Friday, 12 July 2013

MEASURES OF CENTRAL TENDENCY

 ‘Measures of Central Location’ or ‘Measures of Central Tendency’ popularly known as ‘Averages’ serve this purpose. A figure which represents a series of values should obviously be greater than the lowest value and less than the highest value. It should be a value somewhere between these two limits, possibly at the centre where most of the values of the series center. Such a figure is called a Measure of Central Tendency.

Measures of central tendency, very often, are not fully representative of a given set of data. This happens when the extent of variation of individual values in relation to the average, or in relation to the other values is large. As an illustration, let us observe the following three series:


In the first series the arithmetic mean or simply mean (sum of the values divided by the number of values) is 40 and the values of all the items are identical each being equal to 40. The mean fully represents the series in general and the individual items in particular. The data (items or observations) are not at all scattered. In the second series, although the mean is 40, all the observations are not very much scattered as the minimum value of the series is 35 and the maximum value is 43. Hence in case of the second series also the mean is a good representation of the series. Although none of the observations of the series is equal to the mean of the series yet the discrepancy between the mean and any other observation is not so significant. In case of the third series, we observe that all the items or observations of the series are different. This series also has the same mean 40. In case of this series the observations are widely scattered. Clearly, in case of this series the mean neither satisfactorily represents the entire series in general nor the individual items of the series in particular. Thus we have observed that although all the above three series have the same average (arithmetic mean is a method of measuring average) yet they widely differ from one another in terms of their formation. When the extent of variation (deviation or scatteredness) of the individual values (items or observations) of a distribution or series in relation to their average or in relation to the other values is large then measures of central tendency or averages cannot be representative of the distribution. Hence it is important for any investigation not only to know the average of any type (mean, median or mode) but also the scatteredness of the various observations of a distribution.
Measures of Central Tendency or Averages

Definition

An average of a distribution (i.e. a distribution of the values of a variable like height, weight, income etc.) is a representative value of that distribution. This representative value usually lies at the central part of distribution. “A measure of central tendency or an average of a certain distribution is nothing but a representative value of that distribution which enables us to comprehend in a single effort the significance of the whole.”
Unit of average: The unit of average of a distribution is the unit of that distribution.
 Symbol
In order to denote sum (i.e., a total of certain quantities) the Greek letter ‘S’ (capital sigma) is used. For example, if variable x takes the values x1, x2, x3,……..xn then the sum of these values of the variable i.e. (x1, x2, x3,……..xn) is denoted by The symbol means that the lower limit of i is 1 and the symbol S means that all the values of xi for i = 1,2,3…….., n are to added. Thus Again, the symbol implies ‘sum of the values of x’.

Different Types of Measures of Central Tendency

The following three types of measures of central tendency or averages are in use.
(a) Mean
(b) Median and
(c) Mode.
(a) Mean: There are three types of mean, namely,
(i) Arithmetic Mean (A.M.),
(ii) Geometric Mean (G.M.) and
(iii) Harmonic Mean (H.M.)
So far as mean is concerned, we shall discuss arithmetic mean only since this is the most popular technique among the different types of mean.

In unit two, we have already discussed about arithmetic mean and geometric mean. Here we will discuss some special case related to Arithmetic mean.

· Arithmetic mean (A.M.) of grouped frequency distributions: In order to determine the A.M. of a grouped frequency distribution an assumption is made that the observations included in a class represented by a class interval are concentrated around the centre of that class interval. For example, if in a distribution of marks of some students the frequency of the class interval 50-60 is 8 (say) then we assume that each of 8 students is getting marks around 55. Consequently we may approximately take that each of the 8 students is getting 55 marks which is the mid value of the class interval 50-60 and we say ‘8 is the frequency of 55.’ To obtain the A.M. of a grouped frequency distribution, we take the frequencies of the different class intervals to be the frequencies of the mid-values of the corresponding classes. This converts a grouped frequency distribution to a discrete (or ungrouped) frequency distribution. Here by applying the arithmetic mean formula for a discrete or ungrouped frequency distribution we can find the arithmetic mean of a grouped frequency distribution.

Example 1: Find A.M. of the following frequency distribution:


Example 2: Determine mean of the following distribution:


Solution: Since there are three methods of obtaining mean, namely arithmetic mean (A.M.) method, geometric mean (G.M.) method and harmonic mean (H.M.) method, we shall apply A.M. method to find mean of the given distribution. (We shall see subsequently that A.M. is the most popularly used measure of central tendency.)

To find mean by applying arithmetic mean technique we form the following table:

i.e.; The required mean (i.e. average) wage = Rs. 33.40
Note: l1 and l2 imply respectively the lower limit and the upper limit of a class interval.

Alternative methods of finding A.M.:
(a) Assumed Mean Method or Short-Cut Method:


In case of grouped frequency distributions with equal class intervals, h is equal to the length of the class intervals.
A note on assumed mean: Any value of a variable x can be considered to be the assumed mean of x. However, the assumed mean should be taken from the central part of the values
Example 3: Determine arithmetic mean of the following distribution:


Solution: We denote height by the variable x and we take the assumed mean A of x to be 147.



Properties of A.M.:
Property 1: If each value of a variable is increased (decreased) by a constant c, then the A.M. of the new values is increased (decreased) by c. if each value of a variable is multiplied by a constant c then the A.M. of the new values is c times the A.M. of the original values. Again, if each value of a variable is divided by a constant c (c != 0) then the A.M. of the new values is equal to the A.M. of the original values divided by c.


(b) The A.M. of the values of a variable x is 25.
i. If each value is increased by 5, what will be the new A.M.?
ii. If each value is decreased by 7, what will be the new A.M.?
iii. If each value is multiplied by 2.5 what will be the new A.M.?
iv. If each value is divided by 4, what will be the new A.M.?

Solutions:


(c) Two series with 28 and 36 observations have means 2.9 and 5.6 respectively. Find the mean of the combined series.
Solution: Let be the mean of the combined series. Then


Determination of Arithmetic Mean in case of Cumulative Frequency Distributions:
We shall illustrate with the following example how arithmetic mean of cumulative frequency distributions can be determined.

Example 5: The following are the marks obtained by the students of class XII of a certain Higher Secondary School. Find the average marks using arithmetic mean technique.


Solution: The above cumulative frequency distribution should first be converted into a simple frequency distribution as under:
We convert the given cumulative frequency distribution to the simple frequency distribution as follows:



Now arithmetic mean of the data can be obtained by using the Direct Method as under:
Calculation of A.M.:


Note: Student will solve the problem by using step-deviation method.

LET US KNOW
Arithmetic mean is the most popularly used measure of averages. Most people are familiar with this technique. The basic advantages of this measure over other measures of central tendency are: (i) It is comparatively easier to understand and calculate; (ii) It provides a good basis for comparison; (iii) It can be determined for all values namely positive, negative or zero’; (iv) It is amenable for further mathematical treatment. Mainly because of its easiness in understanding and calculating, it has become most popular and hence it is called an ideal average. Usually mean or average means is arithmetic mean. It is widely used in social, economic and business problems.


When we determine the arithmetic mean of series by assigning weights to the different quantities of the series depending upon their relative importance then this arithmetic mean will be called the weighted arithmetic mean. For example, if we want to know the change in the cost of living of a particular community over a period of time then we must assign appropriate weights to the quantities of the different items of consumption of that community while constructing the index. Since all the commodities consumed by them are not of equal importance hence simple mean of the prices of the commodities consumed by them will not reflect the true cost.
Let the weights attached to the quantities be respectively. The weighted mean of these quantities is denoted by and is given by:



Uses of Weighted Mean:
(i) In order to determine the mean of the quantities whose weights are not equal the formula for weighted mean is applied.

(ii) In order to determine the mean of the sub-series of a series, the formula for weighted mean is used.
Example 6: The marks obtained by three students A, B and C in physics, Chemistry and Mathematics out of 100 in each subject in a certain entrance test are:



(i) Rank the three students on the basis of their performance if equal weights are given to the subjects.
(ii) Rank the students if weights are given as below:
Physics: 30%, Chemistry: 20%, Mathematics: 50%
Solution:
Let x, y and z denote the marks obtained by A, B, and C respectively is Physics, Chemistry and Mathematics.

[Recall that when equal weights are given to all observations then it is the case of simple average or average and when different weights are assigned, then it is the case of weighted average. Again, average is usually obtained by arithmetic mean technique.]

From the average marks obtained by A, B and C we find that the ranking positions of A, B and C are 1st, 2nd and 3rd respectively.



We find from above that when the weights are assigned as given to the marks obtained in Physics, Chemistry and Mathematics, the ranking positions of A, B and C becomes 1st, 3rd and 2nd respectively.

Median:
The median of a series or distribution in ascending or descending order is that observation of the distribution which divides the distribution into two equal parts. Thus there are equal number of observations on the right and on the left of the median value.

In order to determine the median of an individual series, first of all we have to observe whether the values (observations) are in a define order or not i.e. whether the values are in ascending or in descending order or not. If the values are not in a define order, then these values are to be arranged either in ascending or in descending order. If there are odd number of values in the series, then the th value from the beginning (and also from the end) will be the median. If the number of values is even then the arithmetic mean of the n/2th value and the th value will be the median.

Example 7: Determine median for the following series:
(i) 77, 73, 72, 70, 75, 79, 78
(ii) 94, 33, 86, 68, 32, 80, 48, 70
Solution:
(i) Arranging the values of the series in ascending order, we get:70, 72, 73, 75, 77, 78, 79
No. of terms in the series = 7 = An odd number
\ The required median th term i.e; 4th term = 75

(ii) Arranging the data (values or observations) in ascending order, we get: 32, 33, 48, 68, 70, 80, 86, 94
No. of term in the series = 8 = An even number



Median of an ungrouped frequency distribution:



Example 8: Determine median for the following distribution:


Solution:
frequency column that 62 and 63 lie between 45 and 65. Since 65 is the cumulative frequency of 24 hence each of the 62th and the 63th terms will be 24.
Hence the required median = Rs. 24.
Note: The 62th term = Rs. 24, and the 63th term = Rs. 24 and their A.M.=

Median of a grouped frequency distribution:
In case of grouped frequency distributions, one may consider the N/2th observation as the median if N is even. When N is odd, N/2 +1 th observation will be the median.


Where L = Lower class limit (lower class boundary) in case of exclusive (inclusive) classification.
f = Frequency i.e. simple frequency of the median class
fc = Cumulative frequency of the class preceeding the median class.
N = Total frequency
I = length of the median class
Note: This formula for obtaining median of a grouped frequency distribution holds only when the distribution is in ascending order. If the distribution is in descending order then it is to be first of all arranged in ascending order in order to apply the above formula.

Example 9: Determine median for the following distribution:

frequency table we find that the 56th term lies in the class 65–70. \ 65–70 is the median class.


i.e. the required median = Rs. 67.92

Advantages and Limitations of Median:

v Advantages:
i. Extreme values do not affect median.
ii. Median is easy to understand; it is easy also to determine.
iii. Median can also be determined graphically
iv. Median can be determined for distributions having open end class intervals.
v Limitations:
i. In order to determine the median of a distribution, the distribution must be arranged in a define order if it is not in an order. This is not needed in other measures of central tendency.
ii. Median of a distribution is not based on all the observations of the distribution.
Uses of median: In order to determine the average of distribution having open-end class intervals median is the best measure. In case of income distribution the use of median gives better results.
Mode: The mode of a distribution is that observation of the distribution whose frequency is the maximum. It is to be noted that mode is not unique which means that a distribution may have more than one modes.

Clearly an individual or single series (distribution) does not have a mode. In many cases of ungrouped frequency distributions, mode can be detected simply by observation. Mode of a grouped frequency distribution is obtained by using the following formula.



Where L = Lower limit/lower boundary of the model class
fo = Frequency of the class preceeding the model class
f1 = Frequency of the model class
f2 = Frequency of the class succeeding the model class
And 1 = Length of the model class

Note: (i) The class (specified by a class interval) whose frequency is the maximum is called the model class.
(ii) The above formula is used when all the classes are of equal length.
Example 10: Determine mode for the following distribution:


Since the frequency of the class 16-20 is the maximum, hence this class is the model class. The class intervals of the given distribution are as per the inclusive method of classification and hence in determining mode we must take the lower boundary of the model class.
Now, Mode

Example 11: Determine mode/modes, if any, of the following series:
i. 3, 4, 5, 2, 3, 4, 1, 6, 4;
ii. 7, 9, 11, 7, 6,5, 9, 13;
iii. 3, 5, 6, 7, 9, 12, 3, 6, 5, 9, 12, 7

Solution:
i. The number 4 is repeated the maximum of number times (thrice). Hence 4 is the mode of the given series.
ii. We find that the frequency of 7 and 9 is the maximum each being equal to 2. Hence the two modes of the series are 7 and 9.
iii. Since the frequency of each observation is the same (being 2 in each case), hence the given series has no mode.

Advantages and Limitations of Mode:

Advantages:
i. In most cases the mode/modes of an ungrouped frequency distribution can be determined simply by observation.
ii. Mode is not affected by extreme values
iii. Mode is easy to understand
iv. Mode can be determined graphically.

Limitations:
i. Mode is not based on all the observations
ii. It is not suitable for further mathematical treatment.
iii. Like arithmetic mean we cannot know the sum of the observations of a distribution if we know the mode and the number of observations of the distribution.

Uses of Mode: The concept of mode is used by manufacturers, businessmen, agriculturists, etc. For example, a manufacture of shoes is interested in the model size of shoes and manufactures them in large quantities. It is useful in industry and business. Weather forecasts are based on mode. The concept of mode is used in socio-economic surveys besides being used in business and commerce.

Relationship among Mean, Median and Mode:
Prof. Kari Pearson has established an empirical (experimental) relationship among mean, median and mode. For any distribution the following relationship known as empirical relationship approximately holds:
Mean – Mode = 3 (Mean – Median)

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