Measures of Central tendanCies calculations of Mean, median and Mode statistiCs topiC- 1
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Measures of Central
tendanCiescalculations of
Mean, median and Mode
statistiCs topiC- 1
1.Measures of Central tendenCy
average
Mathematical
averagePositional
average
Arithmetic
mean
Geometric
mean
Harmonic
mean Median Mode
average
An average is a single figure that represent whole
group.
Mathematical
average
Arithmetic
mean
Geometric
mean
Harmonic
mean
Mathematical average
A.Arithmetic mean-
Generally if we talk about average ,it signifies
arithmetic mean.
It is based on all observations.
It is also known as mean
It is calculated value and not based on the position of
the series.
Features-
Simple Arithmetic mean-
1.Direct method 2.Short cut method
x̄̄̄̄= ∑X
N
x ̄̄= Arithmetic mean.
∑X-sum of the values of
the item of a series.
N= number of observations.
X= A+ ∑d
N
x ̄̄= Arithmetic mean.
A= Assumed mean.
∑d= sum of deviation.
N= number of observations.
Properties of Arithmetic mean-
1.The sum of deviation of item from AM is always zero.
∑(X-X)=0
2.The sum of squared deviation of AM is minimum.
B-Geometric Mean
GM define as nth root of the product of all the n values
of the variable.
If there are two items we take square root .
Two items there and their values are 4 and 9.
GM= n X1.X2.X3………Xn
GM will be- GM = 4*9 = 36 = 6
if there are three items , we take cube root.
Three items and their values 2,4,8
GM = 2*4*8 = 64 = 4
It is based on all the items of the series.
It gives less weight to large items.
It is best measure of ratio change.
C. Harmonic mean
HM is based on reciprocal of the items.
It is the reciprocal of the AM.
HM=N
1/X1 + 1/X2 + 1/X3 ………..+ 1/Xn
It is also called as – sub contrary mean.
2.Median
The median is that value of the variable which divides
the group into two equal parts.
Median is determined by first arranging the series in an
ascending or descending manner.
Median is denoted as M
M= N+1
2
1.If odd number series-
9,5,,3,6,10,12,7.
Arrange items in either ascending or descending order.
Ser.no
1
2
3
4
5
6
7
Items
3
5
6
7
9
10
12
M= N+1
2
= 7+1
2= 4
M= 7
4 7
1.If even number series-
12,16,14,18,24,20.
Arrange items in either ascending or descending order.
Ser.no
1
2
3
4
5
6
Items
24
20
18
16
13
12
M= N+1
2
= 6+1
2= 3.5
M= 17
= 2
= 3rd +4th 18+16
2
34
2=
3.Mode
Mode is define as value which occur most frequently.
8,6,14,12,8,5,10,8,14,3,8,4.
Mode is denoted as Z.
Z= 8
Relationship b/w mean, median ,and mode.
1.Perfectly symmetrical distribution.
In this case x ̄̄,M, and Z. are equal.
+VE-VE
x ̄̄=M=Z
In this case normal
distribution is bell shaped.
In normal distribution mean
is zero and variance is 1.
Relationship b/w mean, median ,and mode.
2. When distribution is positively skewed.
In this case x ̄̄,>M> Z. .
+VE-VE x ̄̄
In this case normal
distribution is +vly skewed..
MZ
Relationship b/w mean, median ,and mode.
3. When distribution is negatively skewed.
In this case x ̄̄,<M< Z. .
+VE-VE x ̄̄
In this case normal
distribution is -vly skewed..
M Z
Empirical Relationship b/w mean, median ,and
mode.
In asymmetrical distribution the difference b/w x ̄̄ and z
is 3 times the difference bw x̄̄ and M.
x ̄̄-z = 3 (x̄̄ - M)
x ̄̄=1/2(3M-Z)
Mean Median Mode
M=1/3(2x ̄̄-z) Z= 3M-2x ̄̄
prev. years ques.
Relationship b/w mean, median ,and mode.
2. When distribution is positively skewed.
In this case x ̄̄,>M> Z. .
+VE-VE x ̄̄
In this case normal
distribution is +vly skewed..
MZ
Empirical Relationship b/w mean, median ,and
mode.
In asymmetrical distribution the difference b/w x ̄̄ and z
is 3 times the difference bw x̄̄ and M.
x ̄̄-z = 3 (x̄̄ - M)
x ̄̄=1/2(3M-Z)
Mean Median Mode
M=1/3(2x ̄̄-z) Z= 3M-2x ̄̄
Measures of dispersion
&skewness
statistiCs topiC-2
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