Collection of Data

Written and reviewed by Devendra Dubey, Mathematics & Commerce faculty at UniDrill (M.Sc. Mathematics, Osmania University · 10 years teaching). Based on NCERT Economics Class XI (Statistics for Economics), Chapter 2.

Where do the numbers come from?

Every table you will ever draw, every average you will calculate and every graph you will plot starts as a number somebody collected. If the numbers are wrong, the cleverest statistics will only give you a precise wrong answer. This chapter is about the very first step of the statistical process: getting good data.

A variable is a characteristic that takes different values, usually written X, Y or Z, and each value it takes is an observation. Data are the values of variables. The chapter's example is the production of foodgrains in India, which varied from 108 million tonnes in 1970-71 to 272 million tonnes in 2016-17. That variation is what statistics studies.

Primary and secondary data

Primary and secondary depend on the user. The Census of India collects data first hand, so for the Census office those data are primary. A student who takes the same figures from the published report uses them as secondary data. The data are the same, and what changes is who is using them.

Survey: asking people

A survey gathers information from individuals. The main tool is a questionnaire (or interview schedule), a list of questions. A questionnaire is typically filled in by the respondent. A schedule is filled in by an enumerator, the field worker who asks the questions. The investigator conducts the enquiry, and the respondent is the person who gives the information.

A good questionnaire

It is short and easy to understand. Questions run from general to specific. Each question is precise and unambiguous. It avoids double negatives and leading questions (questions that hint at the answer), and it does not suggest alternatives inside the question.

Pilot survey. Before the real survey, the questionnaire is tried out on a small group. A pilot survey (pre-testing) detects shortcomings, tests whether questions are clear, shows how enumerators perform, and helps estimate cost and time. It is not used for the final estimate.

Three ways to collect the answers

Worked example: choosing the mode. Match each situation to the best mode.

The wrong choice in the last case would be a mailed questionnaire, since speed is the priority and mail is the slowest and has the lowest response.

Census and sample

The population (or universe) in statistics is the whole set of items under study. It need not be people: it can be farms, firms or houses. There are two ways to collect data from it.

A good sample is smaller than the population but represents it well, so it gives reasonably accurate information at a lower cost and in less time.

The Census of India counts every household every ten years, and it has done so since 1881. The first Census after independence was held in 1951. The last one was held in 2011.

The annual growth rate of population fell from 2.2% (1971-81) to 1.97% (1991-2001) and then to 1.64% (2001-2011).

Worked example: decadal growth. By how much, and by what percentage, did India's population grow between 2001 and 2011?

The common wrong step is to divide by the 2011 figure (the end value). A percentage change is always measured against the starting value.

Two views. For the exam: a census gives the true value and a sample gives an estimate, so a census is more accurate, and a sample is cheaper and faster. Another reading: statisticians point out that a very large complete count can itself contain many mistakes, because it needs a huge team of enumerators who are hard to train and supervise closely. A well designed sample handled by a small, well trained team can sometimes give results that are as reliable in practice. Why it matters: it explains why even a census can contain errors, and why sampling is so widely used.

How a sample is chosen

A sampling frame is the list of all sampling units in the population. The sample is drawn from it in one of two ways.

The lottery method is a random method: each unit is given a ticket, the tickets are mixed, and the required number is drawn. It is a type of random sampling, not of non random sampling.

Worked example: using a random number table. An investigator wants 5 households from a sampling frame of 100 households numbered 00 to 99. How should the sample be drawn?

The result does not depend on the investigator's choice, and that is the point of the table. The common wrong step is to pick the "typical looking" houses by eye, which is judgement sampling and not random.

Beyond the textbook. For the exam: a random sample gives every unit an equal chance of selection, and a bad selection leads to sampling bias. In reality: in 1936 a US magazine collected millions of replies in a poll to predict the presidential election and still got the wrong winner, because its lists of people to contact leaned towards better-off households. Much smaller samples chosen to represent the whole electorate did better. Why it matters: the way a sample is chosen matters far more than how large it is.

Pause and predict: an investigator surveys only the families living on the main road of a village because they are easiest to reach. Is this random sampling?

Check your answer: No. It is convenience sampling, a non random method. Families off the main road had no chance of being chosen, so the sample may not represent the village.

Errors in data

An error is the difference between what the data show and the truth. There are two kinds.

Sampling error is the difference between the sample estimate and the true value for the population. It arises because only some units are studied, and it can be reduced by taking a larger sample.

Worked example: sampling error. Five farmers earn ₹500, ₹550, ₹600, ₹650 and ₹700 a month. A sample of two farmers, those earning ₹500 and ₹600, is drawn. What is the sampling error in the average? Then take a sample of four (₹550, ₹600, ₹650, ₹700).

$$\text{Sampling error} = \lvert \mu - \bar{x} \rvert$$

Here $\mu$ is the population mean and $\bar{x}$ is the sample mean.

The larger sample came closer to the truth, which is why a larger sample reduces sampling error. Note that the error is not always smaller for every possible sample, but taking a larger sample reduces it in general.

Non sampling errors do not depend on the sample size, so a bigger sample or even a census does not remove them. They are more serious for that reason.

Worked example: which error is it? In each case, name the error and say whether a larger sample would fix it.

Pause and predict: a government counts every household in a district (a census) but a few enumerators copy numbers wrongly. Is there sampling error?

Check your answer: There is no sampling error, because every unit was counted. But there is a non sampling error (error in data acquisition), which shows that even a complete count can be wrong.

Where India's data come from

Government agencies gather and publish the data you will meet in later chapters.

The NSS 60th round (January to June 2004) was on morbidity and health care, and its 68th round (2011-12) was on consumer expenditure. Both are examples of sample surveys.

Exam traps

60-second revision

Where this chapter connects

• Previous: Introduction explains why economists need data and what statistics does with them.
• Next: Organisation of Data shows how the raw observations you have collected are classified and arranged.
• Later in the book: Measures of Central Tendency uses the means that appear in the sampling error calculation.