01 · A position
For n = 10, what rank gives Q₁ under this lesson’s rule?
Hint
Calculate 10/4, then round the position up.
Worked solution
10/4 = 2.5, so use rank 3. This is a position, not the numerical value of Q₁.
Understand · explore · practise
Quartiles and percentiles of listed data
Locate quartiles and percentiles in ordered lists and frequency tables, with an explicit rank convention, learner-operated model and original worked practice.
Before you startOrdering data, medians and cumulative frequencies.
01 / Separate position from value
A quartile is a value in the variable’s units.
Quartiles describe positions within an ordered dataset.
Q₁ is a lower-quarter summary; Q₂ is the median; Q₃ is an upper-quarter summary.
Order the observations before selecting positions. With small lists or tied values, the proportions strictly below a quartile need not be exactly 25% or 75%.
Positions are not valuesExplore
Values: 3, 5, 7, 9, 11, 13, 15.
n/4 = 1.75 → rank 2 → Q₁ = 5.
3n/4 = 5.25 → rank 6 → Q₃ = 13.
Median: rank 4 → 9.
This lesson uses the convention stated alongside. Software may use a different quantile definition.
02 / State the rule you use
Different definitions can give different answers.
For the listed-data quartiles in this lesson, calculate r = n/4 for Q₁ and r = 3n/4 for Q₃. If r is not an integer, use the value at rank ⌈r⌉ (round the position up). If r is an integer, average the values at ranks r and r + 1.
For the median, use the central value when n is odd and average the two central values when n is even. Other textbooks, exam instructions and software may specify different quantile rules. Follow the stated rule; do not silently switch conventions.
02 · Integer position
For n = 12, which values give Q₃?
Hint
Calculate 3×12/4.
Worked solution
The position is 9 exactly, so average the values at ranks 9 and 10.
03 / Use a sorted list
Do not preserve the order of collection.
Observed values: 12, 4, 9, 2, 7, 15, 5.Worked example
Ordered: 2, 4, 5, 7, 9, 12, 15
There are seven observations.
Q₁: 7/4 = 1.75 → rank 2 → 4
Round a non-integer position up.
Median = rank 4 = 7; Q₃: 21/4 = 5.25 → rank 6 → 12
Report values, not ranks.
03 · Order first
Find Q₁, median and Q₃ for 18, 3, 11, 7, 15, 5, 9.
Hint
Sort all seven values.
Worked solution
Ordered: 3, 5, 7, 9, 11, 15, 18. Q₁ = 5; median = 9; Q₃ = 15.
04 · Another odd length
For the ordered values 1, 2, 4, 6, 8, 9, 10, 13, 17, find Q₁ and Q₃.
Hint
Positions 9/4 and 27/4 are non-integers.
Worked solution
Round 2.25 up to rank 3 and 6.75 up to rank 7. Q₁ = 4 and Q₃ = 10.
04 / Average neighbouring values
An integer rank triggers the averaging step.
Watch: locate ranks before reading values
Pause, replay or seek freely. The notes explain the same idea and stay in view.
For the eight values 3, 5, 7, 9, 11, 13, 15, 17, Q₁ = (5 + 7)/2 = 6 and Q₃ = (13 + 15)/2 = 14. Neither quartile needs to be an observed value.
05 · Eight observations
Find Q₁, median and Q₃ for 2, 4, 6, 8, 10, 12, 14, 20.
Hint
Average ranks 2–3, 4–5 and 6–7 respectively.
Worked solution
Q₁ = 5; median = 9; Q₃ = 13.
06 · Ten observations
For 1, 3, 4, 6, 7, 8, 10, 12, 14, 19, find Q₁ and Q₃.
Hint
A list with even n does not always have integer quarter positions.
Worked solution
n/4 = 2.5 gives rank 3: Q₁ = 4. 3n/4 = 7.5 gives rank 8: Q₃ = 12.
05 / Locate ranks in cumulative counts
Repeated values occupy multiple positions.
For exact discrete frequency data, use total frequency as n. Cumulative frequencies show which value occupies each rank. An integer quartile position can require two different rows.
07 · Frequency table
Values 1, 3, 5, 8 have frequencies 2, 4, 3, 3. Find Q₁ and Q₃.
Hint
n = 12; cumulative frequencies are 2, 6, 9, 12.
Worked solution
Q₁ averages ranks 3 and 4, both value 3, so Q₁ = 3. Q₃ averages rank 9 (5) and rank 10 (8), so Q₃ = 6.5.
08 · Ties
Eight observations are all equal to 4. Give the quartiles and the percentage strictly below Q₁.
Hint
The quartile values can coincide.
Worked solution
Q₁ = median = Q₃ = 4. None is strictly below Q₁, so the percentage is 0%, not 25%.
06 / Extend the idea carefully
A percentile rule must also be specified.
A pth percentile describes a p% position in the distribution. For the following listed-data exercise only, extend the same rule: r = pn/100; average ranks r and r + 1 when r is an integer, otherwise use rank ⌈r⌉. We use 0 < p < 100. This is one convention, not a universal software definition.
Grouped interpolation uses a different assumption and method, covered next. Do not round a grouped target rank up before interpolating.
09 · A stated percentile rule
For 2, 4, 5, 7, 9, 11, 12, 15, 18, 24, use the rule above to find P₈₀.
Hint
r = 80×10/100 = 8.
Worked solution
Average ranks 8 and 9: P₈₀ = (15 + 18)/2 = 16.5.
10 · A non-integer rank
For the seven ordered values 2, 4, 5, 7, 9, 12, 15, find P₆₀ using the stated rule.
Hint
r = 0.6×7 = 4.2.
Worked solution
Use rank 5, so P₆₀ = 9.
07 / Keep the context and units
Do not promise exact percentages for finite data.
11 · Interpret a quartile
A lower quartile of journey duration is 12 minutes. Does it prove exactly one quarter of journeys took strictly less than 12 minutes?
Hint
Think about ties and finite ranks.
Worked solution
No. It summarises the lower-quarter position, but ties and the chosen finite-data convention can prevent exactly 25% being strictly below it.
12 · Different software answer
A spreadsheet gives a different Q₁ from your hand calculation. What should you check before declaring either wrong?
Hint
There is more than one quantile definition.
Worked solution
Check the ordered data, missing-value handling, total count and the spreadsheet’s quantile convention. Compare results using the same specified definition.
08 / Order, locate, read
Make the convention visible.
- Order values or form cumulative frequencies.
- Count observations, not table rows.
- Apply the stated quartile or percentile rank rule.
- Read the value or average the required neighbouring values.
- Interpret in context and retain the units.
Section 1 of 8 · Separate position from value