Statistics
Statistics summarises a data set. A measure of centre describes a typical value, while spread describes variation. Choose both according to the context and inspect how the data were collected before making a comparison.
What this guide covers
Data displays, mean, median, mode, grouped estimates, range, quartiles, interquartile range, standard deviation and cumulative frequency. The short checks below sample averages and spread; they do not cover every display or establish mastery.
Syllabus reference: 2026 Mathematics 4052, S1, page 11. These are Mentora search groupings, not new official syllabus categories.
The content in this guide was also checked against 2027 G3 Mathematics K310, S1, page 11 on 2026-09-12. This is subject-content alignment, not a claim of complete syllabus coverage.
Worked example
Find the mean, median and range of 3, 5, 5, 7, 20.
The sum is 40 and there are 5 values, so the mean is 8. The middle of the ordered list is 5, so the median is 5. The range is 20 − 3 = 17. The high value 20 pulls the mean above the median; neither summary alone describes the whole distribution.
Cumulative frequency: estimates from grouped data
For 40 journeys, times in minutes have frequencies: 0 < t ≤ 10: 4; 10 < t ≤ 20: 12; 20 < t ≤ 30: 16; 30 < t ≤ 40: 8. Estimate the median and the number lasting more than 25 minutes using the plotted straight-line interpolation.
Cumulative totals at 0, 10, 20, 30 and 40 minutes are 0, 4, 16, 32 and 40. The median is read at cumulative frequency 20: move 4 of the 16 observations through the 20–30 interval, giving 20 + (4/16) × 10 = 22.5 minutes. At 25 minutes the graph gives 24 journeys, so about 40 − 24 = 16 lasted longer. Both are estimates: the exact positions inside each interval are unknown. The class frequencies and their cumulative totals are exact for this dataset.
Box plots: compare centre and the middle half
Delivery times (minutes) have five-number summaries A: 10, 14, 18, 22, 30 and B: 8, 12, 20, 28, 34, ordered as minimum, Q1, median, Q3, maximum. Compare the services.
A has a lower median (18 versus 20 minutes), suggesting a shorter typical delivery. Its interquartile range is 22 − 14 = 8 minutes, versus 28 − 12 = 16 for B, so A has less spread in the middle 50%. B has the lower minimum; this does not make B typically faster. These summaries are given exactly, not estimated from the drawing. Box plots do not reveal the mean or the individual times.
Standard deviation: same mean, different variation
Set A is 4, 4, 6, 6. Set B is 2, 2, 8, 8. Calculate and interpret the population standard deviation of each set.
Both means are 5. For A, squared deviations are 1, 1, 1, 1, so σ = √(4/4) = 1. For B they are 9, 9, 9, 9, so σ = √(36/4) = 3. B is more spread out around the same mean. Divide by n = 4 for these complete listed populations, not n − 1. Standard deviation has the original measurement unit; it is not a percentage or the range.
Try three practice checks
Attempt each on paper before revealing the worked answer. These sample particular skills; they do not diagnose the whole topic or establish mastery.
Check 1
A frequency table has value 2 with frequency 3, and value 6 with frequency 2. Find the mean.
Show worked answer
There are 5 observations and their total is 2 × 3 + 6 × 2 = 18. Mean = 18 ÷ 5 = 3.6, not (2 + 6) ÷ 2.
Check 2
Find the median of 8, 1, 5, 9, 3, 6.
Show worked answer
Order the data: 1, 3, 5, 6, 8, 9. Average the two middle values: (5 + 6) ÷ 2 = 5.5.
Check 3
A distribution has lower quartile 12 and upper quartile 21. Find and interpret its IQR.
Show worked answer
IQR = 21 − 12 = 9. The middle 50% of the distribution spans 9 units; this is not the total range.
Common mistakes to watch for
- Divide a frequency-table total by the sum of frequencies, not the number of rows.
- Sort raw values before locating the median. For grouped data, a midpoint-based mean is an estimate.
- A smaller spread means greater consistency, not automatically a better outcome. Explain whether a high or low value is desirable in context.
If one of these occurs, compare the relevant step with the worked example, then try a different question before drawing conclusions.
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More G3 Math topicsPublished by Mentora · Sources checked 12 Sept 2026
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Original public examples and answers were checked with automated mathematical/editorial QA. No individual author, human reviewer or independent expert review is recorded.