LESSON 2 – MEASURES of SPREAD
(Percentiles and Quartiles)
PERCENTILES divide the ordered data into 100 intervals that have equal numbers of values.
35% of data is less than
or equal to this value
eg. 35th percentile, P35
65% of data is greater than
or equal to this value
PERCENTILE RANK
𝑝(𝑛 + 1)
𝑅=
100
Where 𝑝 is the percentile
𝑛 is the size of the population
𝑅 is the whole number rank of
data point (round decimals down)
PERCENTILE
100(𝐿 + 0.5𝐸)
𝑝=
𝑛
Where 𝑝 is the percentile
𝐿 is the number of data less than the data point
𝐸 is the number of data equal to the data point
𝑛 is the size of the population
𝑅 is the whole number rank of data point
Example ① The list shows the marks for 25 students on a recent test out of 40.
31
28
28
30
20
25
38
40
26
28
36
25
16
21
34
37
30
23
24
36
a) Calculate the 80th percentile.
b) What percentile is a mark of 25?
c) What percentile is a mark of 40?
15
32
21
25
28
QUARTILES
Divide a set of ORDERED data into four groups with equal numbers of values.
The three dividing points are:
1. The first quartile, 𝑄1
2. The median or second quartile, 𝑄2
3. The third quartile, 𝑄3
Interquartile Range
𝑛+1 𝑡ℎ
(
4
3(𝑛+1) 𝑡ℎ
𝑛+1 𝑡ℎ
) position
(
25th percentile
2
) position
(
50th percentile
4
) position
75th percentile
The interquartile range is defined as Q3 – Q1, which is the range of the middle half of the data. This range
provides a measure of spread by showing how closely the data are clustered around the median.
The semi-interquartile range is one half of the interquartile range.
A box-and-whisker plot illustrates Q1, Q2, and Q3. The box is defined by each quartile and shows the
interquartile range while the ends of the “whiskers” represent the lowest and highest values in the data.
Example ① Determine the indicated quantities for each of the following sets:
a) 18, 15, 20, 17, 11, 9, 13, 14, 17, 16, 10, 16
𝑛 =
Rank Data:
Quartile
median (Q2)
Q1
Q3
Q3 – Q1
(interquartile range)
semi-interquartile range
(
𝑛+1
)
2
(
𝑛+1
)
4
3(𝑛 + 1)
4
Position
Value
𝑛=
b) 4, 5, 8, 7, 3, 1, 2, 5, 3
Rank Data:
Quartile
median (Q2)
Q1
(
𝑛+1
)
2
(
𝑛+1
)
4
Position
Value
3(𝑛 + 1)
4
Q3
Illustrate with a box-and-whisker plot.

1

2

3

4

5

6

7

8