MEASURES OF CENTRAL TENDENCY-MEAN

INTRODUCTION:

Ø In statistics, a central tendency is a central value or a typical value for a probability distribution.

Ø It is occasionally called an average or just the center of the distribution.

Ø The most common measures of central tendency are the arithmetic mean, the median, and the mode.

Ø Measures of central tendency are defined for a population (large set of objective of a similar nature) and for a sample (portion of the elements of a population).

SOME DEFINITION:

Simpson and Kafka defined it as “A measure of central tendency is a typical value around which other figures gather”.

Waugh has expressed “an average stand for the whole group of which it forms a part yet represents the whole”.

In layman’s term, measures of central tendency is an AVERAGE. It is a single number of values which can be considered typical in a set of data as a whole.

IMPORTANCE OF CENTRAL TENDENCY:

q To find representative value

q To make more concise data

q To make comparisons

q Helpful in further statistical data

 MEAN:

§  The mean of a set of values or measurements is the sum of all the measurements divided by the number of measurements in the set.

§  The mean is the most popular and widely used. It is sometimes called the  arithmetic mean.

ARITHMETIC MEAN-RAW DATA:

    The arithmetic mean is the sum of a set of observations, positive, negative or zero, divided by the number of observations. If  we have n real numbers x1,x2,x3,x4.......xn, then their arithmetic mean, denoted by xbar  , is given by
xbar=x1+x2+...+xn/n (or) xbar =∑x/n

Example:

Find the arithmetic mean of the marks 72,73,75,82,74 obtained by a student in 5 subject in an annual exam.

Here n=5

  xbar = ∑x/n

            = 72+73+75+82+74/5

            = 376/5 = 75.2

ARITHMETIC MEAN-UNGROUPED DATA:

The mean of the observations x1,x2,x3...xn with frequencies f1,f2,f3,...fn respectively is given by

xbar = ∑fx/∑f

Example:

Obtain the mean of the following data.

x	5	10	15	20	25
f	3	10	25	7	5

Solution:

X	f	fx
5	3	15
10	10	100
15	25	375
2	7	140
25	5	125
Total	∑f = 50	∑f x= 755

Mean = ∑fx/∑f

           =  755/50

           =  15.1

ARITHMETIC MEAN – GROUPED DATA

•         Direct Method

                          xbar  = ∑fx/ ∑f

•         Assumed Method

 xbar = A+ (∑fd/∑f)

•         Step Deviation Method

                        xbar  = A+ (∑fd/∑f)xc

EXAMPLE FOR DIRECT METHOD:

•         From the following table compute arithmetic mean by direct method.

Marks	0-10	10-20	20-30	30-40	40-50	50-60
No. of students	5	10	25	30	20	10

SOLUTION:

Marks	Midpoint (x)	No. of students (f)	fx
0-10	5	5	25
10-20	15	10	150
20-30	25	25	625
30-40	35	30	1050
40-50	45	20	900
50-60	55	10	550
		∑f = 100	∑fx=3300

  x bar  = ∑fx/ ∑f

            =  3300/100

            =  33

  Mean=33

EXAMPLE FOR ASSUMED MEAN METHOD:

•         Let the assumed mean be A=35

Marks	Midpoint (x)	No. of students (f)	d=x-35	(fd)
0-10	5	5	-30	-150
10-20	15	10	-20	-200
20-30	25	25	-10	-250
30-40	35	30	0	0
40-50	45	20	10	200
50-60	55	10	20	200
		∑f = 100		∑fd= -200

x bar= A+ (∑fd/∑f)
        = 35+(-200/100)

       = 35-2

       = 33

Mean=33

EXAMPLE FOR STEP DEVIATION MEAN METHOD:

•         In order to simplify the calculation, we divide the deviation by the width of the class intervals. i.e calculate x-A/c and then multiply by c in the formula for getting the mean of the data.

         xbar = A+ (∑fd/∑f)xc

     width of the class interval is c = 10

Marks	Midpoint (x)	No. of students (f)	d= x-35/10	(fd)
0-10	5	5	-3	-15
10-20	15	10	-2	-20
20-30	25	25	-1	-25
30-40	35	30	0	0
40-50	45	20	1	20
50-60	55	10	2	20
		∑f = 100		∑fd= -20

x bar=A+(∑fd/∑f)xc
        =35-(20/100)x10
        =35-2
        =3

Mean=33

PROPERTIES OF MEAN:

Ø Mean can be calculated for any set of numerical data, so it always exists.

Ø As a set of numerical data it has one and only one mean.

Ø Mean is the most reliable measure of central tendency since it takes into account every items in the set of data.

Ø It is used only if the data are interval or ratio.

Ø It is a quick approximation of the average and an inspection average.

APPLICATIONS OF MEAN:

Ø It helps teachers to see the average marks of the students.

Ø It is used in factories, for the authorities to recognize whether the benefit of the workers is continued or not.

Ø It is also used to contrast the salaries of the workers.

Ø To calculate the average speed of anything.

Ø It is also used by the government to find the income or expenses of any person.

Ø Using this family could balance their expenses with their average income.

CONCLUSION:

Ø A measure of central tendency is a measure that tells us where the middle of a brunch of data lies.

Ø Mean is the most common measure of central tendency. It is simply the sum of the numbers divided by the number of numbers in a set of data. This is also known as average.