# Dibrugarh University - Business Statistics 2012 (Solved)

Answer of Question no. 1 (a).
Dispersion and its characterictics
Dispersion: Dispersion is the measure of variation of items. It measures the extent to which the items vary from central value. Dispersion is also known as average of the second order.
In the words of Brooks and Dick,” Dispersion is the degree of the scatter or variation of the variable about a central value.”
In the words Simpson and Kafka,” The measurement of the Scatterness of the mass of figures in a series about an average is called measure of variation or dispersion.”
Dispersion includes range, mean deviation, quartile deviation, and standard deviation.

The following are the important properties which a good measure of dispersion should satisfy:
a)      It should be simple to understand and easy to compute.
b)      It should be simple to compute.
c)       It should be based on all the items.
d)      It should not be affected by extreme values.
e)      It should be rigidly defined.
f)       It should be capable of further algebraic treatment.
g)      It should have sampling stability.

Answer of Question no. 1 (d).
Measure of Central tendency and Its Properties
In the words of Croxton and Cowden, “An average value is a single value within the range of the data that is used to represent all the values in the series.”
The value of average lies between the maximum and minimum values of the series. That is why it is also called measure of central tendency.

The following are the important properties which a good average should satisfy
a)      It should be easy to understand.
b)      It should be simple to compute.
c)       It should be based on all the items.
d)      It should not be affected by extreme values.
e)      It should be rigidly defined.
f)       It should be capable of further algebraic treatment.

Sample: If a part is selected out of the universe then the selected part or portion is known as sample. Sample is only a part of the universe.
Sample survey: It is a survey under which only a part taken out of the universe is investigated. It is not essential to investigate every individual item of the Universe.

a)      It is cheaper as compared to census survey
b)      While using secondary data, time and labour are saved.
c)       It may also be collected from unpublished form.
d)      If secondary Data are available, they are much quicker to obtain than primary data.

a)      Degree of accuracy may not be acceptable.
b)      Secondary Data may or may not fit the need of the project.
c)       Data may be influenced by personal bias of investigator.

Properties of the Correlation Coefficient
a)      The correlation coefficient is symmetrical with respect to X and Y i.e. rXY=rYX
b)      The Correlation coefficient is a pure number and it does not depend upon the units in which the variables are measure.
c)       The correlation coefficient is the geometric mean of the two regression coefficients. Thus if the two regression lines of Y on X and X on Y are written as Y=a+bx and X=c+dy respectively then bd=r2.
d)      The correlation coefficient is independent of the choice of origin and scale of measurement of the variables, i.e. r remains unchanged if constants are added to or subtracted from the variables and if the variables having same size are multiplied or divided by the class interval size.
e)      The correlation coefficient lies between -1 and +1, symbolically -1≤r≤1.

Plz follow Nag and Chanda book on statistics. This topic is nicely explained

Fisher’s index is regarded as ideal index because:-
a)      It considers both base year and current year’s price and quantity.

b)      It satisfies both time reversal and factor reversal test.

c)       It is based on Geometric mean which is theoretically considered to be the best average of constructing index number.

d)      It is free from bias as it considers both current year and base year price and qty.

Difference between Price index and Quantity index
A measure reflecting the average of the proportionate changes in the prices of a specified set of goods and services between two periods of time. Usually a price index is assigned a value of 100 in some selected base period and the values of the index for other periods are intended to indicate the average percentage change in prices compared with the base period. A quantity index is built up from information on prices of various commodities.
A measure reflecting the average of the proportionate changes in the quantities of a specified set of goods and services between two periods of time. Usually a quantity index is assigned a value of 100 in some selected base period and the values of the index for other periods are intended to indicate the average percentage change in quantities compared with the base period. A quantity index is built up from information on quantities such as the number or total weight of goods or the number of services.

Limitation of index number
a)      Not completely true: - Index number not fully true. The index number simply indicate arithmetical tendency of the temporal changes in the variable.

b)      International comparison is not possible: - Different countries have different bass of index numbers; these do not help international comparisons.

c)       Difference of time: - With the passage of time, it is difficult to make comparison of index number. With the changing time man’s habits.

d)      Limited use: - Index numbers are prepared with certain specific objective. If they are used for another purpose they may lead to wrong conclusion.

e)      Lack of retail price index number: - Most of the index numbers are prepared on the basis of wholesaler prices. But in real life, retail prices are most relevant, but it is difficult to collect retail prices.

Index number is an indicator of changes in prices and quantities. It is a specialized average designed to measure the change in a group of related variables over a period of time. It is also an indicator of inflationary or deflationary tendencies.

Uses of Index number in Trade and commerce
a)      Measurement of change in the price level or the value of money: - Index number can be used to know the impact of the change in the value of money on different sections of the society.

b)      Knowledge of the change in standard of living: - Index number helps to ascertain the living standards of people. Money income may increase but if index number show a decrease in the value if money. Living standard may even decline.

c)       Adjustment in salaries and allowances: - Cost of living index number is a useful guide to the government and private enterprises to make necessary adjustment in salaries and allowances of the workers.

d)      Useful to business community: - Price index numbers serve as a useful guide to the business community in planning.

e)      Information regarding foreign trade: - Index of exports and imports provides useful information regarding foreign trade.

Utility of Time Series Analysis
The analysis of Time Series is of great significance not only to the economist and businessman but also to the scientist, geologist, biologist, research worker, etc., for the reasons given below:
a)      It helps in understanding past behaviors: By observing data over a period of time one can easily understanding what changes have taken place in the past, Such analysis will be extremely helpful in producing future behavior.

b)      It helps in planning future operations: Plans for the future cannot be made without forecasting events and relationship they will have. Statistical techniques have been evolved which enable time series to be analyzed in such a way that the influences which have determined the form of that series to be analyzed in such a way that the influences which have determined the form of that series may be ascertained.

c)       It helps in evaluating current accomplishments: The performance can be compared with the expected performance and the cause of variation analyzed. For example, if expected sale for 1995 was 10,000 refrigerators and the actual sale was only 9,000, one can investigate the cause for the shortfall in achievement. Time series analysis will enable us to apply the scientific procedure for such analysis.

d)      It facilitates comparison: Different time series are often compared and important conclusions drawn there from. However, one should not be led to believe that by time series analysis one can foretell with 100percnet accuracy the course of future events.

(i) Method of Least Square
This method is most commonly used method of measuring trend. It is a mathematical method and a trend line is fitted to the data in such a manner that the following two conditions are satisfied:-
i) the sum of deviation of the actual values from their respective mean is zero.
ii) the sum of square of the deviations of the actual and compute values is least from this line. That is why this method is called method of least square. The straight line trend is represented by the equation:
Y = a + bx
Where, y = denotes the trend values
a = represents the intercept on y axis.
b= represents slope of the trend line.
Merits:
i) This is a mathematical method of measuring trend.
ii) Trend values can be obtained for all the given time periods in the series.
Demerits:
i) This method is more tedious and time consuming.
ii) This method cannot be used to fit the growth curves.

(ii) Simple-average method
Under this method, the given data is divided into two parts. After that an average of each part is obtained which gives two points. Each point is plotted at the mid-point of the class interval covered by the respective part and then the two points are joined by a straight line which gives the required trend line.
Merits:
i) This method is simple to understand as compared to the moving average method and the method of least square.
ii) This is an objective method of measuring trend as everyone who applies this method gets the same result.
Demerits:
i) It is affected by extreme values.
ii) This method assumes straight relationship between the plotted points whether this exist or not.

(a) Future Prediction: Forecasting is a part of human conduct. Businessmen also need to look to the future. Success in business depends on correct predictions. In fact when a man enters business, he automatically takes with it the responsibility for attempting to forecast the future.
To a very large extent, his success or failure would depend upon the ability to successfully forecast the future course of events. Without some element of continuity between past, present and future, there would be little possibility of successful prediction. But history is not likely to repeat itself and we would hardly expect economic conditions next year or over the next ten years to follow a clear cut prediction. Yet, frequently past patterns prevail sufficiently to justify using the past as a basis for predicting the future.
(b) Reducing Uncertainty:  A businessman cannot afford to base his decisions on guesses. Forecasting helps a businessman in reducing the areas of uncertainty that surround management decision making with respect to costs, sales, production, profits, capital investment, pricing, expansion of production, extension of credit, development of markets, increase of inventories and curtailment of loans. These decisions cannot be made off-hand. They are to be based on present indications of future conditions.
(c) Measurement of Risk: However, we should know that it is impossible to forecast the future precisely. There is a possibility of occurrence of some range of error in the forecast. Statistical forecasts are the methods in which we can use the mathematical theory of probability to measure the risks of errors in predictions.

Steps in forecasting
The forecasting of business fluctuations consists of the following steps:
a)      Understanding why changes in the past have occurred: One of the basic principles of statistical forecasting is that the forecaster should use the data on past performance. The current rate and changes in the rate constitute the basis of forecasting. Once they are known, various mathematical techniques can develop projections from them. If an attempt is made to forecast business fluctuations without understanding why past changes have taken place, the forecast will be purely mechanical.

b)      Determining which phases of business activity must be measured: After understanding the reasons of occurrence of business fluctuations, it is necessary to measure certain phases of business activity in order to predict what changes will probably follow the present level of activity.

c)       Selecting and compiling data to be used as measuring devices: There is an independent relationship between the selection of statistical data and determination of why business fluctuations occur. Statistical data cannot be collected and analysed in an intelligent manner unless there is a sufficient understanding of business fluctuations. It is important that reasons for business fluctuations be stated in such a manner that is possible to secure data that are related to the reasons.

d)      Analysing the data: Lastly, the data are analysed in the light of understanding of the reason why change occurs. For example, if it is reasoned that a certain combination of forces will result in a given change, the statistical part of the problem is to measure these forces, from the data available, to draw conclusions on the future course of action. The methods of drawing conclusions may be called forecasting techniques.

Time series analysis
Time series analysis is also used for the purpose of making business forecasting. The forecasting through time series analysis is possible only when the business data of various years are available which reflects a definite trend and seasonal variation. By time series analysis the long term trend, secular trend, seasonal and cyclical variations are ascertained, analysed and separated from the data of various years.
Merits and demerits of time series analysis
 Merits Demerits It is an easy method of forecasting. This method is expensive, difficult and time taking. By this method a comparative study of variations can be made. This method deals with past data only. Reliable results of forecasting are obtained as this method is based on mathematical model. This method can only be used when the data for several years are available.

Methods of Business Forecasting: The following are the main methods of business forecasting.
2.          Time series analysis
3.          Extrapolation
4.          Regression analysis
5.          Modern econometric methods
6.          Exponential smoothing method

Business barometers:  Business indices are constructed to study and analyse the business activities on the basis of which future conditions are predetermined. As business indices are the indicators of future conditions, so they are also known as “business barometers” or ‘economic barometers’. With the help of these business barometers the trend of fluctuations in business conditions are made known and by forecasting a decision can be taken relating to the problem.
The construction of business barometer consists of gross national product, wholesale prices, consumer prices, industrial production, stock prices, and bank deposits. These quantities may be converted into relatives on a certain base. The relatives so obtained may be weighted and their average is computed. The index thus arrived at in the business barometer.
Time series analysis: Time series analysis is also used for the purpose of making business forecasting. The forecasting through time series analysis is possible only when the business data of various years are available which reflects a definite trend and seasonal variation. By time series analysis the long term trend, secular trend, seasonal and cyclical variations are ascertained, analysed and separated from the data of various years.
Extrapolation: Extrapolation is the simplest method of business forecasting. By extrapolation, a businessman finds out the possible trend of demand of his goods and also about the future price trends. The accuracy of extrapolation depends on two factors:
i) Knowledge about the fluctuations of the figures
ii) Knowledge about the course of events relating to the problem under consideration
Thus, there are two assumptions on which extrapolations are based:
i) There is no sudden jump in figures from one period to another
ii) There is regularity in fluctuations and the rise and fall is uniform
In extrapolation, we assume that the variable will follow the established pattern of growth. For the purpose of business forecasting, it is to determine accurately the appropriate trend curve and the values of its parameters.