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Business Statistics Topics Mean Individual Series Direct Method Indirect Method
X =
∑^ X
n
X =A+
∑^ d
n Discrete Series Direct Method Indirect Method
X =
∑^ fx
∑^ f
X =A+
∑^ fd
∑^ 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
∑^ f
X =A+
∑^ fd
∑^ f
X =A+
∑^ fU
∑^ f^
∗ h Example of Table for Continuous Series Marks f Midpoint x fx d=X-A fd U= X − A 10 fu
∑^ f^ ∑^ f^ x^ ∑^ f^ d^ ∑^ f^ u
Median Median Individual Series Discrete Series Continuous Series n + 1 2 Observation N + 1
2 L +
N 2 − Cf f ∗ h
Example of Table for Series Marks f Cf N Mode Mode Continuous Series L+ f 1 − f (^0) 2 f 1 − f 0 − f (^2)
- h Individual Series Just Count which Repeated more Range R=Largest – Smallest Standard Deviation Individual Series Direct Method Actual Mean Method Assumed Mean Method √
∑^ x^
2 n −(
∑^ x
n ) 2 √
∑^ x^
2 n (^) √
∑^ d^
2 n −(
∑^ d
n ) 2 Co-efficient of Variance = σ x
Variance = σ 2 Discrete Series Discrete Series Direct Method Actual Mean Method Assumed Mean Method
∑^ f^ =^53 ∑^ fd^ =−^60 ∑^ f^ d^
2 = 17400
X =A+
∑^ fd
∑^ f
(− 60 ) 53 = 35 – 1. x = 33. Z= L+ f 1 − f (^2) 2 f 1 − f 0 − f (^2)
- h = 30 + 12 − 8 2 ∗ 12 − 8 − 10 ∗ 10 =30 + 4 6 ∗ 10 = 30 + 6. = 36. σ = √
∑^ f^ d^
2
∑^ f^
− (
∑^ fd
∑^ f^ ) 2 = √ 17400 53 −( (− 60 ) 53 ) 2
σ = 18.
s kp =
X − Z σ
33.87−36.
= -0. Bowley’s Co-efficient of Skewness s (^) kb = Q 3 + Q 1 − 2 M Q 3 − Q (^1) Quartile Deviation
C D =
Q 3 − Q (^1) 2 Co-efficient of Quartile deviation
C QD =
Q 3 − Q (^1) Q 3 + Q (^1) Q) Calculate Bowley’s Co-efficient of Skewness Wages Number of Employees 20- 40- 60- 80- 100- 120- 140-
Sol:- Wages f Cf 20- 40- 60- 80-
M = L +
N 2 − Cf f ∗ h
247.5− 195 140
M = 130.
Q (^3) = Position of 3 ( N 4 ) th Observation = (^3) ( 495 4 ) = 371.25th^ Observation Cf is just greater than or equal to 375.25 is 495 and Corresponding Class Interval is 140-
Q 3 = L +
3 N 4 − Cf f ∗ h
371.25− 295 160
Q 3 = 146.
s (^) kb = Q 3 + Q 1 − 2 M Q 3 − Q (^1) = 147.62+75.93− 2 (130.5) 147.62−75.
223.55− 261
s (^) kb =−0. Quartile Deviation
C D =
Q 3 − Q (^1) 2 = 147.62−75. 2 =
2 = 35. Co-efficient of Quartile deviation
C QD =
Q 3 − Q (^1) Q 3 + Q (^1) = 147.62−75. 147.62+75. =
= 0. Semi Average Method Q) The following data gives the sales in Lakhs of a firm Year 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 Sales 38 45 41 53 48 60 56 64 72 68 Sol:-
Sol:- Year Value 3-Yearly Moving Total 3-Yearly Moving Average 1997 1998 1999 2000 2001 2002 2003
1996 1997 1998 1999 2000 2001 2002 2003 2004 0 2 4 6 8 10 12 14 16 Value Average
Least Square Method Q) From the Following data fit the straight line trend and forecast the production figures for the next 2 years of xyz company Year 2006 2007 2008 2009 2010 2011 2012 2013 Production 64 70 82 68 75 88 90 94 Sol:- Year Production X=y-k x 2 xy Trend Value y (^) c = a + bx 2006 2007 2008 2009 2010 2011 2012 2013
∑^ y^ =^631 ∑^ x^ =^0 ∑^ x^
2
= 42 ∑^ xy^ =167.
a =
∑^ y
n
631 8
b =
∑^ xy
∑^ x^
2 =^
42
Trend Value y (^) c = a + bx 2006 = 78.87 + 3.98(-3.5) = 64. 2007 = 78.87 + 3.98(-2.5) = 68. 2008 = 78.87 + 3.98(-1.5) = 72. 2009 = 78.87 + 3.98(-0.5) = 76. 2010 = 78.87 + 3.98(0.5) = 80.
Index Numbers Simple Price Index
Price Relative =
Current Year Price Base Year Price
Aggregate Price Index
P 01 =
∑^ P^1
∑^ P^0
Average Price Relative Index Arithmetic Mean Geometric Mean 1 n
∑^ P^1
∑^ P^0
- 100 Antilog( 1 n ∗∑ log P (^) ) Commodities 20XX 20XX Price Price Weighted Price Index Weighted Aggregate Price Index Laspeyre Price Index Paasche Price Index Fisher Price Index Marshall Edgeworth Method
∑^ P^1 q^0
∑^ P^0 q^0
∑^ P^1 q^1
∑^ P^0 q^1
√^ L^ ∗^ P^ ∑ (^ q^0 +^ q^1 )^ P^1
∑ (^ q^0 +^ q^1 )^ P^0
Weighted Average Price Relative Index Arithmetic Mean Geometric Mean
P (^1) P (^0) ∗ 100 ∗ P 0 q (^0)
∑^ P^0 q^0
Antilog
(
∑^ Vlog^ P
∑^ V^ ) Note : P = p (^0) p (^1)
∗ 100 , V = p 0 q 0
Quality Index Laspeyre Quality Index Paasche Quality Index Fisher Quality Index
∑^ P^0 q^1
∑^ P^0 q^0
∑^ P^1 q^1
∑^ P^1 q^0
∑^ P^0 q^1
∑^ P^0 q^0
∗∑ P 1 q 1
∑^ P^1 q^0
Time Reversal Test P (^01) * P (^10) =
∑^ P^1 q^0
∑^ P^0 q^0
∗∑ P 1 q 1
∑^ P^0 q^1
∗∑ P 0 q 1
∑^ P^1 q^1
∗∑ P 0 q 0
∑^ P^1 q^0
Factor Reversal Test P (^01) * q (^01) =
∑^ P^1 q^0
∑^ P^0 q^0
∗∑ P 1 q 1
∑^ P^0 q^1
∗∑ P 0 q 1
∑^ P^0 q^0
∗∑ P 1 q 1
∑^ P^1 q^0
Example for When Both price and Quantity is Present Commoditie 20XX 20XX^ P^1 q^0 P^0 q^0 P^1 q^1 P^0 q^1
e. Time Reversal Test f. Factor Reversal Test Sol:- Commoditie s 2013 2014 P^1 q^0 P^0 q^0 P^1 q^1 P^0 q^1 Pric e Quantit y Pric e Quantit y Bricks 8 14 6 28 84 112 168 224 Steel 15 12 10 24 120 180 240 360 Timber 12 28 12 54 336 336 648 648 Cement 14 46 4 27 184 644 108 378
∑^ P^1 q^0 =∑^724 P^0 q^0 =∑^1272 P^1 q^1 =∑^1164 P^0 q^1 =^1610
Laspeyers Price and Quantity Index
P 01 =
∑^ P^1 q^0
∑^ P^0 q^0
724 1272
Q 01 =
∑^ P^0 q^1
∑^ P^0 q^0
1610 1272
Paashe Price and Quantity Index
P 01 =
∑^ P^1 q^1
∑^ P^0 q^1
1164 1610
Q 01 =
∑^ P^1 q^1
∑^ P^1 q^0
1164 724
Fisher Price and Quantity Index
P 01 =√ L ∗ P
Marshall-Edgeworth Method
∑ (^ q^0 +^ q^1 )^ P^1
∑ (^ q^0 +^ q^1 )^ P^0
724 + 1164 1272 + 1610 ∗ 100 = 1888 2882 ∗ 100
∑^ P^1 q^0
∑^ P^0 q^0
∗∑ P 1 q 1
∑^ P^0 q^1
∗∑ P 0 q 1
∑^ P^0 q^0
∗∑ P 1 q 1
∑^ P^1 q^0
724 1272 ∗ 1164 1610 ∗ 1610 1272 ∗ 1164 724
Correlation & Regression Analysis Degree of Correlation
Degree of Correlation Positive Negative
Perfect Correlation +1 -
Very High Degree +0.9 -0.
Fairly High Degree Between +0.75 and +0.9 Between -0.75 and -0.
Moderate Degree Between +0.25 and +0.75 Between -0.25 and -0.
Low Degree Between 0 and +0.25 Between 0 and -0.
Zero 0 0
Karl Pearson Coefficient of Correlation
Direct Method n ∑ xy −(∑ x )(∑ y )
√^ n^ (∑^ x^ 2
−(∑ x )
2 ∗√n (∑ y 2
−(∑ y )
2 Format X Y X 2 Y 2 XY
∑^ X^ ∑^ Y^ ∑^ X^
2
∑^ Y^
2
∑^ X^ Y
Actual Mean Method ∑ dxdy
√∑^ d^ x^ 2
∑^ d^ y^
2 Format X Y dx = X- X dy = Y- Y (^) d X 2 d Y 2 dXY
∑^ dX^
2
∑^ dY^
2
∑^ dX^ Y
Assumed Mean Method n ∑ UV −(∑ U )(∑ V )
√^ n^ (∑^ U^ 2
−(∑ U )
2 ∗√ n (∑ V 2
−(∑ V )
2 Format X Y U V U 2 V 2 UV
∑^ U^ ∑^ V^ ∑^ U^
2
∑^ V^
2
∑^ UV
Probable Error = 0.6745 (
1 − r 2
√n^
Q) Find the Coefficient of Correlation from the following X 1 2 3 4 5 6 7 8 9 10 Y 62 56 48 41 36 28 21 16 12 8 Assumed Mean Method Sol: Let Assume X=5, Y= X Y U (X-5) V (Y-40) U 2 V 2 UV 1 62 -4 22 16 484 -
