By CA Anand V Kaku
STATISTICS FORMULAE
No. of class interval × class lengths = Range
LCB = LCL –D/2
UCB=UCL +D/2
Mid-point =LCB+UCB/2 or LCL+UCL/2
Frequency density of a class interval = frequency of that class interval/corresponding class length
Relative frequency and percentage frequency of a class interval=class frequency/ total frequency
ARITHMETIC MEAN
…
Discrete Series
GEOMETRIC MEAN
.
…..
GM=
HARMONIC MEAN
Logarithm of G for a set of
observations is the AM of the
logarithm of the observations; i.e.
∑ log
log
∑
.
Frequency
Distribution
…
…
GM=
∑
1
MODE
the value that occurs the maximum
number of times
∑ log
…..
.
∑
2
X
where,
= LCB of the modal class. i.e. the
class containing mode.
= frequency of the modal class
= frequency of the pre – modal
class
= frequency of the post modal class
C = class length of the modal class
By CA Anand V Kaku
STATISTICS FORMULAE
GEOMETRIC MEAN
ARITHMETIC MEAN
Relationship
variables
HARMONIC MEAN
if z = xy, then
GM of z =(GM of x)×(GM of y)
MODE
ymo=a+bxmo
if z = x/y then
GM of z =(GM of x)/(GM of y)
Weighted Mean
Weighted A.M
∑
.
∑
∑
.
log
∑
Combined Mean
Important points
-
.
∑
-
∑
.
-
the algebraic sum of deviations of a
set of observations from their AM is
zero ∑
0
-
-
-
AM is affected due to a change of
origin and/or scale which implies
that if the original variable x is
changed to another variable y by
effecting a change of origin,
-
-
-
Relation among
Averages
For Given two positive numbers (A.M) x (H.M) = (G.M)2
AM
GM
HM
The equality sign occurs, as we have already seen, when all the observations are equal.
Mode = 3 Median – 2 Mean
STATISTICS FORMULAE
MEDIAN
Discrete Series
/Unclassified Data
Median
1
Size of
2
QUARTILES (Q1,Q2 & Q3)
DECILES (D1,D2 ,D3,…… D9)
PERCENTILES (P1,P2,P3,….,
P99 )
Q1 quartile is given by the
1
value
the D1 Decile is given by the
1
value
the P1 Percentile is given by
the
1
value
Dn Decile is given by the
1
value
Pn Percentile is given by the
1
value
the Qn quartile is given by the
1
value
Group Frequency
Distribution
2
By CA Anand V Kaku
.
X
= lower class boundary of the
median class i.e. the class
containing median.
N = total frequency.
= less than cumulative frequency
corresponding to l1. (Pre median
class)
= less than cumulative frequency
corresponding to l2. (Post median
class)
= being the upper class boundary
of the median class.
C = - = length of the median
class.
yme = a + b xme
∑
is minimum if we choose A
as the median.
.
X
= lower class boundary of the
Quartile class i.e. the class
containing Quartile.
N = total frequency.
p= , ,
for Q1, Q2 ,Q3
respectively
= less than cumulative
frequency corresponding to l1.
(Pre Quartile class)
= less than cumulative
frequency corresponding to l2.
(Post Quartile class)
= being the upper class
boundary of the Quartile class.
C = - = length of the Quartile
class.
X
= lower class boundary of
the Decile class i.e. the class
containing Decile.
N = total frequency.
p= , , , … . ,
for D1, D2
,D3 ,…., D9 respectively
= less than cumulative
frequency corresponding to
. (Pre Decile class)
= less than cumulative
frequency corresponding to
. (Post Decile class)
= being the upper class
boundary of the Decile
class.
C = - = length of the
Decile class.
.
X
= lower class boundary of
the Percentile class i.e. the
class containing Percentile.
N = total frequency.
p=
,
,
,…,
for P1,
P2 ,P3,…, P99 respectively
= less than cumulative
frequency corresponding to
l1 . (Pre Percentile class)
= less than cumulative
frequency corresponding to
. (Post Percentile class)
= being the upper class
boundary of the Percentile
class.
C = - = length of the
Percentile class.
By CA Anand V Kaku
STATISTICS FORMULAE
DISPERSIONS
RANGE (R)
Absolute
Range= Laregest (L) – Smallest (S)
MEAN DEVIATION (M.D)
about A
M.DA= ∑|
MEAN DEVIATION (M.D)
about A.M ( )
M.D about Mean = ∑|
MEAN DEVIATION (M.D)
about Median
M.D about Median = ∑|
Relative
100
Co efficient of Range =
Co efficient of M.D from A =
.D
100
Co efficient of M.D from A.M =
|
STANDARD DEVIATION (σ)
σ
σ
∑ x
|
.D
|
X
|
| Deviation for First n Natural
Standard
numbers ,
QUARTILE DEVIATION
(Qd)
VARIANCE (σ2)
100
Co efficient of M.D from Median
.D
100
=
Co efficient of Variation =
Qd=
Combined Standard Deviation,
Where d1=
σ12=
, d2=
100
Co-efficient of Q.D =
Or
Co-efficient of Q.D =
Variance means Square of Standard Deviation
M.Dy = | | M.Dx
.
M.Dy = | | M.Dx
100 σy = | | σx
X
Standard Deviation for Two numbers,
M.Dy = | | M.Dx
X
n
∑x
n
If
y = a+bx
Ry = | | R x
100
STATISTICS FORMULAE
By CA Anand V Kaku
CORRELATION
KARL PEARSON’S PRODUCT MOMENT
CORRELATION COEFFICIENT
&
Where,
1
For tied ranking,
∑
| || |
1
∑
∑
∑
where
1
where c is concurrent deviation,
∑
X
∑
∑
∑
COEFFICIENT OF CONCURRENT
DEVIATIONS
6∑
∑
&
∑
SPEARMAN’S RANK CORRELATION
COEFFICIENT
∑
6 ∑
∑
12
1
m is one less than number of pairs of
observations
By CA Anand V Kaku
STATISTICS FORMULAE
REGRESSION ANALYSIS
Y depends on X
y= a + byx x
Simple Regression
Equation
Normal Equations
b
x
x
Regression Co
efficient
X depends on Y
x = a + bxy y
b
b
x
b
y
,
b
b
b
y
y
,
b
Intersection
point of these
two lines is ,
b
∑
∑
∑
∑
b
∑
b
∑
∑
∑
∑
∑
b
where
1
By CA Anand V Kaku
STATISTICS FORMULAE
NAME
Binomial
Distribution
CONDITION
Trials are independent
and each trail has only
two outcomes Success
& Failure.
PROBABILITY MASS FUNCTION
n
x
P(X=x) = Cx p q
n-x
Notation
X ∼ B (n,p)
MEAN
VARINACE
μ=np
σ2=npq
Trials are independent
and probability of
occurrence is very small
in given time.
P(X = x) =
X ∼ P (m)
.
If (n+1)p is integer
Mode, μ0 = (n+1)p-1
then
p+q=1
If (n+1)p is non-integer then
Mode,μ0 =Highest Integer in
(n+1)p
For x = 0,1,2,..,n
Poisson
Distribution
Remarks
MODE
μ=m
σ2=m
If m is Integer Mode,μ0 =m-1
e = 2.71828
If m is non- integer Mode,μ0
= Highest Integer in m
!
For x = 0,1,2,..,n
Normal or
Gaussian
Distribution
When distribution is
symmetric
X ∼N (μ,σ2)
/
σ√
For
∞
∞
z=
Mean
=
Median
=
Mode
=μ
σ2
μ
Mean Deviation =
0.8 σ
First Quartile
= μ - 0.675σ
Third Quartile
= μ + 0.675σ
Quartile
Deviation=0.675σ
Point of Inflexion
x=μ - σ and x=μ + σ