Download File - Statistics Notes

Formulae sheet
Arithmetic mean
Individual
Direct
method(actual
mean method)
𝑥̅ =
Short cut
method (from
assumed
mean )
𝑥̅ =a +
Step deviation
method(from
assumed
mean)
Discrete
∑𝑥
𝑛
∑(𝑥−𝑎)
----------------
𝑛
𝑥̅ =
𝑥̅ =a +
𝑥̅ =a +
continuous
∑𝑓𝑥
∑𝑓
∑𝑓(𝑥−𝑎)
∑𝑓
∑𝑓(𝑥−𝑎𝑙 )
∑𝑓
*C
(𝑥 − 𝑎)𝑙 = (x –a)/c
∑𝑓𝑥
̅̅̅̅
𝑥 =
𝑥̅ =a +
𝑥̅ =a +
∑𝑓
∑𝑓(𝑥−𝑎)
∑𝑓
∑𝑓(𝑥−𝑎𝑙 )
∑𝑓
*C
(𝑥 − 𝑎)𝑙 = (x –a)/c
X = mid value
Weighted arithmetic mean
𝑥̅ =
∑𝑤𝑥
∑𝑤
Median
Individual
Discrete
Continuous
(N/2)th class
(𝑁+1)𝑡ℎ
2
Item
(𝑁+1)𝑡ℎ
2
Item
Median = 𝐿 +
(
𝑛
2
− 𝑐𝑓)
𝑓
*c
1
Quartiles
Individual
Discrete
Continuous
(N/4)th class
First quartile
N = total frequency
(𝑁+1)𝑡ℎ
Q1=
4
Item
N = no. of observations
Q1 =
(𝑁+1)𝑡ℎ
4
𝑛
Q1= 𝐿 +
(4 − 𝑐𝑓)
𝑓
*c
N = total frequency
Third quartile
Q3=
3(𝑁+1)𝑡ℎ
4
Item
Q3 =
3(𝑁+1)𝑡ℎ
4
Item
3(N/4)th class
𝒏
Q3= 𝑳 +
(𝟑𝟒 − 𝒄𝒇)
𝒇
*c
Mode
Individual
Discrete
Continuous
𝑓1−𝑓0
Inspection method
Inspection method or with the
help of grouping table
Mode = 𝐿 + 2𝑓1−𝑓0−𝑓2* c
F0- frequency of the class
preceding the modal class
F1- frequency of the modal class
F2- frequency of the class
succeeding the modal class
2
Range
Individual
L-s
L = largest value
s-= smallest value
Discrete
Continuous
L-s
L-s
L = largest variable
s-= smallest variable
L = largest upper limit
s-= smallest lower limit
Inter quartile range
Q3 −
Q1
Quartile deviation
(Q3 − Q1 )/2
Co-efficient of quartile deviation
Q3− Q1
Q3+ Q1
Mean deviation
Individual
Mean deviation =
∑|x − 𝑥̅ |
𝑛
Discrete
Continuous
Mean deviation =
Mean deviation =
∑f|x − 𝑥̅ |
∑f|x − 𝑥̅ |
∑f
∑f
X is the mid value
Co-efficient of Mean deviation
3
𝑴𝑫
𝑥 𝑜𝑟 𝑚𝑒𝑑𝑖𝑎𝑛
̅̅̅
Standard deviation
Individual
Direct
method(actual
mean method)
𝜎=
√∑(𝑋 – ̅̅̅
𝑥)2
𝑛
Discrete
𝜎=
continuous
√∑𝑓(𝑋 – ̅̅̅
𝑥)2
∑𝑓
𝜎=
√∑𝑓(𝑋 – ̅̅̅
𝑥)2
∑𝑓
X –mid value
Short cut
method (from
assumed
mean )
∑𝑑2
∑𝑑 2
𝜎=√
−( )
𝑛
𝑛
∑𝑓𝑑2
∑𝑓𝑑 2
√
𝜎=
−(
)
∑𝑓
∑𝑓
∑𝑓𝑑2
∑𝑓𝑑 2
√
𝜎=
−(
)
∑𝑓
∑𝑓
d=x-a
Step deviation
method(from
assumed
mean)
-----------------
𝜎=√
∑𝑓𝑑′2
∑𝑓
∑𝑓𝑑′ 2
)
∑𝑓
−(
∗𝑐
𝜎=√
∑𝑓𝑑′2
∑𝑓
∑𝑓𝑑′ 2
)
∑𝑓
−(
∗𝑐
d’=x-a/c
Coefficient of standard deviation
𝜎
̅̅̅̅
𝑥
∗ 100
Variance
(SD)2
Lorenz curve (refer notes)
Scatter diagram (refer notes)
4
Karl pearsons co-efficient of correlation

r=

Direct method or actual mean method
̅̅̅
∑(X−x̅)(Y−y)
̅̅̅2 ∑(𝑦 –𝑦)
̅̅̅2
√∑(𝑋 – 𝑥)
Short cut method or assumed mean method
𝑟 =
∑𝑋𝑌−(∑𝑥)(∑𝑦)/𝑛
√∑𝑥 2 –(
(∑𝑥)2
𝑛
) √∑𝑦 2 –(
(∑𝑦)2
𝑛
)
* ( pls. refer the text book pg 96, third formulae)
Spearmans rank correlation method

When ranks are given
r=1–

6∑d2
𝑛(𝑛2 − 1)
When ranks are not given assign rank for the variables and apply the following formula
r=1–

6∑d2
𝑛(𝑛2 − 1)
When variables repeat add the adjustment factor m3-m/12
 rk = 1 - 6﴾∑D2 + m3-m /12 + ------------ )
(refer pg 101 in your text )
n (n2 – 1)
Index number

Simple aggregate method
 p01 = ∑p1/ ∑p0 x 100

weighted aggregate method
Laspeyres
5
P01 =( ∑p1q0 /∑ p0q0 ) x 100
p1 = price of current year
p0 = price of base year
q0 = base year quantity
Paasches
P01 = (∑p1q1 /∑p0q1 ) x 100
p1 = price of current year
p0 = price of base year
q0 = base year quantity
q1 = current year quantity

Simple price relative method
p01 = ∑R/N
R = p1/p0 x 100

Weighted price relative method
 p01 = ∑Rw/∑w
R = p1/p0 x 100


w = weights (if weights are not given take p0q0 as weights
Cost of living index
 P01 =( ∑p1q0 /∑ p0q0 ) x 100

Family budget method
 p01 = ∑Rw/∑w

(laspeyrs method)
(weighted relative)
Production index
 p01 = ∑Rw/∑w

R = p1/p0 x 100
w = weights (if weights are not given take
p0q0 as weight)
6
7