Measures of Dispersion
Importance of Dispersion
Types of measures of
Dispersion
Range
Interquartile range
Variance and Standard
deviation
Importance of Dispersion
In some cases, two sets of data with same mean and same
median, but don’t mean that they have the same dispersion.
E.g. X : 80, 90, 100, 110, 120
Y : 0, 50, 100, 150, 200
Mean of X = 80  90  100  110  120
5
Mean of Y =
 100
0  50  100  150  200
 100
5
Median of X and median of Y = 100
But Y is more dispersed than X.
Types of measure of dispersion
There are three types of measure of dispersion:
- Range
- Interquartile range
- Variance and Standard deviation
Range
What is range?
Ans: The highest score in a distribution minus the lowest score.
Example: There are two sets of data about the amount of rainfall
(mm) in Taipei and Hong Kong.
Taipei
Seoul
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
86 135 178 170 231 290 231 305 244 122 66 71
40 77 83 89 147 168 184 252 209 101 32 13
Range of Taipei = (305 - 66)mm = 239 mm
Range of Seoul = (252 -13)mm = 239 mm
Therefore they have the same range.
400
300
Taipei
200
Seoul
100
Month
N
ov
ep
S
Ju
l
ay
M
M
ar
0
Ja
n
Amount of rainfall
(mm)
Amount of rainfall in Taipei and Seoul
thoughout 1998
Interquartile range
cumulative
frequency
What is interquartile range?
Ans: The difference between the upper and the lower quartiles:
IQR = 3rd quartile - 1st quartile
Example:
Heights of 200 form 5 students
300
200
100
0
155.5 160.5 165.5 170.5 175.5 180.5 185.5
height (cm)
From the graph,
the upper quartile = the 150th value
= 177 cm
the lower quartile = the 50th value
= 168 cm
therefore, the interquartile range = the upper quartile - lower quartile
= 177 cm - 168 cm
= 9 cm
Variance and Standard deviation
What is variance?
Ans: The mean of the squared deviation scores about the mean of a
distribution.
What is standard deviation?
Ans: The square root of the mean of the squared deviation scores about
the mean of a distribution; more simply, the square root of the variance.
Example: The following are two sets of data of an experiment obtained
by two different students.
Student A
Student B
Volume if acid measured (cm^3)
8 12 7 9 3 10 12 11
7 6 7 15 12 11 9 9
12
13
14
11
1). What is the mean volume of acid measured by each student?
Ans: Mean of student A
10  12  11  12  14
= 8  12  7  9  3  10
= 9.8
Mean of student B
= 7  6  7  15  12 10 11  9  9  13  11
= 10
2). What is the standard deviation?
Ans:
Standard deviation of X:
=
(8  9.8) 2  (12  9.8) 2  (7  9.8) 2  (9  9.8) 2  (3  9.8) 2  (10  9.8) 2  (12  9.8) 2  (11  9.8) 2  (12  9.8) 2  (14  9.8) 2
10
= 3.0265
Standard deviation of Y:
= (7  9.8)  (6  9.8)  (7  9.8)  (15  9.8)
2
2
2
2
 (12  9.8) 2  (11  9.8) 2  (9  9.8) 2  (9  9.8) 2  (13  9.8) 2  (11  9.8) 2
10
= 2.757
3). Which set of results is more reliable?
Ans: Y
Conclusion
The range and interquartile range are usually ineffective to measure
the dispersion of a set of data. An useful measure that describes the
dispersion of all the values is the variance or standard deviation.
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