MEASURES OF CENTRAL TENDENCY
In the given set of a scores, there will be a score in which all the
scores are clustered, such a tendency of a scores is called central tendency.
To measure this central tendency we have 3 measures, like Mean, Median and
mode. These are called measures of central tendency.
1. The Arithmetic Mean (M):
The Arithmetic mean is the best known and the most widely used measure
of a central tendency.
From Ungrouped Data
(i)
The
arithmetic mean is the measure resulting from dividing the sum of the scores in
the distribution by the number of scores. Here,
M = ∑X/N Where, X = a score
N=number of scores in a series;
∑=stands for “the sum of.”
Thus, the arithmetic mean or, more
simply the mean of the scores
93, 90, 89, 88, and 86 is
M= 93+90+ 89+ 88+ 86 =
446 = 89.2
5 5
(ii)
There is
another short-cut method. Let us assume that the mean in the above series is
(AM) 89. Find out the deviation of each score from the mean. It will be
|
X
|
d=X –x
|
|
93
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93-89= 4
|
|
90
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90-89= 1
|
|
89
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89-89= 0
|
|
88
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88-89=
-1
|
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86
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86-89= 3
|
|
∑
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1
|
Use the formula: M = X + ∑ X1
N Where, x = assumed mean;
d= deviations from the assumed
.Mean.
. . M=89 + 1/5
M= 89.2
From Grouped Data
(i) Long Method:
C.I Midpoints
(x) f fx
|
10- 14
12
3 36
|
|
15 – 19 17 5 85
|
|
20 – 24 22 9 198
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|
25- 29
27 18 486
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30 – 34 32 11 352
|
|
35 -39
37
5
185
|
|
40 – 44 42 6 252
|
|
45- 49
47
2
94
|
|
50 – 54
52 1 52
|
N = 60 ∑ = 1,740
Mean = ∑f x = 1,740= 29
N 60
M= 29
(ii)Short-cut method:
|
C.I
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f
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x
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d=x-AM/i
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fd
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10-14
|
2
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12
|
-3
|
-6
|
|
15 -19
|
3
|
17
|
-2
|
-6
|
|
20-24
|
4
|
22
|
-1
|
-4
|
|
25- 29
|
6
|
27
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0
|
0
|
|
30 -34
|
5
|
32
|
+1
|
5
|
|
35- 39
|
2
|
37
|
+2
|
4
|
|
40 -44
|
2
|
42
|
+3
|
6
|
N =
24
∑fd = -1
d= x-M/i = 12 – 27 where, x = mid point
5
M = Assumed mean
i=size of the class interval
Mean = AM + ∑f d X i
N
= 27 + (-1/24)5
= 27 – 5/24
= 27 – 0.2
M=26.8
Merits of Arithmetic Mean
1. It is rigidly defined, and a biased
investigator will get the same mean from the series as an unbiased one. Its
value is always definite.
2. It can be accurately determined with the
help of various methods.
3. A mean is an algebraic measure which is
capable of further algebraic treatment. It is useful in computing standard
deviation, correlation, etc.
4. It is simple to follow. It is very
easily understandable.
5. If the number of items in a series is
large, the mean provides a good basis of comparison.
6. A mean has an important property, and
that is that is that the sum of the deviations of all the scores from the mean
is always zero. In this respect, the median does not qualify.
Limitations of the Mean
1. As the skewness of the distribution
increases, the reliability of the mean decreases.
2. Since it is calculated from all the
items of a series, sometimes the abnormal items may considerably affect the mean,
particularly when the number of items is not large. For example, the mean ofRs.
275 is not at all a representative figure of Rs. 1,000, Rs.25, Rs.35 and Rs.
40.
3. If a single item of a series is missing,
the mean cannot be calculated. If out of 500 items, the values of 499 items are
known, the mean cannot be calculated. Other averages, such as the median and
the mode, do not need complete data.
4. A mean gives greater importance to the
bigger items of a series and less importance to the smaller items. One big item
among five, four of which are small, will push up the mean considerably. Thus,
the mean has an upward bias. But the reverse is not true. If in a series of
five items, four of whichare big and one item small, the mean will not be
pulled down very much; e.g., the mean of items 25, 50, 75, 100 and 850 is 220,
while the mean of items 500, 525, 550, 575, and 50 is 440.
5. Sometimes the mean gives fallacious
conclusions; e.g. income of two groups of persons;
A group: Rs.1000, Rs.100,
Rs.75. Rs.25
B group: Rs.325, Rs.300,
Rs.285. Rs.290
For both the groups, the mean is Rs.300.
It would appear that both the groups are economically at the same level, but
this is not really so.
6. Unless the data are very simple, the
mean cannot be located merely by inspection, while the median and the mode can
be.
7. It cannot be accurately determined in
the case of an open-end table. (A open-end table is that which does not provide
exact scores of the extreme 4 or 5 cases.)
8. A mean can be used only with
distributions that give absolute scores. It cannot be used with scores
expressing grades or ranks or order of students, such as A, B,C, D,….
9. A Mean can be a figure which does not
exist in the series at all.
10. A Mean sometimes gives such results as
appear almost absurd; e.g., 3.4 children.
Use the mean…
(i)
When the
scores are distributed symmetrically around the central point, i.e., when the
distribution is not badly skewed. M is the centre of gravity in the
distribution and each score contributes to its determination;
(ii)
When the
measures of a central tendency having the greatest stability and reliability
are wanted;
(iii)
When other
statistics (e.g. S.D, coefficient of correlation) are to be computed later.
2.
The Median (Mdn)
A median is
the average of position. It is often called the counting average. “The median
is the point below which 50% of the score lie.
·
It is the
middle item of the series.
·
It is exactly
half of the total series.
·
It is a point
not a score.
(i)
Find the
median of the scores 24, 21, 30, 18, 27, 28, 30. Here N is odd. Arranging these
seven scores in order of size we get,
18, 21,
24, 27, 28,
30, 30
The median is
27.0, for there are three scores above and three below 27.0
(ii)
Find the
median of the scores 24, 21, 30, 18, 27, 28 Here N is even. Arranging these six
scores in order of size, we get,
18, 21,
24, 27, 28,
30
Here there
are two idle scores, 24 and 27. The average of these two, i.e. (24+27/2 =25.5,
is the median.
In short,
median is the N+1/2th measure in order of size
From Grouped
data
|
C.I
|
f
|
F/Cf
|
|
10-14
|
3
|
3
|
|
15-19
|
5
|
8
|
|
20 -24
|
9
|
17
|
|
25-29
|
18
|
35
|
|
30-34
|
11
|
46
|
|
35 -39
|
5
|
51
|
|
40 -44
|
6
|
57
|
|
45 -49
|
2
|
59
|
|
50 -54
|
1
|
60
|
N = 60
N/2= 60/2 =30
L = 24.5
Median (Md) =L + N/2 - FXi
F = 17fm
Fm = 18
= 24.5 + 30-17 X 5
i=5
18
= 24.5 + 13X5
18
=24.5
+ 3.6
Median = 28.1
Merits of the Median
1. It is an ideal average, for it is
rigidly defined.
2. It is easily understood without any
difficulty.
3. It is not affected by the values of the
extreme items (i.e., skewness) and as such is sometimes more representative
than the mean.
If the income of five persons are Rs.300,Rs.350, Rs.400, Rs.450 and
Rs.10,000, the median in such cases is a better average.
4. In case of an open-end table, where the
values of the extremes are not known, the median can be calculated if the
number of items is known.
5. The median can be used not only with
distributions that give absolute scores such as 25, 30, 32, etc., but also with
scores expressing grades or ranks or order of students, such as A, B, C, D…
6. When the units of measurement are
unequal, the median is preferable. It does not assume equality of units,
whereas the mean does assume it.
7. The median never gives absurd or
fallacious results.
8. It can be located in many cases merely
by an inspection.
Limitations of the Median
1. A median is a non-algebraic measure and
hence not suitable for further algebraic treatment.
2. It cannot be used for computing other
statistical measures such as SD, or coefficient of correlation.
3. The arrangement of items in the
ascending or descending order is sometimes very tedious.
4. When there are wide variations between
the values of different items, a median may not be representive of the series;
e.g.,
Marks: 15, 16, 16, 18, 18, 20, 54, 60, 70, 70, 82
The median is 20, which is not representive of the series.
5. When a median has to be calculated in
continuous series, it requires interpolation. The assumption of the
interpolation that all the frequencies of the class interval are uniformly
spread over their values in the class interval may not be actually true. In
most cases, it will not be true.
6. If big or small items in a series are to
receive greater importance the median would be an unsuitable average, for it
ignores the values of extreme items.
7. A median is more likely to be affected
by the fluctuations of sampling than the mean.
When to use a median?
1. When the exact midpoint---the 50% point
of the distribution--- is wanted;
2. When there are extreme scores which
would markedly affect the mean. They do not disturb the median;
3. When it is desired that certain scores
should influence the central tendency;
4. When the distribution has an upper or
lower class interval of unspecified length;
5. When a distribution is described and
interpreted in terms of percentiles, it is used as a member of the percentile
system.
3. The Mode
The crude or empirical mode (inspection average) is the most quently
occurring score in the distribution.
The mode or the modal value
is the 3rd measure of central tendency of the frequency
distribution.
The most often repeated
score is called Mode.
Example: 66, 68,
69, 70, 70,
70, 71, 71,
72, 73
Here the most often repeated
score is 70. The mode of these score is 70. The above setoff scores has only
one value as a mode therefore is called unimodal.
If two scores are repeated
then we say that distribution is bimodal, if it is three trimodal etc.
Ex: 7, ⑩, ⑥,
8, 13, 9, ⑩, 11,
⑥
When data are grouped into a
frequency distribution, the crude mode is usually taken to be the midpoint of
that interval which contains the largest frequency. For the distribution given
in problem (ii) of mean calculation in short-cut, the mode is 27.0.
A distribution may have more
than one mode. It may be unimodal or bimodal or multimodal.
The true mode is the point
(or peak) of greatest concentration in the distribution. When the distribution
is not badly skewed,the approximation of the true mode can be calculated by the
formula;
Mode=3 median – 2
mean
Or Mo= 3 Mdn – 3 M
Or Direct formula we can use,
Mo = L + fm – f1 X i
2fm – f1 – f2
Where, L ---Lower limit of C.I. where the mode lies.
fm. --- Frequency
of C.I. where the mode lies.
f1 ----first
frequency, the frequency before frequency in which mode lies.
f2 – Second
frequency, after frequency in which mode lies.
i-size of the class interval
Merits of the Mode
1. It possesses the merit of simplicity. In
a discrete series, the mode can be located even by inspection.
2. It is commonly understood. A mode is an
average which people use in their day-to-day expressions; e.g., sale of a
commodity, size of families, etc., are expressed in their mode values.
3. Unlike the mean, it cannot be a value
which is not found in the series.
4. A mode is not affected by the values of
extreme items, provided they adhere to the natural law relating to extremes.
5. For the determination of a mode, it is
not necessary to know the values of all the items of a series.
Limitations of the Mode
1. A mode is ill-defined, indeterminate and
indefinite, and so untrustworthy.
2. It is not based on all the observations
of a series.
3. It is not capable of further
mathematical treatment.
4. It may be unpresentative in many cases.
A slight change in the series may extensively disturb the mode. For example, in
the series, 20, 25, 25, 25, 30, 60, 70, 95, 95, 100,
the mode is 25. If one student gets 26
instead of 25 and one gets 95 instead of 100, the mode will be shifted to 95.
5. The mode is affected by changes in the
grouping scheme, while preparing frequency distributions.
6. In ungrouped data, if no score is
repeated, it may lead to the wrong conclusion that the series has no mode.
7. As there may be 2, 3, or more modal
values, it becomes impossible to set a definite value of a mode.
When to Use a Mode?
Use a mode ----------
1. When a quick and approximate measure of
a central tendency is all that is wanted;
2. When the measure of a central tendency
should be the most typical value.
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