Institutional Planning

Lubna
Toheed

Keywords; Calculation Of mean, Properties of
Mean, Merits of Mean, Demerits of Mean,
Median, Procedure to calculate median, Merits of median, Demerits of
Median

Measures
of central tendency

Introduction                      

Measures of central tendency describe how the data
cluster together around a central point. It is useful for making comparisons
between groups of individuals or between sets of figures. It reduces a large
number of measurements to a single figure, and thus makes comparisons easy.

Unfortunately, there is no single, standard procedure for
determining central tendency. The problem is that no single measure will always
produce a typical, representative value in every situation. Therefore, we use
different ways to measure central tendency. There are three common measures of
central tendency:

1.    The
Mean or Average or arithmetic Mean,

2.    The
median and ,

3.    The
mode.

These three statistical measures are computed differently
and have different characteristics. Here we will discuss ‘Mean’ and ‘Median’
only.

Mean         

The Mean of a distribution
is known is “Arithmetic Mean”. Mean indicate the average value of the group.
The Mean in each group or condition is calculated by adding up all the
scores in a given condition, and then dividing by the number of participants in
that condition.  It can be defined as “
the point on the scale of measurement obtained by dividing sum of all scores by
the number of the scores.

Uses of Mean

1.   
 Mean is used when a reliable and accurate
measure of central tendency is needed.

2.   
 Mean is used when the same scores are
distributed symmetrically around the central point.

3.   
Mean
is used when the greatest stability is required

4.   
Mean
is used when other stastical calculation S.D./ M.D./ A.D. are to be completed.

5. Mean is used when we are having a series with no
   extreme items.

Calculation
Of mean

Mean is calculated by two type of data

(1) 
Ungrouped
Data

(2) 
Grouped
Data

(1)Ungrouped
Data

We use the following formula to find
the mean of ungrouped data:





 

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(1)  
Grouped Data

When the scores are large in number,
the data is first grouped into the frequency distribution table and then the
mean is calculated by using the following formulas:

a)Long Method

 

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A.M = Assumed mean

 = Deviations of mid-point of the class
interval from A.M divided by the size of class interval

F = frequency

I = size of the class interval.

Examples:             

a)Using Long method        

Calculate mean for  the following grouped data:

s.no	Class interval	Frequency
1 2 3 4 5	10-14 15-19 20-24 25-29 30-34	5 4 6 3 2
		N=20

The above data is consisted of class interval and frequencies. The
size of each class interval is 5. It is difficult to know the aggregate of each
class interval. So it is necessary to find out the mid- point of each class
interval by using the following formula:

.

b) Using Short Method

Using short method mean is calculated
by taking the assumed mean from any class interval.

Suppose assumed mean (A.M) remain in
the C.I. 22-24. The mid point of that C.I. is 22, is regarded as A.M.

While calculating of any class interval the formula used
is:

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Properties
of Mean

1.   
The
sum of deviations of all scores in a set from their Arithmetic Mean is “0”.

2.   
The
Sum of deviation from Arithmetic Mean is less then the sum of square of
deviations from any other value.

3.   
If
each score of a series is subtracted by a constant quantity, the Mean will be
as increased or decreased by the same quantity.

4.   
If
each score of a series divided or multiplied by a constant quantity, the mean
will be multiplied or divided by the same quantity.

Merits of Mean

1.   
It
is rigidly defined.

2.   
It
is based on all the values.

3.   
It
is more stable than any other method of  average.

4.   
It
is relatively reliable.

5.   
It
is the center of gravity, balancing the values on either side of it.

6.   
It
has further algebraic treatment capacity.

Demerits of Mean

1.   
It
is highly affected by abnormal values.

2.   
The
loss of even a single observation makes it impossible to compute the aritmetic
mean correctly.

Limitations

Mean
can be very misleading if the distribution differs markedly from the normal and
there are one or two extreme scores in one direction.

Median

The Median is the middle item when the data is listed in
order. It is the point below which remain 50% of ‘N’ and above which remain the
other 50% of N. So it is also called the 5^th percentile. Mean is
called the positional average .The position refer to the place of the value in
a series.

Median is used:

1. When  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. For or articles
   that cannot be precisely measured.

Calculation Of mean

Mean is calculated for two types of
data

(a)  
Ungrouped
Data

(b)  
Grouped
Data

(a)  Ungrouped
Data

(1) 
If
there is an odd number of scores, then the median is simply the middle score,
having an equal number of scores higher and lower than it.

The
formula for the calculation of the 
ungrouped data:

.

Example:

Calculate the
Median from the following distribution table.

C.I.	50-54	55-59	60-64	65-69	70-74	75-79	80-84	85-89
f	8	12	24	16	32	20	11	5

Procedure to calculate median

1.      
Locate
the Median group  by N/2th

2.      
Find
out the cumulative frequencies

3.      
Point
out where N/2th score lies in which class interval

4.      
Apply
the Formula of Median.

S.no	C.I	C.I.	f	cf
1 2 3 4	50-54 55-59 60-64 65-69	49.5-54.5 54.5-59.5 59.5- 64.5 64.5- 69.5	8 12 24 16	8 8+12=20 20+24=44 44+16=60
5	70-74	69.5-74.5	32	60+32=92
6 7 8	75-79 80-84 85-89	74.5-79.5 79.5- 84.5 84.5-89.5	20 11 5	92+20=112 112+11=123 123+5 = 128
			N = 128

Where N/2 is 128/2 = 64

64 frequency remains in the class
interval 69.5- 74.5 where the cumulative frequency is 92.

The Formula is:

Add caption

Merits of median

1.    It
is easily calculated

2.    It
is specially used in open-ended class interval

3.    It
is not affected by the values of extreme items as it focuses only on scores in
the middle of the distribution.

4.    It
is the most appropriate average in dealing with qualitative data

Demerits
of Median

1.    In
calculating Median the data are to be arranged in descending or ascending order

2.    which
is time consuming

3.    Since
it is a positional average, its value is not determined by each and every
observation.

4.    It
may sometimes be indefinite, when the number of items in a median class is
large.

5.    Median
may sometimes be located at the point where the frequency may be quite small.

Limitations

1.    It
is not suitable for further algebraic treatment.

2.    It
can’t be used for computing other statistical measures such as S.D.,
co-efficient of correlation.

3.   
The
main limitation of the median is that it ignores most of the scores, and so it
is often less sensitive than the mean.

4.   
In
addition, it is not always representative of the scores obtained, especially if
there are only a few scores.

Summary

1.   
Central
tendency is the index which represents average performance of the group.

2.   
Measures
of central tendency describe how the data cluster together around a central
point.

3.   
There
are three common measures of central tendency:

a)   
Mean
or Arithmetic mean

b)   
Median

c)   
Mode

4.   
Mean
indicate  the average value of the group.

5.   
Calculation
of mean:

 
If
there is an odd number of scores, then the median is simply the middle score.  
If
there is an even number of scores. In that case, the mean of the two central
values is Median.

 

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a)   
 

b)   
series contain extreme scores.

c)   
distribution
is  open end.

d)   
for
articles that cannot be precisely measured.

  Bibliography

1.   
Biswal,
B. Stastics in Education & Psychology, Dominant Publishers and
Distributers,New Delhi,
2006

2.   
Text
Book of Mathematics for class 10^th, KPK text book board Peshawar.

3.   
http://statistics-in-education-and-psychological-studies.nuvvo.com/lesson/13349-measures-of-central-tendency