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Measures of Central Tendency

MEASURES OF CENTRAL TENDENCY<?xml:namespace prefix = o ns = "urn:schemas-microsoft-com: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

65-69

1

67

67

60-64

3

62

186

55-59

4

57

228

50-54

7

52

364

45-49

9

47

423

40-44

11

42

462

35-39

8

37

296

30-39

4

32

128

25-29

2

27

54

20-24

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'=(X-A)/i

fx'

65-69

1

67

5

5

60-64

3

62

4

12

55-59

4

57

3

12

50-54

7

52

2

14

45-49

9

47

1

9

40-44

11

42

0

0

35-39

8

37

-1

-8

30-39

4

32

-2

-8

25-29

2

27

-3

-6

20-24

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'= X-A/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:schemas-microsoft-com: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 4th student=35

The score of [(N/2)th+1] student ie 5th student=39

Then

Md = 35+39/2= 37

CALCULATION OF MEDIAN FROM GROUPED DATA

Scores

f

F

65-69

1

50

60-64

3

49

55-59

4

46

50-54

7

42

45-49

9

35

40-44

11

26

35-39

8

15

30-39

4

7

25-29

2

3

20-24

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 40-44.

L= 39.5 since median class is 40-44 so exact lower limit of the class interval is 39.5.

F= 15 is the frquency from class interval below the class interval 40-44 is 35-39.

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 + 25-15 * 5

11

=> 39.5+4.55

= 44.05 is the Median

When to use median:

  1. When the mid-point of the given distribution is to be found.
  2. when the series contain extreme scores.
  3. when there is open end distribution it is more reliable than mean.
  4. mean cannot be calculated graphically, median can be calculated graphically.
  5. 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

65-69

1

60-64

3

55-59

4

50-54

7

45-49

9

40-44

11

35-39

8

30-39

4

25-29

2

20-24

1

N =50

Fomula to be used:

Mo = L + f- f1* i

(f-f1)+(f-f2)

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 + 11-9 * 5

(11-9)+(11-8)

=> 39.5 + 2/5* 5

=> 39.5+2

=> 41.5 is the mode.


  1. czeishin saidMon, 11 Oct 2010 15:19:05 -0000 ( Link )

    90-89
    85-84
    80-79
    75-74
    70-69
    65-64
    how about if given is like that, how can i get the midpoint?

    i need the answer asap…
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  2. mohsinrazavirgo saidWed, 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

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  3. mohsinraza_virgo saidWed, 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

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  4. atl_ji saidTue, 09 Sep 2014 11:54:04 -0000 ( Link )

    for the data given find out central 60% limit of the age :
    age no. Of workers
    20-24 5
    25-29 10
    30-34 15
    35-39 18
    40-44 23
    45-49 12
    50-54 7

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