# B.Com 1st Year Statistical Investigation Short Question Answer Notes

**B.Com 1st Year Statistical Investigation Short Question Answer Notes** :- In this post You will Find B.com **Notes** **Study material** Unit Wise Chapter Wise Topic Wise division of the content. This Post is very useful for all the Student **B.A., B.Sc., B.Com., M.A., M.Com. **Planning of statistical Investigation, Census and sampling methods, Collection of primary and secondary data, Statistical errors and approximation, Classification and tabulation of data, Frequency distribution. All topic Notes Available in over site parultech.com.

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In the labyrinth of numbers, where data resides, a** statistical investigation** unfolds—a journey of discovery that transforms the mundane into the magical, the obscure into the comprehensible.

With the heart of a poet and the precision of a watchmaker, we embark upon this odyssey. Each data point, a star in the night sky, twinkles with the promise of revelation. Each equation, a verse in the ballad of patterns and probabilities, sings to the soul of understanding.

We gather our data, like picking petals from a daisy, revealing the hidden truths it holds. In the dance of correlation and causation, we seek the connections, the threads that weave the narrative of statistics. Like the maestro conducting an orchestra, we orchestrate the variables to play in harmony.

As we wander through the forest of hypotheses, we look for the elusive flowers of significance, blossoming in the underbrush. With the compass of mathematical rigor, we navigate the terrain, guided by the North Star of methodology.

The data speaks in whispers, the p-values and confidence intervals, like ancient runes, etching tales of confidence and uncertainty. We consult the oracle of statistical software, unraveling the enigmatic language of graphs and charts, painting a portrait of insights.

And as we reach the denouement of our investigation, we see the conclusion like the final note of a symphony, resonating with the truth. We unveil the findings, not just as numbers on a page but as the revelations of a story told by the universe itself.

In this **statistical investigation,** we do not merely crunch numbers; we dance with the data, converse with the variables, and listen to the heartbeat of knowledge. It is not just a journey; it’s an emotional connection with the mysteries of the empirical world, where numbers transform into poetry and statistics into a sonnet of discovery.

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**Q.1. Describe the preliminary steps you would take in planning a statistical investigation.**

**Ans.** Statistical investigation is the collection of data to serve pre-determined purpose in order to facilitate a proper analysis of the problem to which it is related. Planning refers to the systematic designing of future course of action that facilitates smooth flow of work. The following points must be considered while planning for a statistical investigation:

**1. Objective of the Investigation:** In it, the reason for investigation, its related problems, objectives for the investigation is going to serve, efforts involved, achievements, etc. are carried out.

so the obiectives of the investigation should be clearly defined which is more solution. The investigator should be very much clear about the reasons of undertaking this investigation. Extreme care is required to be taken while defining the objective of the investigation.

**2. Scope of the Investigation:** The scope of investigation is defined after the objective. It defines the limits of the investigation and it can be decided on geographical, political, economic or any other basis.

**3. Period of Investigation:** The period of investigation is of utmost importance as the clarity and accuracy depends on it. The period should not be too short to complete the entire process of investigation. This must be definite, not be insufficient and the period for which data will be collected should be suitable.

**4. Type of Investigation:** There are various types of investigation depending on the nature and objective of the problem under investigation and the type of accuracy expected. It helps us in making a decision whether to collect the data.

(a) From primary or secondary source.

(b) By census or sample investigation.

(C) Through regular or adhoc investigation.

(d) Through postal, personal or both types of investigation.

(e) It is government or private investigation.

(f) Conducting confidential or public investigation.

**5. Unit of Investigation:** It is a measurement that should be taken for collecting the data. It is a quantitative process so suitable unit of enumeration and analysis are to be selected for the study Statistical unit ofinvestigation should be standard, stable, simple, clear, certain and known homogeneous according to the nature, scope and requirement of the investigation. No frequent changes should be possible in the chosen unit.

**6. Collection of Data for Investigation:** It is necessary for successful investigation that the collection of data should be proper and satisfactory. The investigator must determine as what he has to do and how should he proceed. Investigator also has to take a decision as to what should be done to maintain the standard of accuracy in the collection of data.

**7. Editing. Classification and Tabulation of Data:** After the collection, the investigator edit the data so as to consolidate his investigation and hence make use of the data which is relevant to the The investigator classifies the data and tabulate it so as to simplify and arrange the data in an order as rendered necessary by the enquiry to help in interpretation and report writing.

**8. Presentation and Analysis of Data:** The collected data is required to be presented but before it, it is to be analysed with the help of available methods in order to draw conclusion and present them in a manner in which it is most suitable according to the nature of the enquiry.

**9. Preparation of Report:** The report preparation occupies an important position in any statistical investigation because it is the conclusion on the basis of which the decision may be taken. Policy framing, further action, required decisions and implementation depends upon the report that is prepared on the result of statistical enquiry. This job is done by the investigator who prepare the report on the basis of his investigation, edited, classified, tabulated and analysed data. This report should be simple and in lucid language, brief, to the point, clear and also self explanatory.

The report should throw light on the investigated problem, its possible solution and future course of action that helps to know about the changes that the problem may take in future and what would be the possible solution to the problem.

**Q.2. What are the basic sources of business data? Also explain the various methods of collecting data.**

**Ans. Sources of Business Data:** Data sources may be external and/or internal. External data sources may be divided into:

1. Primary sources in which original research and material gathered are summarised. These

may be taken from field studies whether on a sample or census basis or from case studies. 2. Secondary sources which use the data collected through primary sources for certain specific purposes or for general presentation in summary form.

3. Tertiary sources which further distill data from secondary sources. Quite often, however, no distinction is made between secondary and tertiary sources. A large part of the data presented in published sources is initially collected by means of statistical surveys. Statistical material obtained from secondary sources is not always as reliable as that from the primary sources. It is never safe to take published statistics at their face value, without knowing their means and limitations and it is always necessary to criticise arguments that can be based on them.

**Methods of Collecting Data**

The collection of data for various types of investigations may be done with the help of the following two methods: 1. Census method and 2. Sample method.

Every survey involves the collection of the desired information from a population set or the universe, i.e. the totality of the persons, firms or items under study. This population set depends on the purpose of the survey. If an agency is interested in studying the total population in India, it would need to canvas the schedules to all the households in the rural and urban areas in India. If, on the other hand, it is interested, let us say, in learning about the economic conditions of agricultural labourers/ cultivators in a particular district, it will canvas the schedules only with the agricultural labourers or cultivator-households in the district concerned.

A survey which includes every element of the population of universe under study is called a ‘Census survey

A population can be finite or infinite. For example, the population consisting of all bolts produced in a factory on a given day is finite, whereas the population consisting of all possible outcomes (heads, tails) in successive tosses of a coin is infinite.

The census method is generally used in the following cases:

1. When the field of investigation is limited,

2. When more accuracy is desired, and

3. When money spent and time taken for collection of data have no consideration.

The census method is considered to be more reliable because the data are obtained after studying each and every individual of the population.

**Limitations:** This method has the following limitations

- This method is practically not possible in certain cases, e.g. to purchase one kilogram of grapes because it is not feasible to taste the complete lot to test the quality of grapes.
- Due to maximum consumption of time, sometimes the data collected becomes obsolete and loses its consistency.
- Follow-up is very difficult because it is difficult to carry out the investigation again in case of any doubt in the result obtained.

It is often impractical to observe the entire group, especially when it is large. Instena of examining the entire group called the population or universe, one can examine a small part of the group called a sample. If a sample is representative of a population, important conclusions about the population can often be inferred from analysis of the sample. The phase of statistics dealing with conditions under which such inference is valid is called inductive statistics inference. Because such inference cannot be absolutely certain the language of probability is one conclusions. We use the sample method in the following cases:

1. Field of investigation is infinite,

2. Field of investigation is unlimited,

3. Field of investigation leads to fatal ends,

4. Field of investigation leads to devastation, and

5. Economy is considered necessary in investigation.

**1. Lottery Method:** Under this method, all items of the universe are numbered or named on separate slips or cards of paper. These slips are then folded and mixed up in a container or drum. A blind fold selection is then made from the number of slips required to constitute the desired sample size. For the reliability of this method, it is necessary that:

(a) All slips should be homogeneous in shape, size, colour, etc.,

(b) The slips should be throughly shuffled before selection of units for sample, and (c) The work of drawing slips should be done by unbiased person. This method can be explained

by an example. Suppose, there are 3,000 students in a college and a random sample of 100 students is to be drawn, then 3,000 slips numbering from 1 to 3,000 will be prepared and a sample of 100 slips will be drawn one by one. It may be mentioned that the slips must be shuffled after each draw.

**2. By Rotating the Drum:** This method is an improvement over the lottery method. In this method. olins are not prepared for all numbers but round or square wooden, plastic or iron pieces of equal size are used on which digits like 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are written. These pieces are put in the drum and the drum is rotated by mechanical device and the number of sample is formed on the basis of digit written on the piece which is drawn one by one. The digit of the next piece forms the tenth place in the number. This unit-digit of the random number. The digit of the, next piece forms the tenth places are drawn out. This process continues like that and digits for hundreds + method is very popular in drawing the number of lottery ticket for prizes.

**3. Random Numbers:** If the population size is very large, then difficulties arise in lottery method and the method rotate in the drum. Furthermore these methods become time consuming with chances and the method rotate in the drum. Furthermore the of errors. In such cases random number tables are used as alternativs.

**Q.3. Discuss the various methods of collection of primary data.**

**Or Distinguish between primary and secondary primary data and point out their merits and demerits. (2015)**

**Ans. Difference between Primary and Secondary data: **Refer to Section-A, Q.5.

Primary data are collected for the first time through census or sample surveys. Following are the methods of collecting primary data:

1. Direct Personal Interview

In this method, the investigator contacts the source of information directly and personally. This method of collecting data is suitable in the following situations:

(a) When the area of enquiry is limited.

(b) When high degree of accuracy is needed in the results.

(c) When the investigation is difficult and is to be conducted in a heterogeneous area.

(d) When the information collected is of confidential nature.

**Merits:** Merits of this method are:

1. Information collected by this method is accurate and reliable. It can be checked and rechecked and put in the desired form.

2. Alternative supplementary information can be gathered simultaneously.

3. Collected information can be made free from bias and prejudices.

**Demerits:** Demerits of this method are:

1. Such a method can be adopted only when the enquiry is intensive and localised. It cannot be adopted when the field of enquiry is wide.

2. The involvement of personal bias and prejudices is obvious.

3. Investigators should be trained and efficient persons, otherwise the information collected may be unreliable.

2. Indirect Personal Interview

In some cases, the informants cannot be contacted directly. In this situation, an indirect personal enquiry is conducted to get the desired information. The indirect personal investigation is done through some agencies who have some knowledge of the phenomenon under enquiry. For example, an investigator may collect information about cost of cultivation indirectly from the village Pradhan (headman) who have intimate knowledge of such matters. This method is useful in the following cases:

(a) When the area of investigation is large.

(b) When the information cannot be obtained directly from the informants.

(c) This method is generally used by the government or committees and commissions appointed by the government. They collect information about different problems by interviewing the person concerned.

**Merits:** Merits of this method are:

1. It is useful when the area of investigation is wide.

2. It saves labour, time and money.

3. As the information is collected from knowledgeable persons, it is expected to be useful and reliable.

4. The enquiry could be more extensive and different aspects of the problems can also be studied properly.

**Demerits:** Demerits of this method are:

1. The information collected may be subjective and is subject to the personal bias of the persons or agencies from whom it is collected.

2. One has to be very careful about the selection of persons from whom the information is collected, because their knowledge and personal attitude towards the problem greatly affect the quality of data. So one has to be very careful in dealing with such situations.

3. Mailed Questionnaire

In the mailed questionnaire method, a questionnaire in the form of a set of questions is sent by mail to the informants. They are expected to answer the questions and also the additional needed information, whenever it is needed and mail them back to the investigator. This method should be used in the following cases:

(a) When the area under investigation is wide.

(b) When the informants are educated.

(C) When the informants are expected to leave for far away places.

**Merits:** Merits of this method are:

1. It is economical and less time consuming.

2. A larger area can be covered under the investigation as even the people in distant places can be reached without much difficulty.

3. Collected information may be more reliable.

4. It is free from the bias of the investigation as the information is collected from the informants themselves.

**Demerits:** Demerits of this method are:

1. This method can be adopted only in cases when the informants are educated persons.

2. All the informants may not cooperate and the information to be collected may remain incomplete. Sometimes people may not spare time to fill the questionnaire or they may not be willing to return the filled questionnaire.

4. Schedule Through Enumerators

Initially let us make a distinction between a questionnaire and a schedule. The questionnaire is a set of questions the answers to which are recorded by the informant itself, whereas in a schedule answers are recorded by the investigator or an enumerator on his behalf.

In this method the investigators or enumerators approach the informants with a prepared questionnaire and get the replies to the questions. This method is generally used in censuses and large scale surveys. In the case of census, investigators visit every member of the source of information in the zones while, in the case of sample survey, they collect information from those members who have been selected in the sample.

**Merits:** Merits of this method are:

1. This is the only method which is applicable in census and sample survey both.

2. Subjectivity on the part of the interviewer is less in this method.

3. This method is useful where the scope and coverage is very large.

4. Personal contacts increase the reliability of data.

5. Information from an illiterate person too can be obtained.

**Demerits:** Demerits of this method are:

1. The method is quite costly and time consuming.

2. Trained and efficient investigators will be needed to get the information.

3. The method can be used only by big organisations.

**Q.4. Give the published sources of secondary data.**

**Or Point out the various sources of collection of secondary data. Give its limitations. **

**Ans. 1. Published Sources:** Some of the published sources providing secondary data are:

**(a) Government Publications:** Anumber of government, semi-government and private organisations collect data related to business, trade, prices, consumption, production, industries, income, health, population etc. These publications are very powerful sources of secondary data. Central Statistical Organisation (C.S.O.), National Sample Survey Organisation (N.S.S.O.), Office of the Registrar General and Census Commissioner of India, Directorate of Economics and Statistics and Labour BureauMinistry of labour are a few government publications.

**(b) International Publications:** Various governments in the world and international agencies regularly publish reports on data collected by them on various aspects. For example, U.N.O’s statistical year book, demography year book, etc. can be named in this category.

**(c) Semi-official Publications:** Local bodies like District Boards, Municipal Corporations publish periodicals providing information about vital factors like health, births, deaths, etc.

**(d) Reports of Committees and Commissions:** At times, state and central governments appoint committees and commissions with a specific reference to study a phenomenon. The reports of these committees and commissions provide important secondary data. For example, Kothari commission report on education reforms, report of National Agricultural Commission, Wanchoo commission report on taxation and pay commission reports, etc.

**(e) Private Publications:** The following private publications may also be enlisted as the source of secondary data:

**(i) Journals and Newspapers:** Eastern economists, monthly statistics of trade, financial express, economic times are some of the journals and newspapers which regularly collect and publish data on various aspects of business, economics, commerce and trade.

**(ii) Research Publications:** A number of research organisations, university departments and institutes like Indian Statistical Institute (I.S.I.) Calcutta and Delhi, I.C.A.R., N.C.E.R.T., I.C.M.R.. etc. also contribute significantly to the availability of secondary data. (in

**(iii) Publications of Business and Financial Institutions:** A number of business and financial institutions like Chamber of Commerce and Trade Association, Institute of Chartered “Accountants, Sugar Mill Association, Stock Exchanges, Trade Unions and Cooperative Societies, etc. also contribute significantly for the availability of secondary data in the related areas.

**(iv) Articles:** Market reviews and reports also provide data for analysis.

**2. Unpublished Sources:** Here, the data are collected but are not put in published form. But still the data from these sources may be used when needed.

Limitations of Secondary Data

These are as follows:

1. These may not be free from personal bias and prejudices.

2. These may not be adequate.

3. These may not be relevant in the present contest.

4. These may not have the needed accuracy or reliability.

**Q.5. Distinguish between a census and a sample survey. What are the principal steps in a sample investigation?**

**Ans. Census and Sample Survey:** Census method is most appropriate if the population is small and precise information is needed but when the population is very large or the field of investigation is very wide and quick results are needed at low cost, sampling techniques are appropriate.

**Difference between Census and Sampling**

S.No. | Basic of differences | Census method | Sampling method |

1. | Nature of enquiry | All the items lying in the scope of enquiry are studied. | Only selected items are studied. |

2. | Cost | It involves more cost. | It involves less cost. |

3. | Time involved | It is time consuming | It saves the time. |

4. | Organisation | It needs more organisational skill and large force investigators. | It involves less organisational skill. |

5. | Suitability | Method is suitable when area of enquiry is limited. | This method is suitable when area is very wide. |

Principal Steps in a Sample Investigation

The main steps involved are:

**1. Objective of Survey:** It must be clearly explained that the unnecessary data may be avoided.

**2. Population to be Sampled:** Objective should define the population which the survey is intended to cover.

**3. Data to be Collected:** All the data relevant to the purposes of the survey must be verified and that no essential data are omitted.

**4. Degree of Precision Desired:** Degree of precision is desired in the results for which uncertainty has to be reduced.

**5. Methods of Measurement:** The methods like interview method or mail enquiry method are to be adopted.

**6. The Frame:** The list, map or other acceptable material that serves as a guide to the population to be covered is called a frame which must be free from defects

**7. Selection of Sample:** Rough estimates of the size of the sample is made from a knowledge of the degree of precision desired in each sample.

**8. The Pretest:** It is useful to try out the questionnaire and the field methods on a small scale.

**9. Organisation of Field Work:** The staff must receive training in the purpose of the survey ment to be employed and must be adequately supervised in their work.

**10. Summary and Analysis of Data:** A report must be written that gives the findings of the survey related to questions it was meant to answer.

**11. Information Gained for further Surveys:** There must be arrangements to store information as it serve as a guide to improve future sampling. + the completion of the survey as it serve as a guide to improve future sampling.

**Q.6. Discuss the various types of sampling methods.**

**Or Examples the merits and demerits of different methods of sampling.**

**Ans.** The choice of a sampling procedure greatly depends on the purpose of enquiry and the nature of population. Various methods of sampling can be broadly classified as:

(A) Non-random or non-probabil.

(B) Random or probability sampling methods.

(A) Non-random or Non-probability Sampling Methods

In non-random sampling method, the units in the population are selected on subjective basis and no probability law is applied in the selection of a sample from the population. Some of the sampling methods are:

**1. Subjective or Judgement or Purposive Sampling**

Any type of sampling in which the sample selected depends on personal discretion or judgement of the investigator is called a subjective or judgement sampling. This type of sampling is used with a definite purpose in view and as such is not used for general purposes. In this sampling method, the choice of sample items depend exclusively on the judgement of investigator. The investigator includes those items in the sample which he thinks are most typical of the population with respect to the characteristics under study. For example, an investigator may conduct a survey for getting opinions and views on certain issues like family welfare programmes, everyday business problems, liking and disliking of T.V. programmes, etc. by selecting a sample of persons by his own judgement. Thus the success of this sampling method depends on the proper judgement of the investigator in choosing a representative sample, otherwise the application of this method would bring in personal bias. This is the biggest limitation of this sampling method.

**2. Quota Sampling**

This is a restricted type of purposive or judgement sampling. This sampling method is adopted when stratification is difficult and still we wish to derive the benefits of stratification without using probability sampling. Quotas may be fixed for each stratum and then required size of samples are drawn by judgement sampling. This method is usually adopted for opinion polls and for socioeconomic investigations. For example, for an opinion poll about the popularity of a political party, certain quotas may be fixed from people divided into strata like office workers, businessmen and so on. While the method is quite simple, it also suffers from the drawbacks of judgement sampling. More so, the sampling theory cannot be applied as the method is not based on probability sampling. (B) Random or Probability Sampling Methods

Any type of sampling in which every unit of the population has a definite, preassigned probability of being selected in the sample is called probability sampling. Some of the probability sampling methods are as explained below:

**1. Simple Random Sampling**

In random sampling, each unit of the population has a definite preassigned probability of being selected in the sample. Simple random sampling is a particular case of random sampling. In simple random sampling, each unit of the population has an equal and independent chance of being included in the sample. Here independent means that the selection of one unit in the sample does not affect the selection of another unit. Simple random sampling may be with or without replacement according as a unit selected is not or is excluded from further selection.

**Methods of Selecting a Simple Random Sample:** There are two methods of selecting a simple random sample:

**(a) Lottery Method:** This is a popular method of drawing a simple random sample. In this method every unit of the population is numbered or named on separate slips of paper which are identical in size and shape. These slips are then well mixed in a container or a drum and then one draws the required number of slips to constitute the desired sample size.

**(b) Tables of Random Numbers:** In practice, tables of random number can also be used for selecting a simple random sample. Several standard tables of random numbers are available of which some in common use are: (a) Tippet’s random number tables, (b) Fisher and Yates tables and (c) Rand corporation tables. For using these, the population units should be numbered from 1 to N, which makes it possible to determine the range of numbers to be selected. If n is less than 100, two digit numbers are selected, if N is less than 1,000, three digit numbers are selected and so on, As and example, suppose N – 500 and We wish to draw a sample of size n= 10. Then we first identify the 500 units in the population with numbers 1 to 500 tan we take any random number table and choose any page from the table. Starting at any row or column, we select three digit numbers one by one, discarding the numbers greater than 500, until 10 numbers below 500 are obtained. Finally, the population, units bearing these 10 selected numbers will constitute our sample.

**Merits:** These are as follows: 1. Since the selection of items in the sample depends entirely on chance, there is no possibility of personal bias affecting the results.

2. As compared to judgement sampling, random sample represents the population in a better way.

As the size of the sample increases, it becomes increasingly representative of the population.

3. The analyst can easily assess the accuracy of his estimate because sampling errors follow the principle of chance. The theory of random sampling is further developed than that of any other type of sampling which enables the surveyor to provide the most reliable information at the least cost.

**Demerits:** These are as follows:

1. The use of random sampling necessitates a completely catalogued population from which to draw the sample. But it is often difficult for the investigator to have up-to-date lists of all the items of the population to be sampled. This restricts the use of any sampling method.

2. The task of preparing slips is time-consuming and expensive. However, this difficulty can at time be overcome by following regular interval sampling method which enables a random

sample to be drawn without preparing slips.

3. The size of the sample required to ensure statistical reliability is usually large under random sampling than in stratified sampling.

4. From the point of view of field survey, it has been claimed that cases selected by random

compling tend to be too widely dispersed geographically and that the time and cost of collecting data becomes too large.

5. andom sampling may produce the most non-random looking results. For example, thirteen cards from a well-shuffled pack of playing cards may consist of one suit. But the probability of this type of incidence is very very small.

2. Stratified Random Sampling

When a population is heterogeneous in nature, it would not be desirable to use simple random ation eterogeneous in nature, it would not be desirable to use simple random sampling. In such a case, the entire heterogeneous population is divided into a number of homogeneous groups, usually called strata or sub-population. Then units are sampled at random from each of these stratums. The sample, which is the set of all the sampling units drawn from each stratum is called a stratified sample and the technique of drawing the sample is termed as stratified random sampling.

Symbolically, suppose the population of size N1, N2,,,,,,,,,

Nk such that E N; = N. Simple random samples of sizes nj, n2…., nk are drawn respectively from the k,,,, distinct strata. The sample constitutes a stratified random sample of size n = k E n i … En:R i=1

The stratified sampling method is added to the precision of the sample estimates when:

(a) The units within each stratum are more or less homogeneous.

(b) The units in different strata are unlike or heterogeneous.

**Merits:** These are as follows:

1. Since the population is first divided into various strata and then a sample is drawn from each stratum there is little possibility of any essential group of the population being completely excluded. A more representative sample is thus secured. Stratified sampling is frequently regarded as the most efficient system of sampling.

2. Stratified sampling ensures greater accuracy. The accuracy is maximum if each stratum is so formed that it consists of uniform homogeneous items

3. As compared to random sample, stratified samples can be more concentrated geographically. Thus the time and expense of interviewing may be considerably reduced.

**Demerits:** These are as follows:

1. Utmost care must be exercised in dividing the population into various strata. Each stratum

must contain, as far as possible, homogeneous items as otherwise the results may not be

reliable. However, this is a very difficult task and may involve considerable time and expense.

2. The items from each stratum should be selected at random. But this may be difficult to achieve in the absence of skilled sampling supervisors and a random selection within each stratum may not be ensured.

3. Systematic Random Sampling

This is a convenient method when complete list of sampling units called sample frame, is readily available or can be easily prepared. The sampling scheme consists of selecting only the first unit at random and the rest are then automatically selected according to some predetermined pattern. Suppose we have Nunits in the population and a sample of size nis drawn. Then we first determine an integer such that k=N/n, here k is called the sampling interval. Now, in systematic random sampling the first sample unit is selected at random from the first group of k units and thereafter every kth unit is selected.

For example; Suppose N=100 and n=5. Then k=N/n=100/5=20. Now, suppose the first unit selected at random from the first 20 units is 8th, then thereafter every 20th units will be taken in the sample. Thus, the systematic sample of size 5 will consist of units corresponding to number 8, 28, 48, 68 and 88.

A systematic sample gives more precise estimates compared to simple random sample if the units selected within the sample are heterogeneous. However, if there exists some periodicity in the list, we expect the selection of more or less homogeneous units in the sample resulting in biased estimates.

**Merits:** The systematic sampling is more convenient to adopt than the random sampling or the stratified sampling method. The time and work involved in sampling by this method are relatively smaller The results obtained are also found to be generally satisfactory provided that care is taken to see that there are no periodic features associated with the sampling interval. If population is sufficiently large, systematic sampling can often be expected to yield results that are similar to those obtained by proportional stratified sampling.

**Demerits:** Systematic sampling becomes a less representative design than simple random sampling if we are dealing with populations having hidden periodicities. For example, if the sales of every seventh day of the calendar year are included, the sample will contain, say, all Mondays or all Fridays. If there is a definite repetitive weekly pattern in sales (which is usually the case) our sample is not representative at all of sales for the whole year and consequently the sample results may be seriously biased.

**4. Cluster Sampling**

A group of elementary units in the population is called a cluster. When a cluster is taken as a sampling unit, the procedure of sampling is called cluster sampling. This sampling procedure is useful in situations when we observe an unequal concentration of individual units in the population. In such cases, certain blocks or clusters of higher concentration are randomly selected for complete.

enquiry. For example, in verifying the accounts, we may select all sheets from one or more ledgers, or all transaction of one or more weeks in a year showing highest concentration. Further sub-sampling in clusters (when called first stage sampling units or primary units) gives rise to multi-stage sampling as discussed below:

**5. Multi-stage Sampling**

Sometimes sampling is done in stages to reduce the cost of the survey. In this sampling method, the population is divided into first stage sampling units also called primary units. Then a random hirst stage units is made. Further division is made of the first stage sampling units selected and a random sample is then taken from these second stage sampling units. The process can be continued for a number or stages. For example, the country may first be divided into states as first stage units from which a number of states may be randomly selected. Then the selected states can be divided into districts (as second stage units) and then a number of districts can be selected from these second stage units. The selected districts may be further divided into villages (as third stage sampling units) and a number of Villages may be randomly selected from these third stage sampling units. The selected villages can be divided into households as fourth stage sampling units and then a small number of households may be selected. Households will then form the ultimate sampling units in the four stage sampling plan, In case the sampling is restricted to two stages, it is called two stage sampling or sub- sampling.

**Merits:** Multi-stage sampling introduces flexibility in the sampling method which is lacking in other methods. It enables existing divisions and sub-divisions of the population to be used as units at various stages, and permits the field work to be concentrated and yet large area to be covered. Another advantage of the method is that sub-division into second stage unit (i.e. the construction of the second stage frame) need to be carried out for only those first stage units which are included in the sample. It is, therefore, particularly valuable in surveys of underdeveloped areas where no frame is generally sufficiently detailed and accurate for sub-division of the material into reasonably small sampling units.

**Demerits:** However, a multi-stage sample is in general less accurate than a sample containing the same number of final stage units which have been selected by some suitable stage process. 6. Sequential Sampling Sequential sampling consists of a sequence of samples of size drawn randomly from the population, ie, observations are sequentially recorded one by one. Such sampling is generally adopted in statistical quality control. In this sampling, a test may be terminated at any stage of experimentation. Here we randomly draw sample observations one by one and make decisions on the basis of the combined evidence of already drawn observations. In this regard let us consider an example where a lot quality is to be judged by sequential sampling of items from the lot. If the quality of a first few is good enough to accept the lot then we stop testing and terminate the test. Similarly, if he quality of already drawn items at any stage is such that it is bad enough to reject the lot, then we again terminate further sampling and reject the lot quality. However, if the sample information indicate that the quality is neither good nor bad enough to accept or reject the lot, we draw next item from the lot and now on the basis of all the drawn observations we take any of the above mentioned decisions. Thus in sequential sampling. The sample space is divided into three zones. namely:

(a) The region of acceptance.

(b) The region of rejection.

(c) The zone of indifference in which the sampling is continued and observations are drawn from the population one-by-one.

** Q.7. Define classification and explain various ways of classification.**

**Ans. Classification:** ‘Classification is the process of arranging things (either actually or notionally) in the groups according to their resemblances and affinities and give expression to the unity of attributes that may subsist amongst a diversity of individuals’.

Thus, classification is grouping of data according to their identity, similarity or resemblances, e.g. students in a class may be grouped in respect of sex, age, marital status, etc. Similarly, letters in the post office are classified according to their destinations, viz. Delhi, Jaipur, Agra, Kanpur, etc.

**Objectives of Classification**

Following are the main objectives of classifying the data:

1. It condenses the mass of data in an easily assimilable form.

2. It eliminates unnecessary details.

3. It facilitates comparison and highlights the significant aspect of data.

4. It enables one to get a mental picture of the information and helps in drawing inferences.

5. It helps in the statistical treatment of the information collected. Types of Classification Statistical data are classified in respect of their characteristics. Broadly, there are four basic types of classification:

**1. Chronological Classification:** In chronological classification, the collected data are arranged according to the order of time expressed in years, months, weeks, etc. The data are generally classified in ascending order of time.

**2. Geographical or Spatial Classification:** In this type of classification, the data are classified according to geographical region or place. For instance, the production of paddy in different states in India, production of wheat in different countries, etc. For indicating immediate comparison the observations are either classified in the alphabetical order of the reference places or in the order of size of the observation,

**3. Qualitative Classification:** In this type of classification, data are classified on the basis of some attributes or qualities like sex, literacy, religion, employment, etc. Such attributes cannot be measured along with a scale.

**4. Quantitative Classification:** In quantitative classification, the collected data are grouped with reference to the characteristics which can be measured and numerically described such an height, weight, sales, imports, age, income, etc. The first step in the direction of putting observation in some ordered form is to arrange them in ascending or descending order of magnitude.

**Q.8. Write a short note on frequency distribution.**

**Ans. Frequency Distribution:** Arrayed series, however, does not reduce the volume of the data. In order to avoid repetition of the variables of the same magnitude we combine them together and arrange them into two columns under the headings: 1. variables and 2. its frequencies. Frequency of the value of the variable means the number of times that value of the variable is repeated, i.e. the number of times of the occurrence of the value of the variable in any one series. Series represented by a discrete variable are called discrete series. Following are the examples of discrete and continuous frequency distributions:

**1. Discrete Frequency Distribution**

No. Of children | No. Of families |

0 | 10 |

1 | 40 |

2 | 80 |

3 | 100 |

4 | 250 |

5 | 150 |

6 | 50 |

Total | 680 |

**2. Continuous Frequency Distribution**

Weight (Ibs) | No. Of persons |

100-110 | 10 |

110-120 | 15 |

120-130 | 40 |

130-140 | 45 |

140-150 | 20 |

150-160 | 4 |

Total | 134 |

Although the theoretical distinction between continuous and discrete variation is clear and precise, but in practical work it is only an approximation. The reason is that even the most precise instruments of measurement can be used only to a finite number of places. Thus, every theoretically continuous series can never be expected to flow continuously with one measurement touching and w any break in actual observations.

**Formation of Discrete Frequency Distribution: **The process of preparing this type of distribution is very simple. We have just to count the number of times a particular value is repeated which is called the frequency of that class. In order to facilitate counting prepare a column of ‘tallies.

1. We write the marks (called the variables) in the first column in serial order, i.e. from 0 to 10 or from 10 to 0 according to convenience. 2. Again for each value of the variable in the second column we draw vertical bars whenever that particular value of the variable is to be recorded. These are drawn to facilitate counting. After a particular value has occurred four times, for the fifth occurrence we put a cross tally mark (1) on the first four tally marks to give us a block of 5. This technique of putting cross tally marks at every 5th repetition (giving groups of 5 each) facilitates the counting of the number of occurrences of the value at the end. See table below:

**Q. 9. What is tabular presentation or tabulation? How it differs from classification? Also explain the Process to construct a table.**

**Or Distinguish between classification and tabulation.**

**Ans.** Tabular Presentation (Statistical Tables) One of the simplest and most revealing device centing them in a meaningful fashion is the statistical table. A table is a sy data in columns and rows. Rows are horizontal arrangements whereas ose of a table is to simplify the presentation and to facilitate comparison.

**Difference between Tabulation and Classification:** Tabulation is quite different from classification, although both have close relationship. Tabulation logically follows classification. Before the data are put in tabular form, they have to be classified, i.e. the different items having common characteristics must be brought together. It is only after this step that the data are displayed under different columns and rows so that their relationship can be easily understood. Characteristics of a Good Table.

Following are the important characteristics of a good table:

1. The table should be neat and attractive so that it should appeal to the reader om

2. The purpose of the table should be immediately clear. This requires the importance of a clear and concise title.

3. The table should be simple, concise and self-explanatory. It should reveal only those facts which are subject to further statistical investigation and analysis.

4. Each aspect of the data should be clearly brought out.

5. The table should represent the results accurately.

6. The table should facilitate comparisons.

7. The table should not be overloaded with details.

8. A table should always make a specific mention of the source of the data, wherever possible, as well as a note about various data gaps. This will clearly help the user of the table in understanding the limitations of the data and consequently those of the resulting conclusions.

9. Unit of measurement should be clearly indicated. Construction of a Table

Every table should comprise of the parts enumerated as follows:

**1. Title:** Every statistical table should have a title at the top. The title should be self-explanatory and should indicate the nature of the data being presented. It should be clear and short.

**2. Head note:** A table should have a head note below the title. A head note indicates the designation of units in which table values are given.

**3. Stub:** The rows in a table are designated as stub items. The stub items should be complete and clear.

**4. Caption:** The columns in a table are designated as column heads. All the column heads taken together form the caption. The column heads should also be well-defined.

**5. Field (or Body):** The field of table shows all the numerical information to be presented in a table. The relevant information is put against the stubs and caption in a table. This information is presented in the form of the field (or body) of the table.

**6. Footnote:** If at the time of giving details in the table, a complete explanation of an item has not been given, then the same is given by way of a footnote. The footnote is given below the main table.

**7. References:** If the data have been collected from some secondary source, then the source of the data should be disclosed. It is done in the form of giving references at the end of the table. The references should be complete in all respects.

**Q.10. From the data given below, prepare a bivariate frequency distribution with a class interval of 5 marks:**

Roll No. | Marks | Roll No. | Marks | ||

Statistics | Physics | Statistics | Physics | ||

1 | 10 | 21 | 11 | 7 | 21 |

2 | 6 | 23 | 12 | 18 | 3 |

3 | 15 | 19 | 13 | 10 | 15 |

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