Page 1
Page # 47
STATISTICS
1. Arithmetic Mean / or Mean
If x
1
, x
2
, x
3
,.......x
n
are n values of variate x
i
then their A.M. x is defined as
x =
n
x ....... x x x
n 3 2 1
n
x
n
1 i
i ?
?
If x
1
, x
2
, x
3
, .... x
n
are values of veriate with frequencies f
1
, f
2
, f
3
,.........f
n
then
their A.M. is given by
x
=
n 3 2 1
n n 3 3 2 2 1 1
f ...... f f f
f f ...... x f x f x f
? ? ? ?
? ? ?
=
N
x f
n
1 i
i i ?
?
, where N = ?
?
n
1 i
i
f
2. Properties of Arithmetic Mean :
(i) Sum of deviation of variate from their A.M. is always zero that is
? ? ? x x
i
? = 0.
(ii) Sum of square of deviation of variate from their A.M. is minimum
that is ? ? ?
2
i
x x ? is minimum
(iii) If x is mean of variate x
i
then
A.M. of (x
i
+ ?) =
x
+ ?
A.M. of ?
i
. x
i
= ?.
x
A.M. of (ax
i
+ b) = a
x
+ b
3. Median
The median of a series is values of middle term of series when the values
are written is ascending order or descending order. Therefore median,
divide on arranged series in two equal parts
For ungrouped distribution :
If n be number of variates in a series then
Median =
? ?
? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
?
?
? ?
even is n when term 2
2
n
and
2
n
of Mean
odd is n when , term
2
1 n
th th
th
? ? ? ?
=
Page 2
Page # 47
STATISTICS
1. Arithmetic Mean / or Mean
If x
1
, x
2
, x
3
,.......x
n
are n values of variate x
i
then their A.M. x is defined as
x =
n
x ....... x x x
n 3 2 1
n
x
n
1 i
i ?
?
If x
1
, x
2
, x
3
, .... x
n
are values of veriate with frequencies f
1
, f
2
, f
3
,.........f
n
then
their A.M. is given by
x
=
n 3 2 1
n n 3 3 2 2 1 1
f ...... f f f
f f ...... x f x f x f
? ? ? ?
? ? ?
=
N
x f
n
1 i
i i ?
?
, where N = ?
?
n
1 i
i
f
2. Properties of Arithmetic Mean :
(i) Sum of deviation of variate from their A.M. is always zero that is
? ? ? x x
i
? = 0.
(ii) Sum of square of deviation of variate from their A.M. is minimum
that is ? ? ?
2
i
x x ? is minimum
(iii) If x is mean of variate x
i
then
A.M. of (x
i
+ ?) =
x
+ ?
A.M. of ?
i
. x
i
= ?.
x
A.M. of (ax
i
+ b) = a
x
+ b
3. Median
The median of a series is values of middle term of series when the values
are written is ascending order or descending order. Therefore median,
divide on arranged series in two equal parts
For ungrouped distribution :
If n be number of variates in a series then
Median =
? ?
? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
?
?
? ?
even is n when term 2
2
n
and
2
n
of Mean
odd is n when , term
2
1 n
th th
th
? ? ? ?
=
Page # 48
4. Mode
If a frequency distribution the mode is the value of that variate which have
the maximum frequency. Mode for
For ungrouped distribution :
The value of variate which has maximum frequency.
For ungrouped frequency distribution :
The value of that variate which have maximum frequency.
Relationship between mean, median and mode.
(i) In symmetric distribution, mean = mode = median
(ii) In skew (moderately asymmetrical) distribution,
median divides mean and mode internally in 1 : 2 ratio.
? median =
? ? ? ?
3
Mode Mean 2 ?
5. Range
values extreme of sum
values extreme of difference
=
S L
S L
?
?
where L = largest value and S = smallest value
6. Mean deviation :
Mean deviation =
n
| A x |
n
1 i
i ?
?
?
Mean deviation =
N
| A x | f
n
1 i
i i ?
?
?
(for frequency distribution)
7. Variance :
Standard deviation = +
iance var
formula
?
x
2
=
? ?
n
x x
2
i
? ?
Page 3
Page # 47
STATISTICS
1. Arithmetic Mean / or Mean
If x
1
, x
2
, x
3
,.......x
n
are n values of variate x
i
then their A.M. x is defined as
x =
n
x ....... x x x
n 3 2 1
n
x
n
1 i
i ?
?
If x
1
, x
2
, x
3
, .... x
n
are values of veriate with frequencies f
1
, f
2
, f
3
,.........f
n
then
their A.M. is given by
x
=
n 3 2 1
n n 3 3 2 2 1 1
f ...... f f f
f f ...... x f x f x f
? ? ? ?
? ? ?
=
N
x f
n
1 i
i i ?
?
, where N = ?
?
n
1 i
i
f
2. Properties of Arithmetic Mean :
(i) Sum of deviation of variate from their A.M. is always zero that is
? ? ? x x
i
? = 0.
(ii) Sum of square of deviation of variate from their A.M. is minimum
that is ? ? ?
2
i
x x ? is minimum
(iii) If x is mean of variate x
i
then
A.M. of (x
i
+ ?) =
x
+ ?
A.M. of ?
i
. x
i
= ?.
x
A.M. of (ax
i
+ b) = a
x
+ b
3. Median
The median of a series is values of middle term of series when the values
are written is ascending order or descending order. Therefore median,
divide on arranged series in two equal parts
For ungrouped distribution :
If n be number of variates in a series then
Median =
? ?
? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
?
?
? ?
even is n when term 2
2
n
and
2
n
of Mean
odd is n when , term
2
1 n
th th
th
? ? ? ?
=
Page # 48
4. Mode
If a frequency distribution the mode is the value of that variate which have
the maximum frequency. Mode for
For ungrouped distribution :
The value of variate which has maximum frequency.
For ungrouped frequency distribution :
The value of that variate which have maximum frequency.
Relationship between mean, median and mode.
(i) In symmetric distribution, mean = mode = median
(ii) In skew (moderately asymmetrical) distribution,
median divides mean and mode internally in 1 : 2 ratio.
? median =
? ? ? ?
3
Mode Mean 2 ?
5. Range
values extreme of sum
values extreme of difference
=
S L
S L
?
?
where L = largest value and S = smallest value
6. Mean deviation :
Mean deviation =
n
| A x |
n
1 i
i ?
?
?
Mean deviation =
N
| A x | f
n
1 i
i i ?
?
?
(for frequency distribution)
7. Variance :
Standard deviation = +
iance var
formula
?
x
2
=
? ?
n
x x
2
i
? ?
Page # 49
?
x
2
=
n
x
n
1 i
2
i ?
?
–
2
n
1 i
i
n
x
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
n
x
n
1 i
2
i ?
?
– ? ?
2
x
?
d
2
=
n
d
2
i
?
–
2
n
di
?
?
?
?
?
? ?
, where d
i
= x
i
– a , where a = assumed mean
(ii) coefficient of S.D. =
?
?
?
?
?
? ?
x
coefficient of variation =
?
?
?
?
?
? ?
x
× 100 (in percentage)
Properties of variance :
(i) var(x
i
+ ?) = var(x
i
)
(ii) var( ?.x
i
) = ?
2
(var x
i
)
(iii) var(a x
i
+ b) = a
2
(var x
i
)
where ?, a, b are constant.
Read More