Returns to a factor It refers to the behaviour of output when only one variable factor of production is increased in short-run and fixed factors remain constant.
Law of diminishing returns to a factor It refers to a situation in which total output increases at a diminishing rate when more and more variable factor is combined with the fixed factor of production. In this situation, Marginal Product of the variable factor must be diminishing.

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The law of variable proportion states that as we increase the quantity of only one input, keeping other inputs fixed, the total product increases at an increasing rate (convex shape) in the beginning, then increases at diminishing rate (concave shape) and after a level of output ultimately falls.

2. Assumptions of Law of Variable Proportions
(a) Only one input is variable, the other is held constant or fixed.
(b) It is possible to change the proportion in which the factor units are combined.
(c) It assumes a short run.
(d) The state of technology is given and remains unchanged.
(e) Price of factors of production do not change.
3. It can be explained with the help of schedule and diagram:

The money income which a producer gets from the sale of his product is known as revenue of the firm. It is also called sale proceeds of the firm.
The concept of revenue should not be confused with the concept of profit. Profit of a firm is estimated as the difference between revenue and cost related to the production of a commodity (Profit = Revenue – Cost).

Revenue has three aspects:

• Total Revenue
• Average Revenue
• Marginal Revenue

Exclusive method:

 This method is used for those series in which the upper limit of one class becomes the lower limit of the next class interval.Example 0-5, 5-10, 10-20, 20-30. 

Inclusive method:

Under this method, the classes are formed in such a manner that the upper limit of a class interval does not repeat itself as the lower limit of the next class interval.Example 0-5, 6-10, 11-15

A n s w e r

The main method of collecting primary data are as mentioned below:

(i) Direct personal investigation.

(ii) Indirect investigation

(iii) Information through mailed Questionnaire.

(iv) Questionnaire through enumerators.

Following are the remedies to remove distrust statistics: 

• Need of caution. 
• Statistical limitations should be taken into consideration. 
• Self-restraint.
• That is self-control in statistical fallacies. 
• Statistics must be used by experts. 
• Analytical study of data before its use.

the method of studying statistics
A n s w e r
The important statistical methods are as follows:
1. Collection,
2.  summarisation, 
3. presentation, 
4. analysis and 
5. interpretation of data.
 

Data is a tool which helps in reaching a sound conclusion by providing information therefore. For statistical investigation, collection of data is the first and foremost.

We have unlimited wants and limited resources. For example, If we have more pocket money to spend then we can get all the things according to our wish but if we have less pocket money then we have to choose only those things that we want the most. For example, there are two goods in the economy; wheat and shoes. Then we have to choose the good that we want the most because of limited resources and unlimited wants. Scarcity is the root of all economic problems.

In statistics universe (population) refers to an aggregate of all items about which we want to obtain information. Suppose there are 1500 studetns in a particular school. If an investigation relates to all the 1500 students, then 1500 would be taken as universe or population.

On the other hand, if we take out only 150 for investigation, then 150 students will be taken as sample. Sample is only the part of the population or the universe. This part must represent the characteristics of universe.

There is no hard and fast rule about how many class we choose; but as a working rule the number of classes should lie between 5 and 15. It should be noted that the number of classes will be large if we choose small size class intervals and it will be small if the size of class intervals is large.

As an illustration, suppose the range is 70, and we choose classes of width 2 each. We would require 70 ÷ 2 = 35 classes. However, the number of classes would be 14 if the width of each class was 5.

Size of Class Intervals : We may choose all classes of the same width or of different width. In the case of equal class intervals the size of the class interval is determined as soon as we have decided about the number of classes.

Suppose n is the number of classes and all classes are of width h, then n × h = R.

Knowing the range R and number of classes

n we can abtain h = R/n as the width of class interval. If the range is 70 and we choose 10 classes, the width is 7.

Choice of Class Limits : Suppose x is a continuous variable, such that it can take any value in a given range. In that case, it is possible to choose class limits which are not equal to any of the observed values, For example, height of individuals is a continuous variable, even though, in practice, one can measure height to the nearest of the unit value (in centimetres) as 165, 170, 169, 171 .........; or to the nearest of tenth place of decimals as 165.3, 170.4, 168.9, 170.8, ........ We may specify class intervals as 160.55 165.55,165.55 ....... so that none of the observed values of x is equal to any of the class limits.

For constructing histogram of unequal class-intervals, first we note a class of the smallest intervals. Other classes are noted in the increasing order of their intervals. If the size of one class-interval is twice the smallest size in the series, frequency of that class is divided by two. Suppose the class with the smallest interval is 5 – 10 and the class with the largest interval is 10 – 20, the frequency of which is 12. Here the class interval of the bigger class is 10 which is twice as much as the size of the class-interval of the smallest class i.e. 5. The bigger class interval is divided into two parts 10 – 15 and 15 – 20 and accordingly the frequency of the bigger class, 12 would be divided by 2 i.e. 12+ 6. In this way there will be following adjustment.
 

The procedure of selecting random sample of 5 students out of 25 in a class is as follows:

1. Assign a particular number between 1 to 25 to all the 10 students like, 01, 02, 03, 04, 05, 06, 07, 08, 09, 10. ....... 25

2. Select a number randomly. Let us assume that the number selected is 05.

3. Consult the Two-Digit Random Number Table, two numbers successive to the selected random number (i.e. 05) either horizontally or vertically are the remaining two students (i.e. 06 and 07).

Question: Convert the following  data into a simple frequency distribution

5 student obtained less than 3 marks

12 student obtained less than 6 marks

25 student obtained less than 9 marks

33 student obtained less than 12 marks

A n s w e r:

The given statement is false. This is because Statistics deals with both quantitative data as well as with qualitative data. Statistics not only reveals quantitative information but also reflects the qualitative aspect of data like better, worse off, highly skilled, etc.