Index No in Statistics

1. INDEX
NUMBERS
PRESENTED BY:
Mansi Doomra (130550034)
Mehak Puri (130550035)
Nidhi Aggarwal (130550039)
Nikhil Khera (130550040)
Shashank Kapoor (130550052)
Vineet Kumar (130550062)

2.  An
index number measures the relative
change in price, quantity, value, or some
other items of interest from one time
period to another.
A
simple index number measures the
relative change in one or more than one
variable.

3. 
Index Numbers are a specialized type of averages.
- M. Blair

Index Numbers are devices for measuring differences in
the magnitude of a group of related values.
- Croxten and Cowden

An index number is a statistical measure designed to
show changes in a variable or group of related variables
with respect to time, geographical location or other
characteristics.
-Spigel

5. HELPFUL IN
PREDICTIONS
•Index numbers give the knowledge as to what
changes have occurred in the past.
HELPFUL IN
COMPARISONS
•By Index numbers relative changes occurring in
the variables are determined, which simplifies
the comparison of Data.
USEFUL IN
BUSINESS
•Index numbers measures the changes taking
place in the Business World and are useful in
making a
comparative study of the changes.

6. Choice of the base period.
Choice of an average.
Purpose of index numbers.
Selection of commodities.
Data collection.

7. Simple Average of
Price Relatives
Numbers
Index
Un-weighted
Simple
Aggregative
Weighted
Weighted
Aggregative
Weighted Average of
Price Relatives

8. In this method, sum of current year’s prices is divided
by sum of base year’s prices and the quotient is
multiplied by 100. Its formula is:
P01
p1
p0
100
Where,
P01= Index number of the current year.
p1 = Total of the current year’s price of all commodities.
p0 = Total of the base year’s price of all commodities.

9. PRICE (Rs)
2008
COMMODITIES
UNITS
PRICE (Rs)
2007
Sugar
Quintal
2200
3200
Milk
Quintal
18
20
68
71
Oil
Litre
Wheat
Quintal
900
1000
Clothing
Meter
50
60

10. PRICE (Rs)
2008
COMMODITIES
UNITS
PRICE (Rs)
2007
Sugar
Quintal
2200
3200
Milk
Quintal
18
20
Oil
Litre
68
71
Wheat
Quintal
900
1000
Clothing
Meter
50
60
p0
3236
p1
4351
Index Number for 2008:
P01
p1
p0
100
4351
100 34.45
3236
It means the prize in 2008 were 34.45% higher than the previous year.

11. In it, initially the price relatives of all the commodities are
found out. To calculate price relatives, price of current year
(p1) is divided by price of base year (p0) and then, the
quotient is multiplied with 100.
1.
When ARITHMETIC MEAN is used:
P01
2.
p1
100
p0
N
Where N is Numbers Of items.
When GEOMETRIC MEAN is used:
P01
Anti log
p1
log
100
p0
N

12. 3.
When MEDIAN is used:
P01
N 1
Size of
2

13. From the data given below construct the index number
for the year 2008 taking 2007 as base year by using
arithmetic mean.
Commodities
Price (2007)
Price (2008)
P
6
10
Q
2
2
R
4
6
S
10
12
T
8
12

14. Index number using arithmetic mean
Price (2007)
P
6
10
166.7
Q
12
2
16.67
R
4
6
150.0
S
10
12
120.0
T
8
12
150.0
p0
Price (2008)
Price Relative
Commodities
p1
p1
p0
100
p1
100 = 603.37
p0
P01
p1
100
p0
N
603.37
120.63
5

15. When index numbers is constructed taking into
consideration
the
importance
of
different
commodities, then they are called weighted index
numbers.
There are two methods of contructing weighted index
numbers.
1.
Weighted Aggregative Index Numbers.
2.
Weighted Average of Price
Relative Methods.

16. In it, commodities are assigned weights on the basis of the
quantities purchased. Different statisticians have used
different methods of assigning weights, which are as
follows:






Laspeyre’s method.
Paasche’s method.
Fisher’s ideal method.
Dorbish and Bowley method.
Marshall-Edgeworth’s method.
Kelly’s method.

17. This method was devised by Laspeyres in 1871. In this
method the weights are determined by quantities in the
base.
p01
p1q0
p0 q0
100
Paasche’s Method:
This method was devised by a German statistician Paasche
in 1874. The weights of current year are used as base year
in constructing the Paasche’s Index number.
p01
p1q1
p0 q1
100

18. This method is a combination of Laspeyre’s and
Paasche’s methods. If we find out the arithmetic
average of Laspeyre’s and Paasche’s index we get the
index suggested by Dorbish & Bowley.
p01
p1q0
p0 q0
p1q1
p0 q1
2
100
Fisher’s Ideal Method:
Fisher’s ideal index number is the geometric mean of
the Laspeyre’s and Paasche’s index numbers.
P01
p1q0
p1q1
p0 q0
p0 q1
100

19. In this index the numerator consists of an aggregate of
the current years price multiplied by the weights of
both the base year as well as the current year.
p01
p1q0
p1q1
p0 q0
p0 q1
100
Kelly’s Method:
Kelly thinks that a ratio of aggregates with selected
weights (not necessarily of base year or current year)
gives the base index number.
p1q
p01
100
p0 q
Where q refers to the quantities of the year which is selected as
the base. It may be any year, either base year or current year.

20. Given below are the price quantity data,with price
quoted in Rs. per kg and production in qtls.
Find:
(1) Laspeyre’s Index (2) Paasche’s Index (3)Fisher
Ideal Index.
2002
2007
ITEMS
PRICE
PRODUCTION
PRICE
PRODUCTION
BEEF
15
500
20
600
MUTTON
18
590
23
640
CHICKEN
22
450
24
500

21. ITEMS
PRICE
PRODUCT
ION
PRICE
PRODUC
TION
p1q0
p0 q0
7500
p1q1
p0 q1
p0
q0
p1
q1
BEEF
15
500
20
600
10000
MUTTON
18
590
23
640
13570 10620 14720 11520
CHICKEN
22
450
24
500
10800
TOTAL
9900
12000
9000
12000 11000
34370 28020 38720 31520

22. 1.Laspeyre’s index:
p01
p1q0
p0 q0
100
34370
100 122.66
28020
100
38720
100 122.84
31520
2. Paasche’s Index:
p01
p1q1
p0 q1
3. Fisher Ideal Index:
P01
p1q0
p1q1
p0 q0
p0 q1
100
34370
28020
38720
31520
100
122 .69

23. In weighted Average of Price relative, the price relatives for
the current year are calculated on the basis of the base year
price. These price relatives are multiplied by the respective
weight of items. These products are added up and divided by
the sum of weights.
Weighted arithmetic mean of price relative is given by:
P01
Where:
P
PV
V
P
1
100
P0
P=Price relative
V=Value weights =
p 0 q0

24. Quantity index numbers are designed to measure the change in
physical quantity of goods over a given period. These index
numbers represents increase or decrease in physical quantities of
goods produce or sold. The method of construction of quantity
index is same as that of price index.
(1) Simple quantity index numbers
(a) Simple Aggregative Method:
Q01
q1
q0
100

25. (b) Simple Average of Relative Method:
(i) Using A.M.
Q01
q1
100
q0
N
(ii) Using G.M.
log
Q01
Anti log
q1
100
q0
N

26. (2) Weighted quantity index numbers
I. Weighted Aggregate Method
(a) Laspeyre’s quantity index no.:
Q01
q1 p0
q0 p0
100
(b) Paasche’s quantity index no.:
Q01
q1 p1
q0 p1
100

27. (c) Fisher’s quantity index numbers:
Q01
q1 p0
q1 p1
q0 p0
q0 p1
100
(d) Bowley’s quantity index:
Q01
q1 p0
q0 p0
q1 p1
q0 p1
2
100
(e) Marshall’s quantity index:
Q01
q1 p0
q1 p1
q0 p0
q0 p1
100

28. II. Weighted Average of Relative Method:
QW
Q01
Where, Q
W
q1
100
q0
and
W
q0
p0

29. Value is the product of price and quantity. A simple
ratio is equal to the value of the current year divided
by the value of base year. If the ratio is multiplied
by 100 we get the value index number.
V
p1q1
p0 q0
100

30. Various formulae can be used for the construction of index
numbers but it is necessary to select an appropriate/suitable
formula out of them. Prof. Fisher has given the following tests
to select an appropriate formula:


TIME REVERSAL TEST (TRT)
FACTOR REVERSAL TEST (FRT)

31. According to this test, if considering any year as a
base year, some other year’s price index is computed
and for another price index, time subscripts are
reversed, then the both price indicies must be
reciprocal to each other.
TRT is satisfied when:
P01
1
or P01 P
10
P
10
1
Where, P01 is price index for the year 1 with 0 as base and P10
is the price index for the year 0 with 1 as base.

32. Time reversal test permits interchange of price and
quantities without giving inconsistent results, i.e. the
two results multiplied together should give the true
value ratio:
FRT is satisfied when:
Price Index x Quantity Index = Value Index
OR
P01 Q01
p1q1
p0 q0