Business Statistics
Topics
Mean
Individual Series Direct Method Indirect Method
X=
∑X X =A+
∑d
n n
Discrete Series Direct Method Indirect Method
X=
∑ fx X =A+
∑ fd
∑f ∑f
Example of Table for Discrete Series
X f fx d=(X-A) fd
∑f ∑ fx ∑ fd
Continuous Series Direct Method Indirect Method Step-Deviation
Method
X=
∑ fx X =A+
∑ fd X =A+
∑ fU ∗h
∑f ∑f ∑f
Example of Table for Continuous Series
Marks f Midpoint fx d=X-A fd U= fu
x X− A
10
∑f ∑f x ∑fd ∑fu
Median [K]
Median Individual Series Discrete Series Continuous Series
n+1 N+1 N
Observation −Cf
2 2 L+ 2
∗h
f
,Example of Table for Series
Marks f Cf
N
Mode
Mode Continuous Series
f 1−f 0
L+ 2 f −f −f * h
1 0 2
Individual Series Just Count which Repeated more
Range
R=Largest – Smallest
Standard Deviation
Individual Series Direct Method Actual Mean Assumed Mean
Method Method
√ ∑ x2 − ∑ x
( ) √ ∑ x2
√ ( )
2
∑ d2− ∑ d
2
n n n n n
σ
Co-efficient of Variance = x * 100
Variance = σ 2
Discrete Series [U]
Discrete Series Direct Method Actual Mean Assumed Mean
Method Method
√ ( ) √ ∑ f x2
√ ( )
2 2
∑ f x 2 − ∑ fx ∑ f d 2 − ∑ fd
n
∑f ∑f ∑f ∑f
,Continuous Series
Continuous Actual Mean Assumed Mean Step-Deviation
Series Method Method Method
√ ∑ f x2
√ ( ) √ ( )
2 2
∑ f d 2 − ∑ fd ∑ f d 1 − ∑ f d1
2
n
∑f ∑f ∑f ∑f
Karl Pearson’s Co-efficient of Skewness
=3( )
x−M X−Z
skp
σ OR skp = σ
Q) Calculate Karl Pearson’s Co-efficient of Skewness
Wages Number of Workers
0-10 5
10-20 9
20-30 8
30-40 12
40-50 10
50-60 4
60-70 3
70-80 2
[S]
Sol :-
Wages f MV d= X-A d
2
fd fd
2
0-10 5 5 -30 900 -150 4500
10-20 9 15 -20 400 -180 3600
20-30 8 25 -10 100 -80 800
30-40 12 35 0 0 0 0
40-50 10 45 10 100 100 1000
50-60 4 55 20 400 80 1600
60-70 3 65 30 900 90 2700
70-80 2 75 40 1600 80 3200
∑ f =53 ∑ fd=−60 ∑ f d 2=17400
[H]
X =A+
∑ fd
∑f
, (−60)
= 35 +
53
= 35 – 1.13
x = 33.87
f 1−f 2
Z= L+ 2 f −f −f * h
1 0 2
12−8
= 30 + 2∗12−8−10 ∗10
4
=30 + 6 ∗10
= 30 + 6.66
= 36.66
√ ( )
2
σ = ∑ f d 2 − ∑ fd
∑f ∑f
√ ( )
2
(−60)
= 17400 −
53 53
= √ 328.30−1.27
= √ 327.03
σ = 18.08
X−Z
skp =
σ
33.87−36.66
= 18.08
= -0.15
[A]
Bowley’s Co-efficient of Skewness
Topics
Mean
Individual Series Direct Method Indirect Method
X=
∑X X =A+
∑d
n n
Discrete Series Direct Method Indirect Method
X=
∑ fx X =A+
∑ fd
∑f ∑f
Example of Table for Discrete Series
X f fx d=(X-A) fd
∑f ∑ fx ∑ fd
Continuous Series Direct Method Indirect Method Step-Deviation
Method
X=
∑ fx X =A+
∑ fd X =A+
∑ fU ∗h
∑f ∑f ∑f
Example of Table for Continuous Series
Marks f Midpoint fx d=X-A fd U= fu
x X− A
10
∑f ∑f x ∑fd ∑fu
Median [K]
Median Individual Series Discrete Series Continuous Series
n+1 N+1 N
Observation −Cf
2 2 L+ 2
∗h
f
,Example of Table for Series
Marks f Cf
N
Mode
Mode Continuous Series
f 1−f 0
L+ 2 f −f −f * h
1 0 2
Individual Series Just Count which Repeated more
Range
R=Largest – Smallest
Standard Deviation
Individual Series Direct Method Actual Mean Assumed Mean
Method Method
√ ∑ x2 − ∑ x
( ) √ ∑ x2
√ ( )
2
∑ d2− ∑ d
2
n n n n n
σ
Co-efficient of Variance = x * 100
Variance = σ 2
Discrete Series [U]
Discrete Series Direct Method Actual Mean Assumed Mean
Method Method
√ ( ) √ ∑ f x2
√ ( )
2 2
∑ f x 2 − ∑ fx ∑ f d 2 − ∑ fd
n
∑f ∑f ∑f ∑f
,Continuous Series
Continuous Actual Mean Assumed Mean Step-Deviation
Series Method Method Method
√ ∑ f x2
√ ( ) √ ( )
2 2
∑ f d 2 − ∑ fd ∑ f d 1 − ∑ f d1
2
n
∑f ∑f ∑f ∑f
Karl Pearson’s Co-efficient of Skewness
=3( )
x−M X−Z
skp
σ OR skp = σ
Q) Calculate Karl Pearson’s Co-efficient of Skewness
Wages Number of Workers
0-10 5
10-20 9
20-30 8
30-40 12
40-50 10
50-60 4
60-70 3
70-80 2
[S]
Sol :-
Wages f MV d= X-A d
2
fd fd
2
0-10 5 5 -30 900 -150 4500
10-20 9 15 -20 400 -180 3600
20-30 8 25 -10 100 -80 800
30-40 12 35 0 0 0 0
40-50 10 45 10 100 100 1000
50-60 4 55 20 400 80 1600
60-70 3 65 30 900 90 2700
70-80 2 75 40 1600 80 3200
∑ f =53 ∑ fd=−60 ∑ f d 2=17400
[H]
X =A+
∑ fd
∑f
, (−60)
= 35 +
53
= 35 – 1.13
x = 33.87
f 1−f 2
Z= L+ 2 f −f −f * h
1 0 2
12−8
= 30 + 2∗12−8−10 ∗10
4
=30 + 6 ∗10
= 30 + 6.66
= 36.66
√ ( )
2
σ = ∑ f d 2 − ∑ fd
∑f ∑f
√ ( )
2
(−60)
= 17400 −
53 53
= √ 328.30−1.27
= √ 327.03
σ = 18.08
X−Z
skp =
σ
33.87−36.66
= 18.08
= -0.15
[A]
Bowley’s Co-efficient of Skewness
