• Base of statistics
• Central limit theorem
• Testing of hypothesis
• Time series analysis
• Decision theory
• Probability and distribution
• Normal distribution
• Design of experiment
• Correlation and regression analysis
• Statistical quality control
• Index number
• Vital statistics

Types Of Index Number

Example:

Construct index numbers of price from the following data by applying:

1)Laspeyres method

2)Paasche method:

3)Bowley’s method

4) Fisher’s Ideal method

5) Marshall –Edgeworth method

1999 2000

Commodity Price Quantity Price Quantity

A 2 8 4 6

B 5 10 6 5

C 4 14 5 10

D 2 19 2 13

Solution:

 Commodity 1999 1999 2000 2000 Price(Po) Qty. (qo) Price(P1) Qty.(q1) P1q0 P0q0 P1q1 P0q1 A 2 8 4 6 32 16 24 12 B 5 10 6 5 60 50 30 25 C 4 14 5 10 70 56 50 40 D 2 19 2 13 38 38 26 26 SUM 200 160 130 103

1) Laspeyres Method: po1: *100

= 200/160*100 = 125.

2) Paasche’s Method: P01: *100

= 130/103 *100

= 126.21

3) Bowley’s method: P01:

= 2.512/2 *100

=125.6

4) Fisher’s ideal Method: p01:

= 125.6

5) Marshall-Edge worth Method :

= 125.47

Example:

A food product company is contemplating the introduction of a revolutionary new product with new packaging, or replace the existing product at a much higher price (S1), or a moderate change in the composition of the existing product with a new packaging at a small increase in price (S2) or a small change in the composition of the existing product except the word “New” with a negligible increase in price (S3). The three possible states of nature or events are: (1) high increase in sales (N1) , (2) no change in sales (N2) and (iii) decrease in sales (N3) . The marketing department of the company worked out the payoffs in terms of yearly net profits for each of the strategies of the three events ( expected sales). This is represented in the following table:

Strategies N1 N2 N3

S1 700000 300000 150000

S2 500000 450000 0

S3 300000 300000 300000

Which strategy should the concerned executive choose on the basis of

(a) Laplace criterion

Solution:

The payoff matrix is rewritten as follows:

(D) Laplace Criterion:

Since we do not know the probability of the states of nature, assume that they are equal. For this example, we would assume that each state of nature has a one third probability of occurance.

Thus,

Expected return

S1 = (700000+300000+150000)/3 = 383333.33

S2 = (500000+450000+0)/3 = 316666.66

S3 = (300000+300000+300000)/3 = 300000

Since the largest expected return is from strategy S1, the executive must select strategy S1.

Compute the weighted aggregative price index numbers for with as base year using (1) Laspeyre’s Index Number (2) Paashe’s Index Number (3) Fisher’s Ideal Index Number (4) Marshal Edgeworth Index Number.

 Commodity Prices Quantities 1980 1981 1980 1981 A 10 12 20 22 B 8 8 16 18 C 5 6 10 11 D 4 4 7 8

Solution:

 Commodity Prices Quantity p1q0 p0q0 p1q1 p0q1 1980 1981 1980 1981 P0 P1 P0 P1 A 10 12 20 22 240 200 264 220 B 8 8 16 18 128 128 144 144 C 5 6 10 11 60 50 66 55 D 4 4 7 8 28 28 32 32

Laspeyre’s Index Number

Paashe’s Index Number

Fisher’s Ideal Index Number

Marshal Edgeworth Index Number

The Chain Index Number:

Example:

Compute the chain index number with 1995 prices as base from the following table giving the average wholesale prices of the commodities A,B and C for the year 1996.

Average wholesale price(in Rs.)

Commodity 1995 1996 1997 1998 1999

A 20 16 28 35 21

B 25 30 24 36 45

C 20 25 30 24 30

Solution:

Computation of Chain Indices

Relatives based on preceding year

 Commodity 1995 1996 1997 1998 1999 A 100 16/20*100=80 28/16*100=175 35/28*100=125 21/35*100=60 B 100 30/25*100=120 24/30*100=80 36/24*100=150 45/36*100=125 C 100 25/20*100=125 30/25*100=120 24/30*100=80 30/24*100=125
 Total of link Relatives 300 325 375 355 310 Average of link relative 100 108.33 125 118.33 103.33 Chain Index(1995=100) 100 (108.33*100)/100=108.33 (125*108.33)*100=135.41 (118.33*135.41)/100=160.23 (103.33*160.23)/100=165.57

Exmple:

From the data below:

(A) What was the real average weekly wage for each year?

(B) In which year did the employees have the greatest buying power?

(C) What percentage increase in the weekly wages for the year 1998 is required to provide the same buying power that the employees enjoyed in the year in which they had the highest real wages?

solution:

( 1) Real average weekly wage can be obtained by the following formula:

Real wage: (monthly wage/ Price index )*100

CALCULATION OF REAL WAGES

 Year Weekly Take home pay Consumer Price Index Real Wages 1993 109.50 112.8 109.5/112.8*100=97.07 1994 112.20 118.2 112.2/118.2*100=94.92 1995 116.40 127.4 116.4/127.4*100=91.37 1996 125.08 138.2 125.08/138.2*100=90.51 1997 135.40 143.5 135.4/143.5*100=94.36 1998 138.10 149.8 138.10/149.8*100=92.19

(b) Since real wage was maximum in the year 1993, the employees had the greatest buying power in the year.

(c) Percentage increase in the weekly wages for the year 1998 to provide the same buying power that the employees enjoyed in 1993.

Absolute difference = 97.07-92.19 = 6.88

Example:

The prices of 3 commodities for the 5 years are as follows:

 Commodity Prices (per kg) 2001 2002 2003 2004 2005 Rice 20 20 22 25 28 Sugar 11 12 14 27 30 Tea 178 176 174 180 180

Required:

Construct price index numbers using average of relatives’ method, taking 2001 as base year.

Solution:

 Commodity Prices (per kg) 2001 2002 2003 2004 2005 Rice Sugar Tea Total 300 307.97 335.02 471.52 513.85 Mean(Index) 100 102.66 111.67 157.17 171.28

(c) Weighted Index Numbers: This type of index can be further classified into two categories:

(i) Weighted Aggregative Index Numbers: In these index numbers, the quantities produced, sold or bought or consumed during the base year or current year are used as weights. These weights indicate the importance of the particular commodity. Some well-known weighted index numbers are given below:*

(1) Lespeyre’s Index: This index uses base year quantities as weights. For this reason, it is also known as ‘Base Year Weighted Index’:

Here W = Qo

(2) Paasche’s Index: This index uses current years quantity as weights. For this reason, it is known as ‘Current Year Weighted Index’:

Here W = Qn

(3) Fisher’s Ideal Index: This index number is the GM of the Lespeyre’s and Paasche’s index numbers. It is called ‘ideal’ because it satisfies two tests (Time Reversal and Factor Reversal Tests):

(4) Marshall-Edgeworth’s Index: This index number uses the average of the base year and current quantities as weights:

Example:

 Commodities 2001 2005 Price (Rs. / kg) Qty. (kgs) Price (Rs. / kg) Qty. (kgs) Rice 20 100 28 160 Sugar 11 18 30 37 Salt 1 1 5 1.1 Milk 18 57 32 149

Required:

Construct the following price index numbers using 2001 as base year:

(a) Lespeyre’s

(b) Paasche’s

(c) Fisher’s

(d) Marshall-Edgeworth’s

Solution:

 2001 2005 PoQo PnQo PnQn PoQn Qo+Qn Po(Qo+Qn) Pn(Qo+Qn) Po Qo Pn Qn Rice 20 100 28 160 2000 2800 4480 3200 260 5200 7280 Sugar 11 18 30 37 198 540 1110 407 55 605 1650 Salt 1 1 5 1.1 1 5 5.5 1.1 2.1 2.1 10.5 Milk 18 57 32 149 1026 1824 4768 2682 206 3708 6592 Total 3225 5169 10363.5 6290.1 9515.1 15532.5

(a) Lespeyre’s:

(b) Paasche’s:

(c) Fisher’s:

(d) Marshall-Edgeworth’s:

(ii) Weighted Average of Relatives: The formula of weighted average of relatives is:

or

(Arithmetic Mean taken as average); where

or

(Geometric Mean taken as average)

The total value of the commodity is used as weights. If the base year value (PoQo) is used as base, then the formula becomes:

or

If the current year value (PnQn) is used as base, then the formula becomes:

Example
The index of prices for transport is 100 compared to a given base year. The index for housing is 160 (compared to the same base year). Calculate the composite index for transport and housing by using a weighting of 65 for transport and 35 for housing.
Solution:

Composite index = (Total of index * weighting) / Total weighting

Composite index = (100 * 65 + 160 * 35) / (65 + 35) = 12,100 / 100 = 121

Example
The following data are observed:
Beverage: Weight (120); Price index (210)
Clothing: Weight (100); Price index (200)
Food: Weight (80); Price index (190)
Then,
Composite index = (120 * 210 + 100 * 200 + 80 * 190) / (120 + 100 + 80) = 60,400 / 300 = 201.3

* W.A.I.N. is equal to