Compare standard scores, not raw marks: the higher z is the better result relative to the group.
Normal distribution (NF)
About 68% of data lies within 1 SD of the mean, 95% within 2 SD, and 99.7% within 3 SD. The question gives the percentages needed.
Effect of operations (NF)
| Every item | Range, IQR, SD | Variance |
|---|---|---|
| + c | unchanged | unchanged |
| × k (k > 0) | × k | × k² |
Want to check an answer? Standard deviation calculator shows every step.
Worked examples
range = 13 − 4 = 9
mean = 40 ÷ 5 = 8
σ = √[(16 + 1 + 1 + 1 + 25)/5] = √8.8 ≈ 2.97
variance = 8.8lower half 2, 3, 5, 6 → Q₁ = 4
upper half 8, 9, 11, 12 → Q₃ = 10
IQR = 10 − 4 = 6 (median Q₂ = 7)z = (78 − 66)/8 = 1.5.
The mean becomes (8 + 5) × 3 = 39; the SD becomes 2.97 × 3 ≈ 8.90 (adding 5 doesn't change it).
Common mistakes
- Reading sx instead of σx on the calculator.
- Forgetting to order data before finding quartiles.
- Thinking adding a constant changes the SD.
- Comparing two classes by range when there's an extreme value (use IQR).
- Comparing raw scores across tests instead of standard scores.
How the DSE tests it
- A(1): standard deviation from a frequency table, sometimes with an unknown frequency.
- A(2): stem-and-leaf diagram → range, IQR → box-and-whisker diagram → "which class is more dispersed? Explain."
- Paper 2: effect of adding or removing data on mean, range, SD; NF standard scores and the normal distribution.
Questions you might not ask in class
What does standard deviation actually mean?
Roughly, the typical distance of the data from the mean. Small SD: data bunched together. Large SD: spread out.
Why square the differences and then square-root?
If we just added the differences, the positives and negatives would cancel to 0. Squaring makes them all positive; the square root brings the units back.
σx or sx?
σx. The DSE uses the population formula (divide by N).