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:
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| . |
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 5th percentile. Mean is
called the positional average .The position refer to the place of the value in
a series.
Median is used:
- When mid-point 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.
- 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:
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| 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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| https://research-education-edu.blogspot.com/2020/01/measures-of-central-tendency_8.html |
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 10th, KPK text book board Peshawar.
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