Prof. Rajkumar TeotiaInstitute of Advanced Management and Research (IAMR)

Address: 9th Km Stone, NH-58, Delhi-Meerut Road, Duhai,Ghaziabad (U.P) - 201206

Ph:0120-2675904/905 Mob:9999052997 Fax: 0120-2679145e mail: rajkumarteotia@iamrindia.com

Measures of central tendency

A measure of central tendency is a single value that

attempts to describe a set of data by identifying the central position within that set of data. As such, measures of central tendency are sometimes called measures of central location.

The mean, median and mode are all valid measures of central tendency but, under different conditions, some measures of central tendency become more appropriate to use than others.

Measures of central tendency

Mean or Arithmetic mean refers to the sum of all variables divided by the number of variables. It is the most popular technique of finding out the average of values. It is also referred to as arithmetic mean or average in day-to-day life.

 

ARITHMETIC MEAN

For Ungrouped data, arithmetic mean may be calculated by applying any of the following methods:

Direct method

Short-cut method

ARITHMETIC MEAN – FOR UNGROUPED DATA

 It is calculated by the following formula:

Where, X = arithmetic mean of the population N = total number of observations in a population. Σ x = Sum of all observation

Direct method:

X = Σ x N

Example: Find out the arithmetic mean of the following observation 10, 15, 30, 7, 42, 79 and 83

Solution: Symbolically, the arithmetic mean is

Σ x = 10 + 15 + 30 + 7 + 42 + 79 + 83 = 266N = 7

X = Σ x = 266 = 38 N 7

X = Σ x N

In the case of short-cut method, the concept of arbitrary mean is followed. The formula for calculation of the arithmetic mean by the short-cut method is given below:

Where, A = arbitrary or assumed mean d = deviation from the arbitrary or assumed mean (d = X - A) N = total number of observations in a population

Short-cut method:

X = A + Σ d N

Example: Data of imports of a certain firms for the year 2010 are mentioned in the following table:

From the above data calculate average value of imports for these

firms using short-cut method.

firms A B C D E F GValue of imports (Rs)

27 32 34 39 56 62 70

Solution: The formula for calculation of the arithmetic mean by the short-cut method is given below:

X = A + Σ d N

Firms Importsx

d = x – AA = 39

ABCDEFG

27323439566270

-12-7-50

172331

N = 7 Σd = 47

By applying the following formula arithmetic mean will be

= 39 + 47 7

= Rs. 45.71

X = A + Σ d N

For grouped data, arithmetic mean may be calculated by applying any of the following methods:

Direct method Short-cut method Step-deviation method

ARITHMETIC MEAN – FOR GROUPED DATA

In the case of direct method, the following formula is used

Where, X = arithmetic mean of the population N = total number of observations in a population. X = mid-point of various classes f = the frequency of each class

DIRECT METHOD – FOR GROUPED DATA

X = Σ f X N

Example: The following table gives the marks of 58 students in Statistics. Calculate the average marks of this group.

 

Solution:In the case of direct method, the following formula is used

X = Σ fx = 1940 = 33.45 Marks N 58

X = Σ f x N

Marks Mid-pointx

No. of Studentsf

f x

0-10 5 4 20

10-20 15 8 120

20-30 25 11 275

30-40 35 15 525

40-50 45 12 540

50-60 55 6 330

60-70 65 2 130

N = 58 Σ fx = 1940

In the case of short-cut method, the concept of arbitrary mean is followed. The formula for calculation of the arithmetic mean by the short-cut method is given below:

Where,A = arbitrary or assumed meanf = frequency d = deviation from the arbitrary or assumed mean (d = x - A)  When the values are extremely large and/or in fractions, the use of

the direct method would be very cumbersome. In such cases, the short-cut method is preferable. This is because the calculation work in the short-cut method is considerably reduced

SHORT-CUT METHOD FOR GROUPED DATA

Example: The following table gives the marks of 58 students in Statistics. Calculate the average marks of this group. By Short-cut Method

 

Solution:

Here Assumed mean (A) = 35 And (d) = x – A

Marks M id -p o in t

f d = x – A

fd x

0 -1 0 5 4 -3 0 -120 1 0 - 2 0 1 5 8 -2 0 -160 2 0 - 3 0 2 5 1 1 -1 0 -110 3 0 - 4 0 3 5 1 5 0 0 4 0 - 5 0 4 5 1 2 1 0 1 2 0 5 0 - 6 0 5 5 6 2 0 1 2 0 6 0 - 7 0 6 5 2 3 0 60

∑ f d = - 9 0

Following formula will be used for calculating arithmetic mean by step- deviation method

Where, A = arbitrary or assumed mean f = frequency d = deviation from the arbitrary or assumed mean d = X – A i i = class size

STEP-DEVIATION METHOD FOR GROUPED DATA

 

X = A + Σ f d x i N

Example: The following table gives the marks of 58 students in Statistics. Calculate the average marks of this group By Step-deviation method

Solution: Following formula will be used for calculating arithmetic mean by step- deviation method

X = A + Σ f d x i N

MarksMid-point

x fd = x – A

i Fd0-10 5 4 -3 -12

10-20 15 8 -2 -1620-30 25 11 -1 -1130-40 35 15 0 040-50 45 12 1 1250-60 55 6 2 1260-70 65 2 3 6

∑fd = -9

ARITHMETIC MEAN – FOR DISCRETE SERIES (Where frequencies are given)

In the case of direct method, the following formula is used 

Where,

X = arithmetic mean of the population N = total number of observations in a population. x = mid-point of various classes f = the frequency of each class

X = Σ f x N

Short – cut method: In case of Short - cut method the following formula is used

ARITHMETIC MEAN – FOR DISCRETE SERIES (Where frequencies are given)

X = A + ∑ f d

N

X = A + ∑ f d

N

Example- from the following data calculate arithmetic mean by direct method

Solution In the case of direct method, the following formula is used

Marks 5 15 25 35 45 55

No. of students 10 20 30 50 40 30

X = Σ f x N

X = Σ fx = 6300 = 35 N 180

Marks (x) No. of students (f) fx

51525354555

102030504030

50300750

175018001650

N = 180 Σ fx = 6300

Example- from the following data calculate arithmetic mean by Short - cut method

Solution: In case of Short - cut method the following formula is used

Marks 5 15 25 35 45 55

No. of students 10 20 30 50 40 30

X = A + ∑ f d

N

Marks (x) No. of students (f) d = X - A fd

51525354555

102030504030

-35-25-15-55

15

-350-500-450-250200450

N = 180 ∑ f d = - 900

X = 40 - 900 = 35 marks

180

Example- from the following data calculate arithmetic mean by direct method

Solution: - let us first convert less than series into grouped data

ARITHMETIC MEAN – FOR GROUPED DATA (In case of less than series)

Marks less than 10 20 30 40 50 60

No. of students 10 20 30 50 40 30

Marks 0-10 10-20 20-30 30-40 40-50 50-60

No. of students 10 20 30 50 40 30

In the case of direct method, the following formula is used

X = Σ f x = 6300 = 35 N 180

X = Σ f x N

Marks Mid point (x) No. of students (f) fx0-1010-2020-3030-4040-5050-60

51525354555

102030504030

50300750

175018001650

N = 180 Σ fx = 6300

Example- from the following data calculate arithmetic mean by direct method

Solution: - let us first convert more than series into grouped data

ARITHMETIC MEAN – FOR GROUPED DATA (In case of more than series)

Marks more than 0 10 20 30 40 50 60

No. of students 10 20 30 50 40 30 0

Marks 0-10 10-20 20-30 30-40 40-50 50-60

No. of students 10 20 30 50 40 30

In the case of direct method, the following formula is used

X = Σ f x N

Marks Mid point (x) No. of students (f) fx

0-1010-2020-3030-4040-5050-60

51525354555

102030504030

50300750175018001650

N = 180 Σ fx = 6300

X = Σ f x = 6300 = 35 N 180

Example- from the following data calculate arithmetic mean by direct method

ARITHMETIC MEAN – FOR GROUPED DATA (In case inclusive series)

Marks 1-10 11-20 21-30 31-40 41-50 51-60

No. of students 10 20 30 50 40 30

Solution: -

Now applying the following formula X = A + Σ fd X i

N

= 45.5 + (-180 x 10) = 35.5 180

1

 

Marks Mid point (x) No. of students (f) d = x – A i

fd

0.5-10.510.5-20.520.5-30.530.5-40.540.5-50.550.5-60.5

5.515.525.535.545.555.5

102030504030

-4-3-2-101

-40-60-60-500

30

N = 180 Σ fd = - 180

To calculate weighted mean, each variable is multiply by its weight, repeat the multiplication process for all variables and divide this sum by sum of all weights.

OR XW

= Σ WX

Σ W

Weighted Arithmetic mean:-

XW = W1X1

+ W2X2 + W3X3 + ……………… + WnXn

W1 + W2 + W1 + …………………..+ Wn

Where,

XW = weighted arithmetic mean

W = Weight of individual variable Σ W = Total Weight

Example: The performance of a student manager in a business school was

evaluated as follows. Calculate weighted mean

Basis Marks obtained (x) Weights (W)

Class testsPresentationAttendanceClass participationFinal examination

3836152055

101551060

Solution:-

Weighted mean = ΣWX = 4495 = 44.95 ΣW 100

Basis Marks obtained (x) Weights (W) WX

Class testsPresentationAttendanceClass participationFinal examination

3836152055

101551060

38054075200

3300

N = 5 Σx = 164 Σw = 100 ΣWX = 4495

If the arithmetic mean and the number of observation of two or more related groups are known we can calculate the combined arithmetic mean of these groups. The combined mean formula is

Combined mean:

X12….k = N1X1 + N2X2 + N3X3 + ……………………+ NKXK

N1 + N2 + N3 + ………………………+ Nk

Example: the mean marks of 60 students in section A is 40 and the mean marks of 40 students in section B is 45. Find the combined mean of the 100 students in both sections.

Solution: Here, N1 = 60, N2 = 40,

X1 = 40, X2 =45

  By using the following formula combined mean will be  

X12 = N1X1 + N2X2

N1 + N2

= 60 x 40 + 40 x 45 60 + 40

= 42 marks

Some of the important characteristics of the arithmetic mean are:

The sum of the deviations of the individual items from the arithmetic mean is always zero.

i,e ∑ (X – X) = 0

The sum of the squares of deviations of the individual items from the arithmetic mean is always minimum i,e ∑ (X – X )2 is always minimum.

As the arithmetic mean is based on all the items in a series, a change in the value of any item will lead to a change in the value of the arithmetic mean.

In the case of highly skewed distribution, the arithmetic mean may get distorted on account of a few items with extreme values.

 

CHARACTERISTICS OF THE ARITHMETIC MEAN