MEASURES OF CENTRAL TENDENCY<?xml:namespace prefix = o ns = "urn:schemasmicrosoftcom:office:office" />
The statistics, mean median and mode are known to be the most common measures of central tendency. A measure of central tendency is a sort of average or a typical value of the item in the series or some characteristic of members in a group. Each of these measures of central tendency provides a single value o represent the characteristic of the whole group in its own way.
According to Tete measure of central tendency is:
"A sort of average or typical value of the items in the series and its function is to summarize the series in terms of this average value."
Mean represents the average for an ungrouped data; the sum of the scores divide by the total number of the scores gives the value of the mean.
Median is the score or value of that central item which divides the series in exactly two equal halves.
Mode is defined as the size of the variable that occurs most frequently in the series.
Mean
Arithmaetic mean is simply the average of the series represented by M. Mean for an ungrouped data can be calculated by adding all the scores and dividing it by the total number of score. This can be represented by the following equation:
M = X1+X2+X3+X4+…X10
10
or
M= ∑X
N
∑X stands for total of all the scores while N stands for total number of scores.
CALCULATING MEAN OF GROUPED DATA:
Scores 
f 
Midpoint (X) 
fX 
6569 
1 
67 
67 
6064 
3 
62 
186 
5559 
4 
57 
228 
5054 
7 
52 
364 
4549 
9 
47 
423 
4044 
11 
42 
462 
3539 
8 
37 
296 
3039 
4 
32 
128 
2529 
2 
27 
54 
2024 
1 
22 
22 
N= 50 and ∑fX= 2230 M=∑fX/N
M=2230/50= 44.6
Shortcut method {assumed mean method}
Scores 
f 
Midpoint (X) 
x'=(XA)/i 
fx' 
6569 
1 
67 
5 
5 
6064 
3 
62 
4 
12 
5559 
4 
57 
3 
12 
5054 
7 
52 
2 
14 
4549 
9 
47 
1 
9 
4044 
11 
42 
0 
0 
3539 
8 
37 
1 
8 
3039 
4 
32 
2 
8 
2529 
2 
27 
3 
6 
2024 
1 
22 
4 
4 

N =50 


∑fx'=26 
Formula:
M= A+∑fx'/N *i
M= mean
A= assumed mean
f= respective frequency of the mid values of the class intervals
N=total frequency
x'= XA/i
putting the values into the formula
M= 42+26/50*5
M= 42+2.6
M= 44.6
When to use mean:
 When we have to get a reliable and accurate value of central tendency.
 When we are in need to compute further statistics like standard deviation, coefficient of correlationetc.
 When we are having a series with no extreme items.
Median
Median is score or the value of that item which divides the series into two equal parts.
It is represented by <?xml:namespace prefix = st1 ns = "urn:schemasmicrosoftcom:office:smarttags" />Md.
Calculating the median of an ungrouped data:
Let there be a group of 8 students whose scores in test is 17, 47, 15, 35, 25, 39, 50, 44.
First we arrange the scores in ascending order: 15, 17, 25,35,39,44,47,50.
The score of (N/2)th student ie 4^{th} student=35
The score of [(N/2)th+1] student ie 5^{th} student=39
Then
Md = 35+39/2= 37
CALCULATION OF MEDIAN FROM GROUPED DATA
Scores 
f 
F 
6569 
1 
50 
6064 
3 
49 
5559 
4 
46 
5054 
7 
42 
4549 
9 
35 
4044 
11 
26 
3539 
8 
15 
3039 
4 
7 
2529 
2 
3 
2024 
1 
1 

N =50 

The formula to be used :
Md= L + (N/2)F * i
f
Md = Median
L= exact lower limit of the class interval in which the median class lies
N= total number of scores
F= cumulative frequency of the class interval lying below the median class
f= frequency of the class interval in which median class lies
i= size of class interval
so according to the data available:
calculation of median class
formula N/2= 25 frequency closest to 25 is 11 so we can say that the median class lies in class interval 4044.
L= 39.5 since median class is 4044 so exact lower limit of the class interval is 39.5.
F= 15 is the frquency from class interval below the class interval 4044 is 3539.
f= 11 is the frequency of the median class.
i= 5 is the size of the class interval.
Md = 39.5 + 50/2 15 * 5 .
11
=> 39.5 + 2515 * 5
11
=> 39.5+4.55
= 44.05 is the Median
When to use median:
 When the midpoint of the given distribution is to be found.
 when the series contain extreme scores.
 when there is open end distribution it is more reliable than mean.
 mean cannot be calculated graphically, median can be calculated graphically.
 used for articles that cannot be precisely measured.
Median
It is the value of the item which is most characteristic of the and also most repeated.
CALCULATION OF MODE FROM UNGROUPED DATA:
This can be done be simply looking at the given data. For example the series is
35,39,34,35,37,35,38,35,39
here 35 is repeated the maximum times so 35 s the mode.
CALCULATION OF MODE FROM GROUPED DATA:
Scores 
f 
6569 
1 
6064 
3 
5559 
4 
5054 
7 
4549 
9 
4044 
11 
3539 
8 
3039 
4 
2529 
2 
2024 
1 

N =50 
Fomula to be used:
Mo = L + f f1* i
(ff1)+(ff2)
where
Mo =mode
L= exact lower limit of the modasl class
f= frequency of the modal class
f1= frequency of the class interval lying above the modal class
f2= frequency of the class interval lying below the modal class
i= size of class interval
according to given data:
finding of modal class it is simply he class interval with highest frequency.
L= 39.5
f=11
f1= 9
f2= 8
i=5 putting it into formula
Mo = 39.5 + 119 * 5
(119)+(118)
=> 39.5 + 2/5* 5
=> 39.5+2
=> 41.5 is the mode.
3 Comments

mohsinrazavirgo said – Wed, 21 Mar 2012 07:28:57 0000 ( Link )
here your size of interval h=5 so firstly add 90 and 89 =179 then divided by 2 you get 89.5 which is your midpoint of first class now subtract h=5 from 89.5 you get 84.5 which is midpoint of second class and soo on you gets
89.5
84.5
79.5
74.5
69.5
64.5 etc Actions
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mohsinraza_virgo said – Wed, 21 Mar 2012 07:42:58 0000 ( Link )
here your size of interval h=5 so firstly add 90 and 89 =179 then divided by 2 you get 89.5 which is your midpoint of first class now subtract h=5 from 89.5 you get 84.5 which is midpoint of second class and soo on you gets
89.5
84.5
79.5
74.5
69.5
64.5 etc Actions
Post Comments
Post Comments