Central Tendency, Dispersion, Correlation, Regression, Index Number, Time Series

1. Quantitative Applications In Management
Faculty – Mr. Ashu Jain
Course – “Quantitative Applications In Management.”
Programme – MBA-IB; 1st Semester
Amity International Business School

2. 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

3. 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

4. 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

5. Mode
 Continuous Series
 L1 + | f1 – f 0 l * i
| f1-f0 | + | f1-f2 |
 Here L = Lower limit of the Modal Class
1
Interval.
 f = Frequency of Modal class
1

6. Quartile Deviation
 Q.D. = Q – Q / 2
3 1
 Here, Q3 = 3rd quartile
 And, Q1 = 1st quartile

7. 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

8. 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 Mean
And, X = Mid Values of class intervals

9. Variance and Coefficient of Variation
 Variance = (S.D.)²
 Coefficient of Variation = S.D. X 100
Mean

10. Karl Pearson’s Coefficient of
Correlation (Direct Method)
 r = ∑dxdy
N σx σy
 r= ∑dxdy
√∑dx² √∑dy²

11. 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

12. Spearman’s Rank Correlation Method
 When Ranks are not Repeated:-
 rk = 1 - 6 ∑D²
N(N²-1)
 Here D = Rank 1 – Rank 2

13. 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

14. Regression Equations
 Actual Mean Method:-
 X on Y
 X – X = ∑dxdy (Y – Y)
∑dy²
 Y on X
 Y – Y = ∑dxdy (X – X)
∑dx²

15. 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

16. 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

17. Simple Aggregative Method
o 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

18. Chain Base Index
Chain Base Index =
Current year Link Relative x Previous year Chain Index
100

19. Base Shifting
New Base Index Number =
Old Index Number of Current Year x 100
Old Index Number of New Base Year

20. Laspeyre’s Method / Aggregate
Expenditure Method
o P01 = ∑P1Q0 x 100
∑P0Q0

21. Paasche’s Method
o P01 = ∑P1Q1 x 100
∑P0Q1

22. Dorbish and Bowley’s Method
o P01 = ∑P1Q0 + ∑P1Q1
∑P0Q0 ∑P0Q1 x 100
2

23. Marshall-Edgeworth’s Method
o P01 = ∑P1Q0 + ∑P1Q1 x 100
∑P0Q0 + ∑P0Q1

24. Fisher’s Method
o P01 =√ ∑P1Q0 x ∑P1Q1 x 100
∑P0Q0 ∑P0Q1

25. Kelly’s Method
o P01 = ∑P1Q x 100
∑P0Q
Here,
Q = Q0 + Q1
2

26. Weighted Average of Price Relative /
Family Budget Method
 P01 = ∑PV
∑V
 Here, P = Price Relatives
V = P0Q0

27. Components of Time Series
 Secular Trend
 Cyclical Variations
 Seasonal Variations
 Irregular or Random Variations

28. Methods of Measuring Trend
 Free Hand Curve Method
 Semi Average Method
 Moving Average Method
 Method of Least Square

29. Semi Average Method
Annual Change =
Difference of Two Semi Average Values
Difference of Years of Semi Average

30. Method of Least Square
o Equation for Time Series
Y = 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.