Summary Sheet

Session Number :

Subject Expert :

4 Nagesh P.Nagesh P.

Department of Management Studies

S.J. College of Engineering

Mysore – 570 006.

Measures of Central Tendency

A classified statistical data may sometimes be described as

distributed around some value called the central value or

average is some sense. It gives the most representative value of

the entire data. Different methods give different central values

and are referred to as the measures of central tendency.

The important objective of statistical analysis is to determine a

single value representing the characteristics of the entire data.

This single value representing the entire data is called ‘Central

value’ or an ‘average’. Since this value is located at central

point nearest to other values of the data it is also called as

measures of central tendency.

The common measures of central tendency are

a) Mean b) Median c) Mode. These values are very useful not only in

presenting overall picture of entire data, but also for the purpose of

making comparison among two or more sets of data.

Average

Average is a value which is typical or representative of a set of

data. - Murry R. Speigal

Average is an attempt to find one single figure to describe whole of

figures. - Clark & Sekkade

Functions of an average

It represents complex or large data.

It facilitates comparative study of two variables.

Helps to study population from sample data.

Helps in decision making.

Represents single value for a series of data.

It establishes mathematical relationship.

Characteristics of a typical average

It should be rigidly defined and easily understandable.

It should be simple to compute and in the form of

mathematical formula.

It should be based on all the items in the data.

It should not be unduly influenced by any single item.

It should be capable of further mathematical treatment.

It should have sampling stability.

Types of average Average or measures of central tendency are of following types.

1. Mathematical average

a. Arithmetical mean

i. Simple mean

ii. Weighted mean

b. Geometric mean

c. Harmonic mean

Arithmetic mean

Arithmetic mean is also called arithmetic average. It is most

commonly used measures of central tendency. Arithmetic

average of a series is the value obtained by dividing the total

value of various item by its number.

Arithmetic average are of two types

a. Simple arithmetic average

b. Weighted arithmetic average

Simple arithmetic average (Mean)

Arithmetic mean is simply sometimes referred as ‘Mean’.

Ex: Mean income, Mean expenses, Mean marks etc.

Simple arithmetic mean is equal to sum of the variable divided

by their number of observations in the sample.

n

x

n

x...............xxxx n321

Arithmetic mean can be computed by.

a. Direct method b. Short cut method.

Ex: (For Direct Method)

1. Six month income of departmental store are given below. Find mean income of stores.

Month Jan Feb Mar Apr May June Income (Rs.) 25000 30000 45000 20000 25000 20000 n = Total No. of items (observations) = 6

Total income = xi = (25000 + 30000 + 45000 + 20000 +

20000) = 140000

Mean income = 33.23333.Rs6

140000

n

xi

Shortcut method Steps of this method is given below.

Step 1: Assume any one value as a mean which is called

arbitrary average (A).

Step 2: Find the difference (deviations) of each value from

arbitrary average. D = xi – A

Step 3: Add all deviations (differences) to get d.

Step 4: Use the following equation to compute the mean value.

n

dAx

Ex: Find the mean marks obtained by the students for the data given below.

20 25 20 22 20 21 23 25 22 18 Let A = 20 and n = 10 Marks D=(xi–20)

20 0 25 5 20 0 22 2 20 0 21 1 23 3 25 5 22 2 18 -2

d = 16

1. Mathematical characteristics of mean a. Algebraic sum of deviations of all observations from their

arithmetic mean is zero i.e. (xi - x ) = 0.

b. The sum of squared deviations of the items from the mean is a minimum.

d2 = minimum

c. Since n

xx . If any two values are given, third value

can be computed.

d. If all the items of a sets are increased / decreased by any constant value, the arithmetic mean will also increases / decreases by the same constant.

2. Weighted arithmetic mean

The weighted mean is computed by considering the

relative importance of each of values to the total value. The

arithmetic mean gives equal importance to all the items of

distribution. In certain cases, relative importance of items is not

the same. To give relative importance, weightage may be given

to variables depending on cases.

Weighted arithmetic mean computation

Let x1, x2, x3, ………… xn are the variables and

w1, w2, w3, ………… wn are the respective weights

assigned. Then weighted mean wx is given by below equation.

w

xw

w............www

wx......wxwxwxx

n321

nn332211w

i.e., weighted average is the ratio of product of all values and

respective weights to sum of weights.

Ex: Compute simple weighted arithmetic mean and comment on them.

Designation Monthly salary

(Rs) (x) Strength of cadre (w)

xw

General Manager 25000 10 250000 Mangers 19000 20 380000 Supervisors 14000 10 140000 Office Assistant 10000 50 500000 Helpers 8000 25 200000

(N = 5) Total x=76000 x = 115 x=1470000

a. Simple arithmetic mean = 15200.Rs205

79000

N

x

b. Weighted arithmetic mean = 70.7170.Rs205

1470000

x

xw

In this example, simple arithmetic mean does not accounts the difference in salary range for various staff.

Ex: Comment on performance of students of two universities given below. University Bombay Madras

Course % of

pas (x) No.of(w)

students (000) wx

% of pas (x)

No. of(w) students

wx

MBA 71 3 213 81 5 405 MCA 83 2 166 76 3 228 MA 73 5 365 58 3 174 M.Sc. 75 2 150 76 1 76 M.Com. 70 2 140 81 2 162 Total () x=372 w=14 wx=1034 x=372 w =14 wx=1045

a. Since x is same, simple arithmetic average for both

universities. = 4.745

372

N

x

b. Weighted mean for Bombay University = 86.7314

1034

w

wx

c. Weighted mean for Madras University = 64.7414

1045

w

wx

Discrete Series

Frequencies of each value is multiplied with respective

size to get total number of items is discrete series and their total

number of item is divided by total number of frequencies to

obtain arithmetic mean. This can be done in two methods one

by direct or by short cut method.

Ex: Calculate the mean for following data. Value (x) 1 2 3 4 5 Frequency(f) 10 15 10 9 5

Steps: 1. Multiply each size of item by

frequency to get fx

2. Add all frequencies (f = N)

3. Use formula N

fx

f

fxx

to get mean value.

67.249

163

N

fdAx

Continuous series

In continuous frequency distribution, the individual value

of each item in the frequency distribution is not known and the

mid points of various class intervals are written down to replace

the class interval. In continuous series the mean can be

calculated by any of the following methods.

a. Direct method

b. Short cut method

c. Step deviation method

a. Direct method

Steps of this method are as follows

1. Find out the mid value of group or class.

Ex: For a class interval 20-30, the mid value (m) is

252

50

2

3023

.

2. Multiply the mid value ‘m’ by frequency ‘f’ of each class

and sum up to get fm.

3. Use N

fmx

formula to get the mean value.

Ex: Compute the mean for following data.

Age group (CI)

No. of persons (f)

Mid point m

fm

0 – 10 5 5 25 10 – 20 15 15 225 20 – 30 25 25 625 30 – 40 8 35 280 40 – 50 7 45 315 Total f = 60 = N fm = 1470

Mean age = 24560

1470

N

fm

f

fm

x = 24.5

b. Short cut method

Steps of above method are described below.

1. Find the mid value of each class

2. Assume any of the mid value as arbitrary average (A).

3. Multiply the deviation (differences) ‘d’ by frequency ‘f’.

Using the formula N

fdAx

to find the mean value.

Ex: Find the mean age of patients visiting to hospital on a particular day for the following data.

Age group CI

No. of patients (f)

Mid value m

d=(m–25) fd

0 – 10 5 5 -20 -100 10 – 20 15 15 -10 -150 20 – 30 25 25 0 0 30 – 40 8 35 10 80 40 – 50 7 45 20 140 Total f=60=N fd = –30

Let Arbitrary average = A = 25

Mean age N

fdAx

5.42

2

125

60

3025x

5.24x

c. Step deviation method

In this method, after finding deviation from arbitrary

mean, it is divided by a common factor. Scaling down the

deviation by a ‘step’ will reduce the calculation to

minimum.

Step deviation method is described below.

1. Find out the mid value ‘m’ and select the arbitrary men ‘A’.

2. Find the deviation (d) of mid value of each from ‘A’.

3. Deviations ‘d’ are divided by a common factor d'.

4. Multiply d' of each class by frequency ‘f’ to get fd' and sum

up for all classes to get fd'.

5. Using the formula CxN

'fdAx

(where, C is a common

factor) to calculate the mean value.

Ex: Find the mean age of following data. Age (CI) No. of

persons ‘f’ Mid

value ‘m’ (d=m–A) (d=m–25) d'=

10

d

fd'

0 – 10 5 5 -20 -2 -10 10 – 20 15 15 -10 -1 -15 20 – 30 25 25 0 0 0 30 – 40 8 35 10 1 8 40 – 50 7 45 20 2 14 Total f=60=N fd'= -3

Let A = 25 and C = 10

CxN

'fdAx

10x

60

)3(25x

2

125x 5.24x