Download Measures of Dispersion/ Variability

Measures of Dispersion/
Variability
Dr Faris Al Lami
MB ChB PhD
Measures of dispersion & variability
• They measure the variability in the values of
observations in the set.
• They also called measures of variation, spread
and scatter.
Measures of dispersion & variability
• If all values are the same the dispersion is
zero.
• If the values are homogenous and close to
each other the dispersion is small.
• If the value are so different the dispersion is
large.
Measures of dispersion
•
•
•
•
•
Range: Is the difference between the largest
and smallest value
R=XL- XS
R=Range
XL= largest value,
XS= smallest value
Properties of the range:
ØSimple to calculate
ØEasy to understand
ØIt neglect all values in the center and depend
on the extreme value, extreme value are
dependent on sample size
Properties of the range:
ØIt is not based on all observations
ØIt is not amenable for further mathematic
treatment
Øshould be used in conjunction with other
measures of variability
Variance:
The mean sum of squares of the deviation from the mean.
e.g. if the data is: 1,2,3,4,5.
The mean for these data=3
the difference of each value in the set from the mean:
1-3= -2
2-3= -1
3-3= 0
4-3= 1
5-3= 2
•
The summation of the differences =zero
•
Summation of square of the differences is not zero
Variance:
• Population Variance (sigma
squared)
2
2
∑(X- μ)
• α =---------------N
2
α= sigma squared(pop.var)
X=observation value
μ= population mean
N=population size
2
∑x =summa on of squared
2
(∑ X)=squared of summa on
2
2
2
• α =[ N ∑x – (∑ X) ] / N.N
Variance:
• Sample Variance
_2
2 ∑ (X- X )
• S=---------------OR
n-1
2
2
[ n∑X – (∑X) ]
s= ---------------------n(n-1)
2
2
• S= sample variance
• n= sample size
Variance:
• Variance can never be a negative value
• All observations are considered
• The problem with the variance is the squared
unit
Standard deviation (SD):
•
It is the square root of the variance
•
SD=√sigma square= ± sigma(α)---- for
population
2
•
Sd= √S = ± S----for sample
Standard deviation (SD):
• The standard deviation measured the
variability between observations in the
sample or the population from the mean of
that sample or that population.
• The unit is not squared
• SD is the most widely used measure of
dispersion
Standard Error of the mean(SE)
•
•
•
It measures the variability or dispersion of
the sample mean from population mean
It is used to estimate the population mean,
and to estimate differences between
populations means
SE=SD/√ n
Coefficient of variation (CV):
•
•
•
•
It expresses the SD as a percentage of the
mean
CV= S /mean X100
(mean of the sample)
It has no unit
It is used to compare dispersion in two sets
of data especially when the units are
different
Coefficient of variation (CV):
• It measures relative rather than absolute
variation
• It takes in consideration all values in the set
EXERCISE
• For the same 15 patients in the previous
example , calculate measures of dispersion.
Pat. no
Distance
(mile)(X)
2
Pat. no
Distance
(mile)(X)
2
X
1
2
3
4
5
9
11
3
X
25
81
121
9
5
12
144
13
12
144
6
7
13
12
169
144
14
15
15
5
225
25
8
6
36
Total
141
1575
9
10
11
12
13
7
3
15
169
49
9
225
Range
R=XL- XS
=15-3
=12 mile
Variance & sd
2
2
2
n∑X – (∑X)
s= ---------------------n(n-1)
2
=(15)(1575) – (141) / 15 x 14
2
=17.8 mile
sd= √17.8 = ± 4.2 mile
Standard Error
• SE=SD/√ n
•
=4.2/√15 = 4.2/3.87 = 1.085 mile
Coefficient of Variation
• CV= S /mean X100
•
= 4.2 mile/ 9.4 mile X 100%
•
=44.7%
EXERCISE
The following are the hemoglobin values
(gm/dl) of 10 children receiving treatment for
hemolytic anemia:
9.1,10.0, 11.4, 12.4, 9.8,8.3, 9.9, 9.1, 7.5, 6.7
Compute the sample mean, median, variance,
and standard deviation
EXERCISE
• A sample of 11 patients
admitted to a
psychiatric ward
experienced the
following lengths of
stay, calculate measures
of central tendency and
dispersion.
No.
length
No.
length
1
29
7
28
2
14
8
14
3
11
9
18
4
24
10
22
5
14
11
14
6
14
total