All three answer the same question: what value sits at a given share of the way through?
Find the quartile
For the sorted data 3, 7, 8, 12, 15, 18, 21, what is Q1?
The median is 12, the 4th of seven values. The lower half is 3, 7, 8, and its middle is 7. D is the median itself, which marks Q2. A is the minimum. Once the median is found, each quartile is just the median of one half — which is why sorting comes first.
| Measure | Cuts the data into | Number of cuts |
|---|---|---|
| Quartiles | 4 parts | 3 (Q1, Q2, Q3) |
| Deciles | 10 parts | 9 (D1 to D9) |
| Percentiles | 100 parts | 99 (P1 to P99) |
They overlap: Q1 = P25 = D2.5, Q2 = median = P50 = D5, and Q3 = P75. Different names for the same cuts.
3, 7, 8, 12, 15, 18, 21
sorted, n = 7
Q2 = 12
the 4th value
lower half 3, 7, 8 → Q1 = 7
exclude the median
upper half 15, 18, 21 → Q3 = 18
same again
A quartile is a position. Applying any of this to an unsorted list produces a number with no meaning at all.
position of Pk = k(n + 1)100
n = 19, find P30
sorted data
30(20)100 = 6
the 6th value
P30 = the 6th number in the list
read it off
When the position is not a whole number — say 6.5 — take the value halfway between the 6th and 7th. That is exactly what averaging the two middle values does for a median.
A score of 72 in a class of 40
30 scored below it
3040 = 0.75
the share below
percentile rank = 75
× 100
Being at the 75th percentile means you scored higher than 75% of the group — it says nothing about getting 75% of the marks. A student at the 90th percentile might have scored 55 out of 100 on a hard paper.
When a half has no single middle, its quartile is the mean of its two middle values.
Find Q1, Q2 and Q3 for the sorted data 3, 5, 9, 12, 15, 19, 24, 27.
Q2 = (12 + 15) ÷ 2 = 13.5
n = 8: between the 4th and 5th
lower half 3, 5, 9, 12 → Q1 = (5 + 9) ÷ 2 = 7
four values, two middles
upper half 15, 19, 24, 27 → Q3 = (19 + 24) ÷ 2 = 21.5
the same in the top half
With an even n the data splits cleanly into two halves of four, and nothing is left over. Q3 − Q1 = 14.5 is the spread of the middle half.
Picking 5 or 9 for Q1 because "there must be a number in the list" gives an answer that is a quarter of the way up for neither. With an even half there is no single middle, so Q1 = 7 sits between the two.
For the first two, use these 15 sorted scores: 12, 15, 18, 20, 21, 24, 27, 29, 30, 33, 35, 38, 40, 44, 47.
A position that is not whole: P40
40(15 + 1)100 = 6.4
between the 6th and 7th values
6th = 24, 7th = 27
a gap of 3
P40 = 24 + 0.4 × 3 = 25.2
0.4 of the way along
Different: the decimal part of the position tells how far to move between two neighbours.
A decile: D7
7(15 + 1)10 = 11.2
deciles divide by 10, not 100
11th = 35, 12th = 38
a gap of 3
D7 = 35 + 0.2 × 3 = 35.6
same as P70
Different: Dk sits at k(n + 1)10, so D7 and P70 are the same cut.
Unsorted, with repeats: Q1 and Q3 of 9, 4, 7, 4, 12, 15, 7
4, 4, 7, 7, 9, 12, 15
sort first; keep both 4s and both 7s
median 7; lower 4, 4, 7 → Q1 = 4
the median is left out of both halves
upper 9, 12, 15 → Q3 = 12
the middle of the top half
Different: a repeated value counts every time it appears, and Q1 can equal the minimum.
These three numbers, plus the minimum and maximum, are the five-number summary — everything a box plot draws.
Expect Q1 or Q3 of a short list (often with an even number of values), a percentile rank from "how many scored below", and a "what does the 80th percentile mean" item. The fastest route is sort, find the median, then take the median of each half. The 10-second check: Q1 ≤ Q2 ≤ Q3, and about a quarter of the values sit below Q1.
Interpret the rank
A student's score is at the 60th percentile. What does that mean?
A percentile is about rank within the group: 60% of the others scored lower. A and C confuse it with a score — on a difficult paper the 60th percentile might be 42 marks out of 100. D misreads the direction and also treats a percentage as a count, which would only match if the group had exactly 100 people.
Hint: percentiles compare a student with the others who sat the test, not with the total marks available.
