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Lec Location (MEAN & Median).ppt

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					MEASURES OF
 LOCATION


      Zubair Latif
  Main Campus UVAS
        Lahore
 MEASURE OF CENTRAL TENDENCY

The diagrammatic representation of a set of data can give
  us some impression about its distribution. Even then
  there remains a need for a single quantitative measure
  which could be used to indicate the center of the
  distribution . An average is a single value which
  represent a set of data or a distribution as a whole. It
  is more or less central value round which the
  observations in the set of data or distribution tend to
  cluster. Such a central value is also called a measure
  of central tendency or measure of location.

                                                      2
 MEASURE OF CENTRAL TENDENCY

TYPES OF AVERAGES


1.   Arithmetic mean

2.   Median

3.   Mode

4.   Geometric mean

5.   Harmonic mean




                           3
MEASURE OF CENTRAL TENDENCY
Arithmetic Mean
It is obtained by dividing the sum of all the observations
     by the total number of observations. It is denoted by
     " X ".
For ungrouped data
                 x1 + x2 + x3 + ...xn  X
              X=                     =
                          n            n
  For grouped data

              f 1 x1 + f 2 x2 + f 3 x3 ... f k xk  fx
           X=                                    =
                     f 1 + f 2 + f 3 ... f k       f
                                                         4
Example
The following data is the final plant height (cm) of thirty plants of wheat.
   Construct a frequency distribution

87 91     89       88        89         91      87               92
   90     98       95        97         96      100            101        96
          98       99        98         100     102              99
          101      105       103        107     105      106
   107    112




                                                                      5
 MEASURE OF CENTRAL TENDENCY


For ungrouped data


            x1 + x2 + x3 + ...xn  X
         X=                     =
                     n            n

                2929
            X =      = 97.63
                 30


                                       6
    FREQUENCY          DISTRIBUTION         TABLE

  Class Class       Mid            Tally    Frequ   fx
                                            ency
  Limits Boundaries Points
                                             (f)
 86---90     85.5---90.5    88   /////        6     528
 91---95     90.5---95.5    93   ////         4     372
 96---100   95.5---100.5    98   ////////    10     980
101---105   100.5--105.5   103   /////        6     618
106---110   105.5– 10.5    108   ///          3     324
111---115   110.5--115.5   113   /            1     113
 MEASURE OF CENTRAL TENDENCY

For grouped data

               f 1 x1 + f 2 x2 + f 3 x3 ... f k xk  fx
           X =                                    =
                      f 1 + f 2 + f 3 ... f k       f


                      fx 2935
         X =                   97 .83
                     f    30


                                                          8
Median

         9
Median
  The median is a value that divides a set of data in to
  two equal parts after arranging the values in ascending
  or descending order of magnitude.

For ungrouped data

(i) When "n" is odd then the median is
          [(n+1)/2]th observation.

(ii)When "n" is even then the median is
          1/2[(n/2) + (n/2+1)]th observation.



                                                  10
Median
Example
Given below are the marks obtained by 20 students.
      53        74      82     42     39      28     20
      81        68      58     67     54      93     70
      30        61      55     36     37      29.
Find Median
Solution:
First arrange the data in ascending order of magnitude
       20       28      29     30     36      37     39
       42       53      54     55     58      61     67
        68      70      74     81     82      93
Median = 1/2[(n/2) + (n/2+1)]th observation.
Median = 1/2[ 10 + 11]th observation = 1/2[ 54 +55]
Median = 54.5
                                                          11
Median
Example
Given below are the marks obtained by 9 students.
     45       32     37
     41       48     36
     46       39      36
Find Median
Solution:
First arrange the data in ascending order of magnitude
       32       36      36
       37       39      41
        45      46      48
Median = [n + 1]/2th observation.
Median = [9 + 1]/ 2th observation = 5th observation
Median = 39
                                                     12
Median
     For grouped data

            h n
Median = l + ( - c)
            f 2
l=     Lower class boundary of the class containing median
h = Class interval of the class containing median*
f=     Frequency of the class containing median
n=      Total number of observations
c = Cumulative frequency of the class preceding the class
         containing median.
                                                      13
Median
  Example: Estimate the Median
   Daily   No. of     c. f          class
Income Rs Work                   boundaries
           ers( f )
  5-----24    4       4       4.5---24.5
 25-----44    6       10      24.5---44.5
 45-----64   14       24      44.5---64.5
 65-----84   22       46      64.5---84.5
 85----104   14       60     84.5---104.5
 105---124    5       65     104.5---124.5
 125---144    7       72     124.5---144.5
 145---164    3       75     144.5---164.5    14
  Median
 Step -1 : Calculate n/2, its helps which class
            containing the median.

        n/2 = 75 / 2 = 37.5 see this observation in
    cumulative frequency column
Step- 2: The class containing the median is 64.5 ---- 84.5
Step- 3: l = lower class boundaries containing the median
       =    64.5
Step- 4: h = class interval of the class containing the median
       =    20
Step- 5: f = frequency of the class containing median = 22
Step- 6: c = cumulative frequency of the preceding class = 24

                                                        15
    Median



                 h     n
   Median = l +    (      - c)
                 f     2
                20     75
Median = 64.5 +    (       - 24)
                22      2
          = 76. 77

                             16

				
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Description: Location (MEAN & Median).ppt