Central tendency may be defined as a value of the variety which is thoroughly representative of the series or the distribution as a whole. The measure of this character are known as averages.
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Averages:
Types of Averages:
Following are the different types of averages which are in common are:
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1. Arithmetic Mean
2. Median
3. Mode
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4. Geometric Mean
5. Harmonic Mean
6. Weighted Mean
Arithmetic Mean:
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This is the only form of average which is of practical importance. Its value depends upon all the observations.
If x1, x2, x3,……… xn are the N observations.
The mean x will be given by the following expression:
X = 1/N (x1 + x2 + x3 +……. + xn) = Ʃx/N
Method of Computation —
(A) Computation from the raw data.
Example 1:
Find out the arithmetic mean of the following fifteen numerical quantities.
62, 85, 103, 107, 109, 110, 114, 115, 126, 151, 154, 161, 165, 170, 175.
Here A.M = Ʃx/N
= 62 + 85 +…………..+ 175 / 15
= 1970/15 = 127.133
(B) Computation from the frequency distribution
(i) Discrete series:
Example 2:
The arithmetic mean of the following discrete distribution of wages of 100 workers has been computed by calculating the value of N and Zfx
Merits of Arithmetic Mean:
1. It is rigidly defined.
2. It based on all the observations.
3. It can be easily calculated.
4. Its algebraic treatment is specially easy and definitely possible.
Demerits:
1. It is very much affected by the values at extremes and hence, often abnormal results are obtained due to some abnormal value.
2. If the items of the beginning and end of the series are not known, arithmetic mean can not be calculated.
Uses of A.M.:
1. It is calculated to compare two or more series with respect to certain character.
2. It is used in calculating the standard deviation of the data.
3. It is calculated, for calculating the regression coefficient and the correlation coefficient.
Median:
When all the observations are arranged in ascending or descending order of magnitude, the middle one is known as the median.
If N is the total number of observations, the value of the (N+1 / 2)th item will be called median (N + l /2) ,which stands for the serial order of the median is known as median number.
Method of Computation —
(A) Computation from the raw data.
Example 1:
(When the number of item is odd).
Find the median of the following figures:
10, 11, 13, 11, 13, 10, 13, 11, 12, 15, 12
Here, median number = 11+1/2 = 6 Arranging the figures in the ascending order of magnitude, we get
10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 15
Here 6th figure is 12.
Therefore, the median is 12.
Example 2:
(When the number of item is even).
Find the median of the following figures:
10, 7, 11, 9, 8, 10, 11, 7, 9, 12
Here, median number = 10+1/2 = 5 1/2
But, serial order of a term can never be fractional. In such a case there are two middle terms 5th and 6th. The mean of these two terms will be median.
Arranging the terms, we get
7, 7, 8, 9, 9, 10, 11, 11, 12
Here, 5th term = 9
6th term = 10
Median = 9+10/2 = 9.5
(B) Median for grouped data:
We first prepare the cumulative frequency table. Let the total number of observations be n. We determine the median class i.e., the class in which (n/2)th observation lies, then we use the formula
Median = l + h/f1 (n/2 – F0)
Where,
l — is the lower limit of the median class,
n — is the total frequency,
h — is the range (upper limit – lower limit) of the median class;
f0 — is the cumulative frequency of the class just before the median class and f1 is the frequency of the median class.
Merits of Median:
1. It is easy to calculate.
2. It is not affected by the extreme values.
3. It can be located graphically.
4. It is best suited for open-end classes.
Demerits of Median:
1. It can be located precisely, when the items are grouped.
2. It is not based on all the observations in the given data.
3. It may not be a true representative of a given data, when the items vary greatly in magnitude.
4. For finding out the median of a data, it is necessary to make an array in ascending or descending order. So when the number of item is quite large, this operation becomes tedious.
Mode:
This is the value of the variable which occurs most frequently or whose frequency is maximum. Also, if several samples are drawn from a population, the important value which appears repeatedly in all the samples is called the mode.
Mode for Individual Data:
Example:
The marks obtained in a test by 12 students in a class are given below:
37, 23, 16, 19, 34, 23, 5, 27, 36, 23, 20, 38 Find the modal mark.
Solution:
Arranging the data in ascending order.
5, 16, 19, 20, 23, 23, 23, 27, 34, 36, 37, 38
Obviously, 23 occurs the maximum number of times
Mode = 23
Mode for Grouped Data
Mode = l1 + f2 / f1+ f2 x i
Where, l1, = lower limit of the modal class
f1 = Frequency of the next lower class
f2 = Frequency of the next higher class
i = width of the class interval
Merits of the Mode:
1. It is easily comprehensible.
2. It can be obtained simply by inspection. Even if it is to be computed, it can be computed very easily.
3. This is the value whose expectation is the greatest in whole of the statistical series.
4. Neither the extremes are needed in its computation nor it is affected by them.
Demerits:
1. In various cases, there is no single and well defined mode.
2. Its computation is not based on all the observations.
3. It is unsuitable for any algebraic treatments.
4. When there are more than one mode in the series, it becomes difficult and takes much time to compute it.
5. In any set of observations, where, there are very small differences among the items of the same size, mode does not prove to be a good representative.
Use of mode:
Its use is increasing everyday especially in the field of business.
Measures of dispersion:
The measures that are constructed to indicate the spread of the data with respect to the average in a set of observations are called the measures of dispersion.
Range:
Range is the simplest measure or dispersion. It is the difference between the highest and the lowest terms of a series of observations.
Mean Deviation:
If the deviations of all the observations from their mean are calculated, their algebraic sum will be zero. When this sum is always zero, it is impossible to get the average of these deviations.
For the overcome this difficulty these deviations are added irrespective of plus or minus sign and then the average is calculated.
The deviations without any plus or minus sign are known as absolute deviations. The mean of these absolute deviations is called the mean deviation about the mean.
It is defined by the following expression:
Mean deviation about the mean
Where, x = Deviation from the mean = x-x
\x\ = Absolute deviation
N= Number of observations.
Standard Deviation:
Its calculation is based on the deviations from the arithmetic mean.
In case of mean deviation the difficulty that the sum of the deviations from the arithmetic mean is always zero is solved by taking these deviations irrespective of plus or minus signs. But here, that difficulty is solved by squaring them and taking the square root of their average.
It is defined by the following formula
Where, x = An observation or variate value
m = Arithmetic mean of the population
N = Number of given observations.
Variance:
Variance is the square of the standard deviation variance = (S.D.)2
Characteristics:
l. If all the variate values are the same, S.D. is equal to zero.
2. S.D. is least affected by fluctuations of sampling.
3. Its computation is based on all the observations.
4. It is affected by the change of scale, but not affected by the change of origin.
Uses:
S.D. is used in computing statistical quantities like, regression coefficients, correlation coefficient, etc.
Coefficient of Variance:
This is also a relative measure of dispersion. It is important on account of the widely used measure of central tendency and diversion i.e., Arithmetic Mean and S.D.
It is given by –
C.V = S.D. / A.M. x 100