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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