CENTRAL TENDENCY AND MEAN AND TYPES IN STATISTICS

Introduction
01
Facts of central tendency
Objectives & Functions
03
Properties of central
tendency
Definition & meaning
02 Meaning by various
scientists..
Requisites of central
tendency
04 How to identify good central
tendency?
TABLE of contents

intro-duction
1. A measure of central tendency is a summary statistic that represents the center point or
typical value of a dataset.
2. The term is first found in the mid-1690s in the writings of Edmund Halley (1656-1742),
3. It has been used to summarize observations of a variable since the time of Galileo (1564-
1642). Carl Friedrich Gauss (1777-1855)

Statistical constants which enable us
to comprehend in a single effort the
significance of the whole
Plainly speaking, an average of a statistical series
is the value of the variable which is representative
of the entire distribution
-By Prof. Bowley

Measures of central tendency is a single
value that is used to represent an entire
set of data.
meaning
Measures of central tendency is also
known as “ An average”.
The three most commonly used
measures of central tendency are
Median
Mode
Arithmetic Mean

Objectives
To present huge data in a
summarized form.
To Facilitate comparison.
To facilitate further Statistical
analysis.
To trace precise relationship.
To help in decision making.

Characteristics of central tendency
It should be rigidly defined.
It should be based on all
observations.
It should not be affected much by
extreme values.
It should be least affected by
fluctuations of sampling.
It should be easy to understand and
compute.
It should be capable of further
statistical analysis.

Definition 01
Merits and demerits 03
Grouped & ungrouped
mean
02
properties
04
TABLE of contents

Arithmetic Mean of a set of
observations is their sum divided
by the number of observations.
If n numbers, x1,x2,…,xn, then their
arithmetic mean average

In case of the frequency
distribution xi /fi
where fi is the frequency of the
variable xi

There are three different ways of calculating the
Arthimetic Mean
Calculating am of individual series
Direct Method
Short Cut Method
Step Deviation Method

Direct method short cut method
Let us take an example by taking the expenditure
of some families as
30, 70, 40, 20 and 60.
By using Direct Method formula:
N=5
=30+70+40+20+60/5
=44
Average Daily Expenditure is 44
By using Short Cut Method:
A is the assumed mean
dx stands for the deviation of the items from the
assumed mean (x-A).
Let A=40
dx=(x-A)=(-10)+30+0+(-20)+20 = 20
N=5
=40+20/5
=40+4
=44
Average Daily Expenditure is 44

Let us take same example by taking the
expenditure of some families as
30, 70, 40, 20 and 60.
By using Short Cut Method formula:
X C
= 40+⅖ x 10
=40+2+2
=40+4
=44
Average Daily Expenditure is 44.
x dx=(x-A) dxI=dx/C
30 -10 -1
70 30 3
40 0 0
20 -20 -2
60 20 2
Total 2

Arthimetic Mean
Calculating am of discrete series
Direct Method
Short Cut Method
Let us take an example
Calculate the Arthemetic Mean for the following data:
Wages 20 30 40 50 60 70 80
No. of persons 5 2 3 10 3 2 5

Direct method
Wages(x) No. of
people (f)
fx
20 5 20x5=100
30 2 30x2=60
40 3 40x3=120
50 10 50x10=500
60 3 60x3=180
70 2 70x2=140
80 5 80x5=400
N=30 Σfx=1500
x= 1500/30
= 50
Average Wage is 50.

Short cut method
x= 300/30 + 40
= 10+40
= 50
Average Wage is 50.
x f dx=(x-A) fdx
20 5 20-40=20 -100
30 2 30-40=-10 -20
40(A) 3 40-40=0 0
50 10 50-40=10 100
60 3 60-40=20 60
70 2 70-40=30 60
80 5 80-40=40 200
N=30 Σfdx=300

Step deviatoan method
By using Step Devaition Method formula:
x= 25+ 150/30 x 5
= 25 + 5 x 5
= 25 + 25
= 50
Average Wage is 50.
x f dx=(x-A) dxI=dx/c fdxI
20 5 20-25=-5
-1 -5
30 2 30-40=-10
1 2
40 3 40-40=0
3 9
50 10 50-40=10
5 50
60 3 60-40=20
7 21
70 2 70-40=30
9 18
80 5 80-40=40
11 55
N=30 ΣfdxI=150

Arthimetic Mean
Calculating am of continuous series
Direct Method
Short Cut Method
Let us take an example
Calculate the Arthemetic Mean for the following data:
Marks 0-10 10-20 20-30 30-40 40-50 50-60 60-70
Students 5 12 30 45 50 37 21

Direct method
x= 8180/200
= 40.9 OR 41
Average Marks is 41.
x f Mid
value
fx
0-10 5 5 25
10-20 12 15 180
20-30 30 25 750
30-40 45 35 1575
40-50 50 45 2250
50-60 37 55 2035
60-70 21 65 1365
N=200 Σfx=8180

Short cut method
x= 35+ 1180/200
= 40.9 OR 41
x f midvalue dx fdx
0-10 5 5 5-35= -30 -150
10-20 12 15 15-35= -20 -240
20-30 30 25 25-35= -10 -300
30-40 45 35(A) 35-35= 0 0
40-50 50 45 45-35= 10 500
50-60 37 55 55-35= 20 740
60-70 21 65 65-35= 30 630
N=200 Σfdx=1180

Step deviatoan method
By using Step Devaition Method formula:
x= 35+ 118/200 x 10
= 40.9 OR 41
x f midvalu
e
dx=(x-A) dxI=dx/c fdxI
0-10 5 5 -30 -3 -15
10-20 12 15 -20 -2 -24
20-30 30 25 -10 -1 -30
30-40 45 35(A) 0 0 0
40-50 50 45 10 1 50
50-60 37 55 20 2 74
60-70 21 65 30 3 63
N=200 ΣfdxI=
118

● It is based on all observations
● Its value always definite and it Is
rigidly defined.
● It is capable of further algebraic
treatment.
● Arithmetic mean is least affected
by fluctuations of sampling.
merits
Progress:

demerits
It is effected by extreme values .
It cannot be obtained graphically.
It cannot be computed for
qualitative data such as honesty,
intelligence etc.

properties
Mean can be calculated for any set
of numerical data, so it always
exists.
A set of numerical data has one and
only one mean.
Mean is the most reliable measure
of central tendency since it takes
into account every item in the set of
data.
It is greatly affected by extreme or
deviant values (outliers)
It is used only if the data are
interval or ratio.

THANKS a lot!
Do you have any questions?
CH.NARESH
Department of Statistics
Govt. college (A),
Rajahmundry

CENTRAL TENDENCY AND MEAN AND TYPES IN STATISTICS

CENTRAL TENDENCY AND MEAN AND TYPES IN STATISTICS

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