# Lec Location (MEAN & Median).ppt by ZubairLatif

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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.

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MEASURE OF CENTRAL TENDENCY

TYPES OF AVERAGES

1.   Arithmetic mean

2.   Median

3.   Mode

4.   Geometric mean

5.   Harmonic mean

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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
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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

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MEASURE OF CENTRAL TENDENCY

For ungrouped data

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

2929
X =      = 97.63
30

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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

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Median

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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.

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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
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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
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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.
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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

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Median

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

16

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