# Dibrugarh University - Business statistics 2009 (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c)       It may also be collected from unpublished form.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.

Regression line
A line of regression by the method of “least square” shows an average relationship between variables under study. This regression line can be drawn graphically or derived algebraically. A line fitted by method of least square is known as the line of best fit. There are two regression lines:-
Regression line of x on y: - Regression line of x on y is used to predict x for a given value of y. The regression equation of x on y is x=a+by.
Regression line of y on x: - Regression line of y on x is used to predict y for a given value of x. The regression equation of y on x is y=a+bx

Two regression lines:
There are two lines of regression: - x on y and y on x. For these lines, the sum of the square of the deviations between the given values and their corresponding estimated values obtained from the line is least as compared to other line. One regression line cannot minimise the sum of squares for both the variables that is why we are getting two regression lines. (We get one regression line when r = +1 and Two regression lines will be at right angles when r = 0.)

Coefficient of Correlation
To measure the degree of association or relationship between two variables quantitatively, an index of relationship is used and is termed as co-efficient of correlation.
Co-efficient of correlation is a single number that tells us to what extent the two variables are related and to what extent the variations in one variable changes with the variations in the other.
The co-efficient of correlation is always symbolized either by r or p (Rho). The notion 'r' is known as product moment correlation co-efficient or Karl Pearson's Coefficient of Correlation. The symbdl 'P' (Rho) is known as Rank Difference Correlation Coefficient or Spearman's Rank Correlation Coefficient.
Range of Coefficient of Correlation
The measurement of correlation between two variables results in a maximum value that ranges from -1 to +I, through zero.

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.

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.

Components of time series
1. Secular trend: - Secular trend is a long term trend which has the basic tendency to grow or decline over a period of time. It may be due to population change technological progress, large scale shifts in consumer tastes, discovery of new things, etc.
2. Seasonal variation: - Seasonal variations are those periodic movements in business activity, which occur regularly every year and have their origin in the nature of the year itself. It may be due to climate weather conditions, customs, traditions and habits, festivals, etc.
3. Cyclical variation: - The term cycle refers to the recurrent variations in time series that usually last longer than a year and are regular neither in amplitude nor in length. Cyclical fluctuations are long-term movements that represent consistently recurring rises and declines in activity. It has four important characteristics:
i) Prosperity
ii) Decline
iii) Depression
iv) Improvement

4. Irregular variation or irratic movement: - It is the variation in business activities, which do not repeat in a definite pattern. Floods, earthquakes, strikes and wars cause it.

Models of time series
In Traditional time series analysis, it is ordinarily assumed that there is a multiplicative relationship between the components of time series.
Symbolically, Y=T X S X C X I
Where T= Trend
S= Seasonal component
C= Cyclical component
I= Irregular component
Y= Result of four components.
Another approach is to treat each observation of a time series as the sum of these four components Symbolically, Y=T + S+ C + I

Determining the probability of two events occurring, use the additive law of probability. For example if asked "What is the probability of drawing a 4 or a 7 from a shuffled deck of cards?", Eight different cards would satisfy this condition, 4 (4s) or 4 (7s). So the probability of this event is 8/52. The equation below properly states the additive law of probability.  p(A or B)= p(A) + p(B) - p(AB)
Multiplicative Law of Probability
The multiplicative law of probability is used to calculate the probability that two events will occur in sequence. Remember that the additive law was used to find the probability that one of two different events would occur on a single trial. These two probabilities have key words to look for. In the additive law, you will always see something like the following: Find the probability that event (A) or event (B) will occur. The word or is key. With the multiplicative law, the word and is key. Here is a typical problem requiring the multiplicative law: What is the probability of drawing a 4 and then a 7 from a shuffled deck of cards.
The multiplicative law states that the probability of the occurrence of event A and then event B, and then event C ... and so on is the product of the separate simple probabilities of those events. The following equation expresses the multiplicative law of probability's formula.
P(A and B and C) = P(A). P(B). P(C)

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.

(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.

Sales Forecasting
In general terms, forecasting means “A statement made about the future”. So, Sales forecasting is the estimation of sales made for the future. Sales forecast is an estimate of sales in rupees or in units for future period. A sales forecast is the prediction of sales volume that a company can estimate to achieve in specified period of time in future.
According to American marketing Association, “Sales forecast is an estimate of sales in dollars (Rupees in India) or physical units for a specified future period”.
Sales forecasting is the basis of all business activities. All the business activities may it a sales related matter, production related matter, finance, advertising, etc depend on sales forecasting. Any business firm starts its plan with sales forecasting .Sales forecasting is a self-assessment tool for a company. The future direction of the company depends on sales forecasting.  The importance of Sales forecasting can be stated as follows:
a)      Overstocking and under stocking of materials can be maintained by a good inventory control.

b)      With the help of sales forecasting, sales opportunities can be found out on the basis of the forecast.

c)       All the activities in an organization are controlled on the basis of forecasting.

d)      Advertising and sales promotion expenses are based on sales forecasting.

e)      Sales forecasting is also important in the field of personnel department. The number of sales persons, executives etc can be increased or decreased on the basis of sales forecasting.

Demand Forecasting
Forecasting is the process to predict how the future needs, which include the requirement in the size of the quantity, quality, time and location that is required in order to meet the demand for goods or services.
Demand Forecasting is the demand for products that are expected to be realized for a certain period in the future. Inforecasting we must pay attention to forecasting procedures should be implemented, namely :
a)      Determining the purpose of forecasting.

b)      Choose the item “independent demand” to be predicted. Plot data intoscatter diagram.

c)       Selecting “forecasting method” in accordance with the pattern of data for its intended purpose.

d)      Counting errors are to be performance of each method used, can be known.

e)      The selection of the best method, which has the smallest error rate.

f)       Make predictions of future demand, then perform the test verifies that the forecasting results carried out a representative to the past data