Measures of Dispersion for Ungrouped Data · 8.2.5
Comparing dispersion of data sets
Students select a suitable measure of dispersion, range, interquartile range, variance or standard deviation, to compare two or more sets of ungrouped data, then interpret the values in context to draw a conclusion, such as which data set is more spread out or which one is more consistent.
The official learning standard (8.2.5)
“Compare and interpret two or more sets of ungrouped data, based on the appropriate measures of dispersion, hence make conclusion.”
What it means
Students select a suitable measure of dispersion, range, interquartile range, variance or standard deviation, to compare two or more sets of ungrouped data, then interpret the values in context to draw a conclusion, such as which data set is more spread out or which one is more consistent.
How it is examined
Paper 2 questions usually give two data sets, often with equal or stated means or medians, and ask students to calculate a dispersion measure for each, compare the values, and conclude which set is more consistent or more widely spread, with the conclusion clearly linked to the context given.
Worked example
Runner A's times (in seconds) for 5 races are 40, 42, 41, 43, 39. Runner B's times are 35, 45, 38, 48, 39.
Both runners have the same mean time. Compare their consistency using the standard deviation.
- Mean of A = (40+42+41+43+39) ÷ 5 = 205 ÷ 5 = 41 s. Mean of B = (35+45+38+48+39) ÷ 5 = 205 ÷ 5 = 41 s (equal, as given).
- Deviations from the mean for A: −1, 1, 0, 2, −2. Squared deviations: 1, 1, 0, 4, 4, giving a sum of 10.
- Variance of A = 10 ÷ 5 = 2, so standard deviation of A = √2 ≈ 1.41 s.
- Deviations from the mean for B: −6, 4, −3, 7, −2. Squared deviations: 36, 16, 9, 49, 4, giving a sum of 114.
- Variance of B = 114 ÷ 5 = 22.8, so standard deviation of B = √22.8 ≈ 4.77 s.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Which measure should I use to compare two data sets, range, IQR, or standard deviation?
Standard deviation or variance is most common when full raw data is given, since it uses every value. Use the interquartile range if outliers are present or only quartiles are known.
Range is only a rough, less reliable measure of spread.
Can I compare standard deviations directly if the two data sets have different means?
Yes, standard deviation measures spread only, not the average value, so it can still be compared directly. Just remember your conclusion should describe consistency of spread, and should not be confused with which set has the higher mean.
Why might a data set with a smaller range still have a larger standard deviation?
Range only depends on the two extreme values, while standard deviation considers every value's distance from the mean. A data set can have a small range but many values spread unevenly around the mean, giving it a comparatively larger standard deviation.