6.6 Measures of Dispersion

6.6 Measures of Dispersion

(A) Determine the range of a set of data
1. For an ungrouped data,
Range = largest value – smallest value.
2. For a grouped data,
Range = midpoint of the last class – midpoint of the first class.

Example 1:
Determine the range of the following data.
(a) 720, 840, 610, 980, 900
(b)
Time (minutes)
1 – 6
7 – 12
13 – 18
19 – 24
25 – 30
Frequency
3
5
9
4
4

Solution: 
(a)
Largest value of the data = 980
Smallest value of data = 610
Range = 980 – 610 = 370
(b)
Midpoint of the last class
= ½ (25 + 30) minutes
= 27.5 minute

Midpoint of the first class
= ½ (1 + 6) minutes
= 3.5 minute

Range = (27.5 – 3.5) minute = 24 minutes


(B) Medians and Quartiles

1. The first quartile (Q1) is a number such that 1 4 of the total number of data that has a value less than the number.
2. The median is the second quartile which is the value that lies at the centre of the data.
3. The third quartile (Q3) is a number such that 3 4 of the total number of data that has a value less than the number.
4. The interquartile range is the difference between the third quartile and the first quartile.

Interquartile range = third quartile – first quartile


Example 2:




























The ogive in the diagram shows the distribution of time (to the nearest second) taken by 100 students in a swimming competition. From the ogive, determine
(a) the median,
(b) the first quartile,
(c) the third quartile
(d) the interquartile range
of the time taken.

Solution:


(a)  1 2  of 100 students = 1 2 ×100=50 From the ogive, median, M=50.5 second (b)  1 4  of 100 students = 1 4 ×100=25 From the ogive, first quartile,  Q 1 44.5 second (c)  3 4  of 100 students = 3 4 ×100=75 From the ogive, third quartile,  Q 3 = 54.5 second

(d)
Interquartile range
= Third quartile – First quartile
= 54.5 – 44.5
= 10.0 second