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MEASURES OF SPREAD - RANGE & INTERQUARTILE RANGE

Range, Quartiles, and Interquartile range of data

March 16, 2021

Updated on 05th September 2025, added: an interquartile range for an odd number of observations.

The Range and the Inter-Quartile Range (IQR) are the measures that describe the spread of data.

EXAMPLE  2

The heights (mm) of a different variety of plants are as follows.

65, 32, 40, 27, 35, 49, 53, 33, 47, 56, 45, 30, 48, 67, 56, 37, 56, 41, 38, 52

Let’s arrange the data in ascending order.

27
30
32
33
35
37
38
40
41
45
47
48
49
52
53
56
56
56
65
67

RANGE

The range of a data set is the difference between the highest (maximum) and lowest (minimum) values.

Range = Maximum value - Minimum value

Range = 67 - 27 = 40 mm

QUARTILES

The 2nd Quartile (Q2) or the middle quartile is the median. It is the middle value that divides the data into two equal parts.

27
30
32
33
35
37
38
40
41
45
47
48
49
52
53
56
56
56
65
67
Median
(45 + 47) ÷ 2 = 46 mm

The 1st Quartile (Q1) or the lower quartile (QL) separates the lower quarter of values from the rest of the values. This is the value at the middle point of the first half of the data.

27
30
32
33
35
37
38
40
41
45
Lower Quartile (QL)
(35 + 37) ÷ 2 = 36 mm

The 3rd Quartile (Q3) or the upper quartile (QU) separates the upper quarter of values from the rest of the values. This is the value at the middle point of the second half of the data.

47
48
49
52
53
56
56
56
65
67
Upper quartile (QU)
(53 + 56) ÷ 2 = 54.5 mm

INTER QUARTILE RANGE (IQR)

It is the difference between the upper quartile (Q3 or QU) and the lower quartile (Q1 or QL)

IQR = QU - QL

IQR = 54.5 - 36 = 18.5 mm

EXAMPLE  3

The heights (mm) of 19 plants are as follows.

60, 22, 42, 25, 30, 50, 52, 33, 47, 57, 43, 31, 28, 55, 46, 48, 54, 51, 45

Let’s arrange the data in ascending order.

22
25
28
30
31
33
42
43
45
46
47
48
50
51
52
54
55
57
60

The 2nd Quartile (Q2) or the middle quartile is the median. It is the middle value that divides the data into two equal parts.

22
25
28
30
31
33
42
43
45
46
47
48
50
51
52
54
55
57
60
Median = 46 mm

Here, the number of observations is odd. Tukey’s hinges include the median in each half of the observations when the number of observations is odd. Therefore, the observations can be divided into two equal parts (10 observations on either side).

The 1st Quartile (Q1) or the lower quartile (QL) is the value at the middle point of the first half of the data.

22
25
28
30
31
33
42
43
45
46
Lower Quartile (QL)
(31 + 33) ÷ 2 = 32 mm

The 3rd Quartile (Q3) or the upper quartile (QU) is the value at the middle point of the second half of the data.

46
47
48
50
51
52
54
55
57
60
Upper quartile (QU)
(51 + 52) ÷ 2 = 51.5 mm

INTER QUARTILE RANGE (IQR)

It is the difference between the upper quartile (Q3 or QU) and the lower quartile (Q1 or QL)

IQR = QU - QL

IQR = 51.5 - 32 = 19.5 mm

 

 

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