Download Measurement PPT

Measures of Dispersion
Outcomes
• By the end of this lecture, the student will be
able to Know definition, uses and types of
statistics.
Measures of Dispersion
These are methods which used for measuring
variability (or homogenicity) of observations.
a) The Range:
It is defined as the highest observation the
lowest observation. It is a simple measure,
easily and quickly obtained. Sometimes we
cannot differentiate between the amount of
variation among different groups if they have
equal largest and smallest observations.
This results from the fact that this methods
neglects all intermediate o
This results from the fact that this
methods neglects all intermediate
observations.
e.g.
1st group: 9 7 5 3 1 range = 9-1 = 8
2nd group: 9 3 4 3 1 range = 9-1 = 8
b) The mean absolute Deviation:
It is defined as the average of the absolute
deviation of each observation from the
arithmetic mean
N.B. Absolute deviation means difference between
two quantities and this difference is given a +ve
sign always. It is denoted by

xi  x

Mean absolute Deviation =
n
 The range is a good measure of dispersion but it
does not have good mathematical properties. Ex :
xi


xi  x
x
1
5
4
3
5
2
5
5
0
7
5
2
9
5
4
n  25

 x  x  12
i
The mean Absolute deviation= 12/5 = 2.4
c) The variance:
2
(S )
• It equals the mean of the squared deviations of observations
from their arithmetic mean.
 x  x
 i




S2 
n

2
We use (n-1) instead of n as a correction for small values.
 x  x

 i


n 1

So, S2 =
2
Mathematically this equation equals to:

x 


2
s 
2
x
2
i
i
n 1
n
This formula is better and is easier in computation.
It is the on commonly used.
d) The Standard Deviation: (S)
It is defined as the positive Square route or the variance.
It should always be defined in the same unit as the
original variables.
I. For ungrouped data:

x 


2
s
x
2
i
i
n 1
n
Ex:
xi
xi
6
5
4
8
3
26
36
25
16
64
9
150
x
x
i

x 


2
2
s
x
2
i
i
n 1
n

26 
150 
2
5
5 1
150  135.2

4
 3.7
s2 
2
i
S 
3.7  1.92
II. Computation of the standard deviation from grouped data:
a. Using the long method:
Steps





Determine the mid point for each interval x j.
Find the product f j x j for each interval and
the sum of these products  f j x j.
Find the product
f j x 2j for each interval by multiplying x j
by the corresponding f x value and then find the sum
j
j
2
of these product
fx

j
j
 Find the variance S2 from the formula:
Standard Deviation
 f x
j
S2 
2
j



f j xj
 fj
 f 1
j
S 
s2

2
Ex :
Age in
years
Frequency
fj
Mid point
xj
fj xj
fj xj2
10-
3
12.5
37.5
468.75
15-
7
17.5
122.5
2143.75
20-
6
22.5
135
3037.5
25-29
4
27.5
110
Total (Σ)
20
405
8675
 f x
j
S2 
2
j


 f
fj xj
 fj
1
j
405 
2
8675 -
20
20  1
S  24.9
S 
S 
24.9
S  5 ye ars

2
‫‪Assignment‬‬
‫‪Student Name‬‬
‫امل محمد احمد احمد‬
‫اميره اسعد يوسف‬
‫اميره صالح مرشدي‬
‫انجي عبد الموجود‬
‫اوركيد اشرف السيد‬
‫ايمان سعيد محمد‬
‫ايمان محمدي يوسف‬
‫ايه رجب عبد العزيز‬
‫ايه حماده عطيه‬
‫بنوب عوض ناجي‬
‫‪Title‬‬
‫‪The Standard Deviation‬‬
References
• Biostatistical analysis: Jerrold H. Zar