Quantitative Applications In Management

Faculty – Mr. Ashu Jain

Course – “Quantitative Applications In Management.”

Programme – MBA-IB; 1st Semester

Amity International Business School

Arithmetic Mean (Direct Method)

Individual Series x¯ = ∑X / N Here ∑X = Sum of variables And N = Number of Items

Discrete Series x¯ = ∑fX / ∑f Here ∑f = Total no of Frequencies

Continuous Series x¯ = ∑fX / ∑f Here X = Mid values of class intervals

Arithmetic Mean (Short cut Method)

Individual Series x¯ = A + ∑dx / N Here ∑dx = Sum of deviations taken from

assumed mean A = Assumed Mean

Discrete Series x¯ = A + ∑(fdx) / ∑f Here ∑fdx = Sum of Multiplication of Frequency

with deviations taken from assumed mean

Continuous Series x¯ = A + ∑(fdx) / ∑f

Median Individual Series

N+1 / 2th Item Here N = Total No. of Items arranged in ascending or descending order.

Discrete Series ∑f+1 / 2th Item

Here (∑f+1 / 2th) Item will be judged on the basis of cumulative frequency.

Continuous Series N / 2th Item L1 + N/2 – C.F. * i

F Here L1 = Lower limit of Median class N/2 = Median item C.F. = Cumulative Frequency preceding class interval F = Frequency against Median class interval i = Gap of Median class interval

Mode

Continuous Series

L1 + |f1 – f0l * i

| f1-f0 | + | f1-f2 | Here L1 = Lower limit of the Modal Class

Interval. f1 = Frequency of Modal class

Quartile Deviation

Q.D. = Q3 – Q1 / 2

Here, Q3 = 3rd quartile And, Q1 = 1st quartile

Mean Deviation / Average Deviation

Individual Series M.D. =( ∑ldxl ) / N

Here, dx = X – Mean / Median / Mode

Discrete Series, Continuous Series M.D. =( ∑f ldxl ) / ∑f

Here, dx = X – Mean / Median / Mode

Standard Deviation

Individual Series S.D. = √∑dx² / N

Here, dx = X – Actual Mean

Discrete Series S.D. = √∑fdx² / ∑f

Here, dx = X – Actual Mean Continuous Series

S.D. = √∑fdx² / ∑f

Here, dx = X – Actual MeanAnd, X = Mid Values of class intervals

Variance and Coefficient of Variation

Variance = (S.D.)²

Coefficient of Variation = S.D. X 100

Mean

Karl Pearson’s Coefficient of Correlation (Direct Method)

r = ∑dxdyN σx σy

r = ∑dxdy√∑dx² √∑dy²

Karl Pearson’s Coefficient of Correlation (Short cut / Assumed Mean Method)

o r = ∑dxdy - ∑dx∑dy

N √∑dx² - (∑dx)² √∑dy² - (∑dy)²

N N

r = ∑fdxdy – (∑fdx)(∑fdy)

N

√∑fdx² - (∑fdx)² √∑fdy² - (∑fdy)²

N N

Spearman’s Rank Correlation Method

When Ranks are not Repeated:-

rk = 1 - 6 ∑D²

N(N²-1)

Here D = Rank 1 – Rank 2

Regression Equations General Form:-

X on Y

X – X = r σx (Y – Y) σy

• r σx = bxy = Regression Coefficient of Equation X on Y σy

Y on X

Y – Y = r σy (X – X) σx

• r σy = byx = Regression Coefficient of Equation Y on X

σx

Regression Equations Actual Mean Method:-

X on Y

X – X = ∑dxdy (Y – Y) ∑dy²

Y on X

Y – Y = ∑dxdy (X – X) ∑dx²

Regression Equations Assumed Mean Method:-

X on Y X – X = ∑dxdy - ∑dx∑dy (Y – Y) N ∑dy² - (∑dy)²

N

Y on X Y – Y = ∑dxdy - ∑dx∑dy (X – X) N ∑dx² - (∑dx)²

N

Regression Equations Assumed Mean Method ( Continuous Series ) :-

X on Y X – X = ∑fdxdy - ∑fdx∑fdy (Y – Y) N x ix ∑fdy² - (∑fdy)² iy

N

Y on X Y – Y = ∑fdxdy - ∑fdx∑fdy (X– X) N x iy ∑fdx² - (∑fdx)² ix

N

Simple Aggregative Methodo P01 = ∑P1 x 100

∑P0

Here, P01 = Price Index for the Current year ∑P1 = Total of Current year Prices ∑P0 = Total of Base year Prices

P01 = ∑(P1/ P0 x 100)

N

Here, P01 = Price Index for the Current year ∑P1 = Current year Price ∑P0 = Base year Price N = Total Number of Years

Chain Base Index

Chain Base Index =

Current year Link Relative x Previous year Chain Index

100

Base Shifting

New Base Index Number =

Old Index Number of Current Year x 100

Old Index Number of New Base Year

Laspeyre’s Method / Aggregate Expenditure Method

o P01 = ∑P1Q0 x 100∑P0Q0

Paasche’s Method

o P01 = ∑P1Q1 x 100∑P0Q1

Dorbish and Bowley’s Method

o P01 = ∑P1Q0 + ∑P1Q1

∑P0Q0 ∑P0Q1 x 100

2

Marshall-Edgeworth’s Method

o P01 = ∑P1Q0 + ∑P1Q1 x 100

∑P0Q0 + ∑P0Q1

Fisher’s Method

o P01 =√ ∑P1Q0 x ∑P1Q1 x 100∑P0Q0 ∑P0Q1

Kelly’s Method

o P01 = ∑P1Q x 100

∑P0Q

Here,Q = Q0 + Q1

2

Weighted Average of Price Relative / Family Budget Method

P01 = ∑PV ∑V

Here, P = Price Relatives V = P0Q0

Components of Time Series

Secular Trend

Cyclical Variations

Seasonal Variations

Irregular or Random Variations

Methods of Measuring Trend

Free Hand Curve Method

Semi Average Method

Moving Average Method

Method of Least Square

Semi Average Method

Annual Change =

Difference of Two Semi Average Values

Difference of Years of Semi Average

Method of Least Square

o Equation for Time SeriesY = a + bX

To calculate a and b, Solve the following Equations:

∑Y = aN + b∑X

∑XY = a∑X + b∑X²

Here,

Y = Given Data i.e. Sales or Profit etc.

X= Years in terms of Units like 1,2,3 etc.