Measure of central tendancy, Slides of Business Statistics

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Measures of Central

Tendency: Mean, Mode,

Median

Introduction:

 Measures of central tendency are statistical measures which describe the position of a distribution.  They are also called statistics of location, and are the complement of statistics of dispersion, which provide information concerning the variance or distribution of observations.  In the univariate context, the mean, median and mode are the most commonly used measures of central tendency.  (^) computable values on a distribution that discuss the behavior of the center of a distribution.

Definition

Simpson and Kafka defined it as “ A measure

of central tendency is a typical value around which

other figures congregate”

Waugh has expressed “An average stand for the

whole group of which it forms a part yet represents

the whole”.

1. Arithmetic Mean Arithmetic mean is a mathematical average and it is the most popular measures of central tendency. It is frequently referred to as ‘mean’ it is obtained by dividing sum of the values of all observations in a series (ƩX) by the number of items (N) constituting the series. Thus, mean of a set of numbers X1, X2, X3, ………..Xn denoted by xx̅ and is defined as

Example : Calculated the Arithmetic Mean DIRC Monthly Users Statistics in the University Library Month No. of Working Days Total Users Average Users per month Sep-2011 (^24 11618) 484. Oct-2011 (^21 8857) 421. Nov-2011 (^23 11459) 498. Dec-2011 (^25 8841) 353. Jan-2012 (^24 5478) 228. Feb-2012 23 10811 470. Total 140 57064

• = 407.

Disadvantages of Mean: 

It is affected by extreme values.



It cannot be calculated for open end

classes.



It cannot be located graphically



It gives misleading conclusions.



It has upward bias.

2.Median

Median is a central value of the distribution, or

the value which divides the distribution in equal

parts, each part containing equal number of items.

Thus it is the central value of the variable, when the

values are arranged in order of magnitude.

Connor has defined as “ The median is that value of

the variable which divides the group into two equal

parts, one part comprising of all values greater, and

the other, all values less than median”

Calculation of median – Continuous series For calculation of median in a continuous frequency distribution the following formula will be employed. Algebraically,

Example: Median of a set Grouped Data in a Distribution of Respondents by age Age Group Frequency of Median class(f) Cumulative frequencies(cf) 0-20 15 15 20-40 32 47 40-60 54 101 60-80 30 131 80-100 19 150 Total 150

Advantages of Median:



Median can be calculated in all distributions.



Median can be understood even by common people.



Median can be ascertained even with the extreme

items.



It can be located graphically



It is most useful dealing with qualitative data

Disadvantages of Median:  It is not based on all the values.  It is not capable of further mathematical treatment.  It is affected fluctuation of sampling.  In case of even no. of values it may not the value from the data.

Croxton and Cowden : defined it as “the mode of a

distribution is the value at the point armed with the

item tend to most heavily concentrated. It may be

regarded as the most typical of a series of value”

The exact value of mode can be obtained by the

following formula.

Z=L

1

Monthly rent (Rs) Number of Libraries (f) 500-1000 5 1000-1500 10 1500-2000 8 2000-2500 16 Example: Calculate Mode for the distribution of monthly rent Paid by Libraries in Karnataka