# lec-18 dispersion.ppt

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```					Measures of
Dispersion
Measures of Dispersion

Tariq Mahmood Bajwa UVAS, LHR.   2
Measures of Dispersion

The scatter of the values about their centre
is called dispersion and any measure
indicating the amount of scatter about
the centre is called a Measure of
Dispersion.
The individual observations of a variable
tend to scatter about their centre. The
highest degree of concentration is that all
the observations are of same size. The
scatter in this case would be zero and
mean will be exactly same as the
individual values of the variable.
Tariq Mahmood Bajwa UVAS, LHR.       3
Types of Measures of Dispersion

There are two main types of measures of
dispersion:
1. Absolute Measure of Dispersion
2. Relative Measure of Dispersion
Absolute Measure of Dispersion
The absolute measure of dispersion
measures the variation present among the
observations in the unit of the variable or
square of the unit of the variable.
Tariq Mahmood Bajwa UVAS, LHR.       4
Types of Measures of Dispersion

Relative Measure of Dispersion
The relative measure of dispersion
measures the variation present among the
observations relative to their average. It
is expressed in the form of a ratio,
coefficient or percentage. It is
independent of the unit of measurement.

Tariq Mahmood Bajwa UVAS, LHR.     5
Types of Measures of Dispersion

The commonly used measures of absolute
dispersion are:
1. Range
2. Quartile Deviation
3. Mean (Average) Deviation
4. Variance and Standard Deviation

Tariq Mahmood Bajwa UVAS, LHR.     6
Types of Measures of Dispersion

Their corresponding measures of relative
dispersion are:
1. Coefficient of Range
2. Coefficient of Quartile Deviation
3. Coefficient of Mean (Average)
Deviation
4. Coefficient of Variation (CV)

Tariq Mahmood Bajwa UVAS, LHR.      7
Range

If X1, X2, …, Xn are n observations of a
variable X, with X1 and Xn as the
smallest and largest observations
respectively.

Then its range is defined as:

Range = Xn - X1

Tariq Mahmood Bajwa UVAS, LHR.      8
Range
Example: The following data set shows the
weekly TV viewing times, in hours.
Calculate range and range coefficient of variation.
25, 41, 27, 32, 43, 66, 35, 31, 15, 5,
34, 26, 32, 38, 16, 30, 38, 30, 20, 21.
Range  X n   X 1  66  5  61hours
Range X n   X 1 66  5
Semi Range                                   30.5hours
2             2          2
X n   X 1
Mid Range                     35.5
2
66 - 5
Co - efficient of Range               0.859
66  5
Tariq Mahmood Bajwa UVAS, LHR.                       9
Quartile Deviation

If X1, X2, …, Xn are n observations of a
variable X, with Q1 and Q3 as their first
and third quartiles respectively, then
their Quartile Deviation (QD) is as:
IQR  Q3  Q1
IQR Q3  Q1
SIQR  QD      
2     2
Q3  Q1
MIQR 
2
Tariq Mahmood Bajwa UVAS, LHR.          10
Quartile Deviation
Example:
Calculate QD and coefficient of QD of
above data set shows the weekly TV
viewing times, in hours.

Q1  22.0 h Q 3  36.5 h IQR  36.5 - 22.0  14.5 h
IQR Q 3  Q1
SIQR  QD                   7.25 h
2        2
Q3  Q1
MIQR               29.25 h
2
SIQR Q3  Q1
Co - Efficient of Q.D                0.248
MIQR Q3  Q1
Tariq Mahmood Bajwa UVAS, LHR.            11
Mean Deviation
If X1, X2, …, Xn are n observations of a
variable X, with m as their average
(mean, median or mode), then their
mean deviation, denoted by MD, is
defined as:

MD 
X                               i   m
n
Tariq Mahmood Bajwa UVAS, LHR.            12
Mean Deviation

Example:
Calculate MD and coefficient of MD.

X
 X  605  30.25 h
n              20

MD 
X      i   m
175
 8.75 h

n        20
M.D.       8.75
Coefficient of MD                      0.289
Average Used 30.25

Tariq Mahmood Bajwa UVAS, LHR.       13
X    X-Mean |X-Mean|                    X     X-Mean |X-Mean|
25    -5.25               5.25          34     3.75     3.75
41    10.75              10.75          26     -4.25    4.25
27    -3.25               3.25          32     1.75     1.75
32    1.75                1.75          38     7.75     7.75
43   12.75               12.75          16    -14.25   14.25
66   35.75               35.75          30     -0.25    0.25
35    4.75                4.75          38     7.75     7.75
31    0.75                0.75          30     -0.25    0.25
15   15.25               15.25          20    -10.25   10.25
5   -25.25              25.25          21     -9.25    9.25
Continue                          605     0      175.00
Tariq Mahmood Bajwa UVAS, LHR.                     14
Variance and Standard Deviation
The Variance is defined as the mean of the
squared deviations from mean. The
population variance is denoted by σ2 where
as sample variance is denoted by S2 and
defined as
For ungrouped data
 (x -  )            2

 =
2
For population
N
2   (x - x )2

S =                                    For sample
n
Tariq Mahmood Bajwa UVAS, LHR.                    15
Variance and Standard Deviation

For grouped data

 f (x -  )               2

 =
2
For population
N
2      f (x - x )2
S =                        For sample
n
Standard deviation
The positive square root of the variance is
called Standard Deviation. It is denoted by
σ (S for sample)
Tariq Mahmood Bajwa UVAS, LHR.    16
Variance and Standard Deviation
Co-efficient of Variation.

The standard deviation is an absolute measure
of dispersion its relative measure of dispersion
is called co-efficient of variation (CV) and is
defined by :

S .D.        s
CV        100    100
Mean         X

Tariq Mahmood Bajwa UVAS, LHR.     17
Variance and Standard Deviation
Example: Find Variance, S.D and
Co-efficient of Variation.
X     2     3     6      8     11                              30

(X-6)2       16           9               0        4   25        54

2  (x - x )2                           54
S =                                  =      = 10.8
n                                 5
(x - x )2
S=                                   =       10.5 = 3.286
n
S                            3.286
C.V =          x 100 =                      x 100 = 54.76 %
x                              6
Tariq Mahmood Bajwa UVAS, LHR.                               18
Variance and Standard Deviation
Example:-
Find Variance, S.D and Co-efficient of Variation.

Class          f          X          (X-X)      (X-X)2   f(X-X)2
20---24         1         22               -17    289      289
25---29         4         27               -12    144      576
30---34         8         32               -7     49       392
35---39        11         37               -2      4        44
40---44        15         42               3       9       135
45---49         9         47               8      64       576
50---54         2         52               13     169      338
TOTAL          50                                          2350

Tariq Mahmood Bajwa UVAS, LHR.                            19
Variance and Standard Deviation
Example:-
X  39
 f(x - x )2 2350
S2                    47
n         50
 f(x - x )2
S =                 = 47 = 6.85
n
6.85
C.V =         x 100 = 17.56
39
Tariq Mahmood Bajwa UVAS, LHR.   20

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