BUSINESS STATISTICS

UNIT II

Time Series Analysis: - A time series is a set of statistical observation arranged in

chronological order.

Utility of Time series analysis:-

(i) It gives a general description of past behavior.

(ii) It helps in forecasting the future behavior on the basis of past behavior.
(iii) It facilitates comparison.
(iv) It helps in the evaluation of current accomplishments.

Objectives of Time Series Analysis:-

(i) Understanding past behavior:- Through time series analysis of any phenomenon,
we can identify the nature that phenomenon represented by the sequence of
observation. We can understand the past behavior. Time series analysis of sales gives
us idea of sales in past few years. On the basis of which we can estimate sales in
successive years also. Time series analysis in order to understand the underlying
structures that produce the observations.
(ii) Seasonal Variations: - The component responsible for the regular rise and fall in the
magnitude of the time series is called seasonal Variation.
(iii) Cyclic Variations: - Refers to the long term oscillations about trend line.

Irregular trend: - Refer to such variations in a time series which do not repeat in a different
pattern.

(i) Chance Variation

(ii) Episodic variations.

By: Dr C.K.Dwivedi
Ph.D(UGC NET)
BUSINESS STATISTICS

Time series Model:-There are two types of model:-

Additive Model Multiplicative Model

Additive Model:- According to additive model the time series may be decomposed as:-

Ut=Tt + St + Ct + Rt

Where, Ut= original value at time t.

Tt = Trend value at time t.
St = Seasonal variations at time t.
Ct = Cyclic variation at time t.
Rt = Random fluctuations at time t.
Here we assume that the various components are additive.

Multiplicative Model: - According to multiplication model, we can assume that the various
components in a time series operate proportional to the general level of the series. Thus the
multiplicative model is:-

Ut=Tt X St X Ct X Rt

Estimation & forecasting: - Time series analysis helps in future prediction on the basis of post
behavior. Forecasting can be done in the following areas through time series analysis:-

(i) Financial and economic forecasting.

(ii) Sales forecasting and inventory control.
(iii) Statistical estimation of parameters of time series models.

Measurement of Trend:-

Moving Average Least square

Graphical Method Semi-Average
Method
method

Straight Line Parabolic

Trend Trend

By: Dr C.K.Dwivedi
Ph.D(UGC NET)
BUSINESS STATISTICS

(i) Free Hand Method (Graphical Method):- This is the simplest method of studying
trend. Steps for obtaining a straight line trend by this method are:-
(a) Plot the given time series on a graph paper.
(b) Examine the direction on the trend on the basis of plotted points.
(c) Draw a straight line which will best fit to the data according to personnel
judgment. This line gives the direction of trend.

Merits:-

(a) It is the simplest method of measuring trend

(b) It is very flexible
(c) Easy to calculate.

Demerits:-

(a) It is based on the personnel judgment of the investigator.

(b) Since it is highly subjective, it cannot be left to inexperienced and untrained people.
(c) It is a time consuming process.
(d) It cannot be used for future predictions as it is based on personal judgment.

Semi-Average Method: - The given data is divided into 2 groups preferably with same no. of yr.

Merits:-

(i) It is simple to understand then the other models such as moving, method of least
square.
(ii) It is an objective method. Everyone will get the same answer by this Method.

Demerits:-

(i) It always assumes the straight line relationship the plotted points regardless of the fact
whether that relationship exist or not.
(ii) Since it is based on the arithmetic mean, limitations of arithmetic mean shall
automatically apply.

Moving average Method: - This method is used when seasonal variations, cyclical variations
and irregular variations are involved.

The 3- yearly moving average shall be computed as follow:-

abc bcd cd e

, , 
3 3 3
By: Dr C.K.Dwivedi
Ph.D(UGC NET)
BUSINESS STATISTICS

The 5 yearly moving averages shall be computed as follow:-

abcd e bcd e f cd e f  g

, , 
5 5 5
The 7 yearly moving averages shall be computed as follow:-

abcd e f  g bcd e f  g h cd e f  g hi

, , 
7 7 7
Merits:-

(a) It is simple to understand and easy to compute

(b) It is a flexible method.
(c) It is effective if the trend of the series is irregular.
(d) Cyclic fluctuations are automatically eliminated if the period of moving average
coincides with the period of cyclic fluctuations.

Demerit:-

(a) Trend values cannot be computed for all the years.

(b) There is no fixed rule for selecting the period of moving average.
(c) This method cannot be used in forecasting as it does not involve any mathematical
function.

Least Square Method:- (i) Straight Line Method:-

Yc  a  bX
Normal Equations:-

 Y  na  b X
 XY  a X  b X
2

By: Dr C.K.Dwivedi
Ph.D(UGC NET)
BUSINESS STATISTICS

Merits:-

(i) It is a mathematical method of determining trend.

(ii) The straight line obtained by this method is called line of best fit.
(iii) Trend values can be determined for all the years which are not possible in moving
average method.
(iv) Once we obtain the line of best fit, we can determine the trend values for future years
also.

Demerits:-

(i) It is difficult to select the type of trend to be fitted.

(ii) It is a time consuming method.
(iii) Prediction can make only on long term variations.
(iv) It is not a flexible method.
(v) This method cannot be used to fit growth curves like logistic curves.

Parabolic Trend:-

Yc  a  bX  cX 2
Normal Equations:-

 Y  na  b X  c X 2

 XY  a X  b X  c X 3
2

X Y  a  X 2  b X  c  X 4
2 3

By: Dr C.K.Dwivedi
Ph.D(UGC NET)
BUSINESS STATISTICS

Index Number: - Index no. is used to measures the changes in some quantity which we cannot
observe directly.
OR
Index no. is a series of numbers by which changes in the magnitude of a phenomenon are
measured from time to time or form place to place.

Characteristics of Index No.:-

(i) Relative Measurement:- The basic characteristics of index no. is that it is a relative
or comparative measurement. The fact is that changes are measured by index no. and
changes can be measured only in relative or comparative term.
(ii) Specialized Average: - An average is a single figure which represents a group of
figures. Index no. is also an average because through this one figure the trend of data
is represented.
(iii) Measurement of changes not capable of Direct measurement:- Index no. is used in
measuring such mixed and complicated changes, which cannot be measured directly
such as price level, cost of living, changes in economic activities.
(iv) Measurement of common characteristic of a group of items:- Index no. expresses
the common characteristics of a group of items.
(v) Comparison on the basis of time or place:- The technique of index no. is used to
measure the relative changes either on the basis of time or on the basis of place.
(vi) Universal Use:- The technique of index no. was develop for the measurement of
changes in prices.

Limitations of Index no:-

(i) Based on Samples:- Because it is very difficult task to include all items in its
calculation. Thus the accuracy of the result will depend on the proper size and method
of sampling.
(ii) Indicator of Average trend:- Index no. indicates only average trend of changes.
Hence they should be interpreted keeping this limitation into consideration.
(iii) Limitation of construction:- There may be confusion on account of errors or lack of
precaution in the construction of index no. such errors may arise in the selection of
base year, determination of weight, use of average and formula etc.
(iv) Impact of Specific objectives
(v) Ignorance of change in qualitative facts

Weighted Index No. (i) Weighted Aggregative Index:-There are six types of weighted
aggregative method.

(1) Laspeyres Method:- In this method, quantities of base year are used as weights for
constructing price index.

P01  pq 1 0
X 100
p q 0 0
On the other hand, base year price is used as weight for constructing quantity index no.

By: Dr C.K.Dwivedi
Ph.D(UGC NET)
BUSINESS STATISTICS

Q01  p q 0 1
X 100
p q 0 0
Where:- P1= price in current year.
P0 = price in base year.
q0= Quantity in base year.
q1= quantity in current year.
P01= price index no.
q01= quantity index no.

(2) Paasche’s Method:- Current year quantity is used as weight for constructing price index
& vice- versa for quantity index.

P01  pq 1 1
X 100
p q 0 1

 
pq 1 1
Q01 X 100
pq 1 0

(3) Dorbish & Bowley’s Method:- This method involves the simple arithmetic mean of the
two indices- Laspeyres and Paasche’s.

LP
P01 
2
  p1q0 pq 
  1 1

p q p q 
P01  
0 0 0 1 
X 100
2

  p0 q1 pq 
  1 1

p q pq 
  X 100
0 0 1 0
Q01
2
(4) Fisher Ideal Index No.:- This index is obtained by taking the geometric mean of
Laspeyres and Paasche’s.

P01  LxP

By: Dr C.K.Dwivedi
Ph.D(UGC NET)
BUSINESS STATISTICS

  p1q0  p q  X 100
P01  
1 1
X
  p0 q0  p q  0 1

  p0 q1 pq 
Q01  
1 1
X  X 100
  p0 q0 pq 1 0

Formula is known as ideal because of the following reasons:-

(i) It involves the geometric mean which is theoretically considered to be the best
average for constructing index no.
(ii) Both current year prices and quantities are taken into account.
(iii) It satisfies both the time reversal test and factor reversal test.
(iv) It is free from bias.

Time Reversal Test:-

P01 X P10=1
LHS:-
pq 1 0
x
pq 1 1

P01 = p q 0 0 p q 0 1

p q 0 1
x
p q 0 0

P10 = pq 1 1 pq 1 0

pq 1 0
x
pq 1 1 p q 0 1
x
p q
0 0

P01 X P10 = 0 p q 0 p q0 1
x pq 1 1 pq1 0

=1 Ans.

(ii). Factor Reversal Test:- P01 X Q01 =

pq 1 1

p q 0 0

pq 1 0
x
pq 1 1
x
p q 0 1
x
pq = pq
1 1 1 1
Hence Proved
p q 0 0 p q0 1 p q 0 0 pq p q
1 0 0 0

(5) Marshall Edgeworth Method:- In this method both current year and base year prices
and quantities are taken into account.

P01 
 p (q 1 0  q1 )
X 100
 p (q 0 0  q1 )

By: Dr C.K.Dwivedi
Ph.D(UGC NET)
BUSINESS STATISTICS

Q01 
q ( p
1 0  p1 )
X 100
q ( p
0 0  p1 )

(6) Kelly’s Method: - Average of quantities of base year and current year is taken average
for constructing price index.

P01 
 p q X 100
1

p q 0

Q01 
 pq X 100 1

 pq 0

q1  q0 p1  p0
Where q  p
2 2

Fixed Base Index No.

Case 1: - Fixed Base (Single Year)

Let the years be 0,1,2,3,……….,k

With Prices P0, P1,P2………Pk

p1 p p3
Then
I 01  *100 , I 02  2 *100 , I 03  *100
p0 p0 p0
Case 2: -Fixed base (Average of several years)

Let the years be 1,2,3,……….,m

With Prices P1,P2………Pm Then

p1 p p
I 01  *100 , I 02  2 *100 , I 03  3 *100
p p p
Case 3 :- Chain Base:-

Let the years be 0,1,2,3,……….,k

With Prices P0, P1,P2………Pk respectively for a commodity. Then, the price relatives based on
previous year P01, P12, P23 …………………………P(K-1)K

By: Dr C.K.Dwivedi
Ph.D(UGC NET)
BUSINESS STATISTICS

P1[Current year price]

Link Re lative of Current Year  X 100
P0 [Pr evious Year Pr ice]

p1 p p3
P00  100 , P01  *100 , P12  2 *100 , P23  *100
p0 p1 p2
Conversion of chain Index to Fixed Index:-

(i) For the first, the value of the fixed base index will be taken same as that of chain base
index. If the index no. are to be constructed by taking first year as base, then the index
for the first year is taken as 100.

(ii) Current Year fixed base index

Current year chain base indexX Pr evious Year fixed Base index
=
100

I 00  I00  100 , I 01  I 01 , I 01 * I12 I 02 * I 23

I 02  , I 03 
100 100
Merits of chain base method:-

(i) Link relatives obtained by chain base method are concerned with making comparison
with previous period and not with any distant past.
(ii) This method enables the new items to be included and old ones to be deleted in order
to make the index more representative.
(iii) Chain index is used in constructing consumer price index and wholesale price index.
(iv) Index no. is free from seasonal variations.

Demerits chain base method:-

(i) It is complicated and is time consuming.

(ii) Link relatives are % of previous year figures.

Conversion of Fixed Index to chain Index:-

Fixed base index of the current year

 *100
Fixed base index of the previous year

I 01  100 I 02 I 03
I12  *100 I 23  *100
I 01 I 02

By: Dr C.K.Dwivedi
Ph.D(UGC NET)
BUSINESS STATISTICS

Cost of Living Index No.

Aggregate Expenditure Method (base on Laspeyres Method)

P01  pq 1 0
X 100
p q 0 0

Family Budget Method: -

Steps: -

P1
1. Calculate price relatives I  *100 it is called group Index No.
P0

2. Calculate V =p0q0 to be taken as Weight and Obtain V

3. Multiply I by corresponding V and Obtain  IV
4. Divide  IV by  V to find the required Index No.

P01 
 IV
V

By: Dr C.K.Dwivedi
Ph.D(UGC NET)