Measures of Dispersion for Ungrouped Data · 8.2.6
Solving problems on dispersion
Students apply range, interquartile range, variance and standard deviation to multi-step, often real-life problems, for example, recovering a missing data value from a given mean before finding dispersion, or combining calculation with interpretation. This standard draws together all the dispersion skills from the topic into one connected task.
The official learning standard (8.2.6)
“Solve problems involving measures of dispersion.”
What it means
Students apply range, interquartile range, variance and standard deviation to multi-step, often real-life problems, for example, recovering a missing data value from a given mean before finding dispersion, or combining calculation with interpretation. This standard draws together all the dispersion skills from the topic into one connected task.
How it is examined
Paper 2 typically presents higher-order, multi-step questions set in real-life contexts such as sports scores, sales figures or exam marks, combining computation of dispersion measures with justification or comparison. Paper 1 may test a single-step application, such as finding the effect of a data change on the standard deviation.
Worked example
The number of customers served by a cashier over 6 days were 18, 22, 25, 19, 21 and k. Given that the mean number of customers is 21, find the value of k, then determine the range and standard deviation of the data.
- Total customers over 6 days = mean × number of days = 21 × 6 = 126.
- Sum of the 5 known values = 18 + 22 + 25 + 19 + 21 = 105.
- k = 126 − 105 = 21.
- Complete data set: 18, 22, 25, 19, 21, 21. Range = 25 − 18 = 7.
- Deviations from the mean (21): −3, 1, 4, −2, 0, 0. Squared deviations: 9, 1, 16, 4, 0, 0, giving a sum of 30.
- Variance = 30 ÷ 6 = 5, so standard deviation = √5 ≈ 2.24.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I find a missing data value if I only know the mean?
Use total = mean × number of values to find the overall total, then subtract the sum of the known values. The remainder is the missing value.
Always check your final answer makes sense within the given data.
Do I need to memorise the variance and standard deviation formulas for the exam?
Yes. You should be able to recall variance = Σ(x − x̄)² ÷ n and standard deviation as its square root, and apply them confidently, since these formulas are not usually printed on the exam formula list.
What if a problem gives quartiles instead of the full raw data set?
Use the quartiles directly to find the interquartile range (Q3 − Q1) instead of range or standard deviation, since you cannot calculate those without every individual data value.