What statistics is: data, variables, scales of measurement and how data are collected

What this chapter is for

This chapter answers the first line of the syllabus, “Definition, Importance and scope of Statistics”. It teaches what the word statistics means, what statistics is used for and what it cannot do, and the raw material every later chapter works on, which is data. You will learn to tell the kinds of data apart, to name the scale on which each is measured, and to choose a sensible way of collecting data for an inquiry of any size.

By the end you will be able to define statistics in both of its senses, classify any variable as qualitative or quantitative, discrete or continuous, and nominal, ordinal, interval or ratio, say why the scale decides which averages you may use, and plan how to collect data for a small, a large or a national inquiry. The chapter is long, so here is its map. The first three sections explain what statistics is, what it does and where it stops. The next six teach data, variables, scales and the ways data are collected. Three more let you practise, first by watching, then with help and then alone. The next follows Pakistan’s 2023 census from the first form to the approved count, and the last sections prepare you for the exam.

The paper has set this ground twice in Part-II since 2016. CE-2024 gave ten marks to the methods of collecting data, and CE-2025 gave ten to defining statistics and its importance.⁶⁷ Its terms also turn up in the MCQs nearly every year.

Before you start

The words

• Data. Facts recorded as numbers or as categories, such as the ages of a family, the marks of a class or the province each person lives in. One fact is a datum, though most people use “data” for one or many.
• Observation. One recorded value, such as one student’s mark.
• Variable. A feature that can differ from one person or thing to the next, such as age, sex, income or province. The section on data and variables teaches its kinds.
• Population. In statistics, every person or thing the inquiry is about, such as every household in Pakistan or every bulb a factory made last month. It need not be people.
• Sample. A part of the population, chosen to stand for the whole.
• Census. A count or a study of every member of the population, not a sample.
• Survey. An inquiry that collects data from people or things, by questions or by measurement. A sample survey uses only a sample.
• Enumerator. The person who goes from house to house and records the answers in a census or survey.
• Questionnaire. A printed or on-screen list of questions that the respondent, the person answering, fills in.
• Primary data and secondary data. Data collected first-hand for the inquiry at hand, and data that someone else collected and published earlier.

The bodies

• The Pakistan Bureau of Statistics (PBS). The federal body that takes Pakistan’s census and runs its main surveys of households, labour, prices and the economy. The chapter on Pakistan’s statistical system teaches how it is organised.
• The Council of Common Interests (CCI). The council of the federal and provincial governments under the Constitution. It approves the results of each census before they are used.
• NADRA, the National Database and Registration Authority. The body that issues identity cards and keeps the registration of births and deaths.

The story in plain words

1. People have always counted things a ruler needs to know, such as how many people, how much grain and how many soldiers.
2. Statistics began as those numbers about a state, and the word itself comes from the Latin for “state”.
3. Over time it also became a method, a set of ways to collect numbers, sum them up and draw conclusions from them.
4. The numbers can be about qualities, like a person’s province, or about quantities, like a person’s income.
5. Some numbers only name things, some put them in order, and some can be added and divided. Which kind you have decides what you may do with them.
6. You get data either by collecting them yourself or by using what someone else collected.
7. A country collects its biggest set of data in a census, and checks the census with a smaller sample survey.

1. Statistics in two senses, numbers and a method

Start with a sentence of the kind a newspaper prints. Pakistan’s population in the 2023 census was 241,499,431, and 60.65 per cent of people aged ten and above were literate. Both numbers in that sentence come from the census report, and both are statistics.¹ They are facts about many people, expressed as numbers. Now think of the work behind them. Someone had to decide what to ask, train more than 120,000 enumerators, record the answers, check them, add them up and decide how far to trust the total.¹ That work is also statistics. The word has two meanings, and a good answer gives both.

In the plural sense, statistics are numerical facts about a group, collected in an orderly way and set beside each other for comparison. The population of each province, Pakistan’s literacy rate, the wheat crop of each year and the prices in the consumer price index are statistics in this sense. Merriam-Webster gives this meaning as “a collection of quantitative data”.⁴

In the singular sense, statistics is the science of collecting, organising, presenting, analysing and interpreting data, and of drawing conclusions from them when the data are uncertain. The same dictionary gives this meaning as “a branch of mathematics dealing with the collection, analysis, interpretation, and presentation of masses of numerical data”.⁴ This whole subject teaches statistics in the singular sense. Its methods are what turn a pile of forms into a sentence like that one.

The word itself shows where the subject began. It came into English from the German Statistik, the “study of political facts and figures”, which goes back through New Latin to the Latin status, meaning state.⁴ Statistics began as the facts a government needed about its people, its land and its wealth. That is still one of its largest uses, and it is why a federal exam includes it.

What makes a number a statistic

Not every number is a statistic in the plural sense. A single fact about one person, “Ali is 23 years old”, is not. Five things turn numbers into statistics. They are about an aggregate, a group of many people or things, not one. They are affected to a marked extent by many causes, not one. They are counted or estimated to a reasonable standard of accuracy. They are collected in a systematic way for a stated purpose. They are placed beside each other so they can be compared. The average age of the 241.5 million people counted in 2023, set beside the average age in 2017, meets all five. Ali’s age meets none of them.

This is why two recent MCQs asked what statistics deals with, and gave “Aggregative facts” as one option against “Individuals”, “Particular facts” and “Isolated items”.⁷⁸ Statistics deals with aggregates.

2. What statistics is used for

The importance of statistics is easiest to see in the work it does, so take the users one by one. Each is a field where decisions are made about many people or many things at once, and where guessing would be costly.

The government is the oldest user. Pakistan’s Constitution says the seats of the National Assembly “shall be allocated to each Province and the Federal Capital on the basis of population in accordance with the last preceding census officially published”.² So the census decides how many members each province sends to the Assembly, Punjab 173 of the 326 general and women’s seats under Article 51 today.² The government also plans schools, hospitals, roads and power from statistics of population and need, sets its budget from statistics of revenue and spending, and watches inflation through the price indices of PBS. The chapters of Part-II on official statistics teach these in detail.

Business and industry use statistics to forecast demand, to set prices, and above all to control quality. A factory making medicine or cement cannot test every tablet or every bag, so it tests a sample and judges the whole batch by it. Banks and insurance companies use statistics to price risk. A market research firm asks a sample of customers and plans a product from their answers.

Agriculture uses statistics to estimate each year’s crops and to test new seeds and fertilisers. A new variety of wheat is grown beside the old one on many plots, and a test decides whether its higher yield is real or only chance. The design of such field trials is a whole head of this syllabus, the design of experiments.

Medicine and public health test new drugs and vaccines by comparing a group that receives them with a group that does not. They track the spread of disease, and they measure a country’s health through its birth, death and infant mortality rates.

Economics measures national income, unemployment, poverty and prices, and tests its theories against the data. Psychology, education and sociology measure attitudes, abilities and social conditions through surveys and tests.

How statistics extends scientific knowledge

CE-2025 asked how statistics “can help the extension of scientific knowledge”.⁷ The answer is that statistics is how science learns from evidence that is never perfect. A scientist states an idea, collects data, and then must decide whether the data support the idea or could have come about by chance. Measurements vary, samples differ, and no two plots of wheat or two patients are alike. Statistics gives the rules for that decision. It tells the scientist how large a sample to take, how to design the experiment so its answer is fair, and how much confidence to place in the result. Without it, every claim would rest on the scientist’s impression. With it, two scientists who disagree can look at the same data and the same test, and see the same answer.

3. What statistics cannot do

A good answer on the importance of statistics also says where it stops. Its limits follow from what it is.

First, statistics deals with groups, not individuals. The literacy rate of a district says nothing certain about any one person in it. A doctor who reads that a drug cures 90 per cent of patients still cannot say whether it will cure the next one.

Second, it deals with what can be counted or measured. Honesty, love or the beauty of a poem cannot be measured directly. Statistics can study them only through something that can be recorded, such as a score on a test or an answer on a scale from 1 to 5, and the result is only as good as that stand-in.

Third, its results are true on average and in the long run, not in every case. A statement such as “rainfall in July averages so many millimetres” will be wrong for many single Julys. That is why a CE-2024 MCQ asked whether statistical results are “Absolutely correct”, “Not true”, “True on average” or “Universally true”, with “True on average” the answer.⁶

Fourth, statistics can be misused. Numbers chosen with care can support almost any claim, by picking a convenient year, leaving out an awkward group or comparing things that are not alike. The method protects only those who use it honestly and read it carefully. Many later chapters show the misuse beside the proper use.

Fifth, only someone trained can use it safely. A table of averages handed to someone who does not know what a sample is, or why a difference might be chance, is more likely to mislead than to inform.

4. Data and variables

Every statistical inquiry starts with data, and data are always the values of some variable. Think of a census enumerator at a door in Multan. For each person in the household she records age, sex, marital status, whether the person can read and write, and the level of schooling. Each of those is a variable, because it varies from person to person. The values she writes down, “34”, “female”, “married”, “literate”, “matric”, are the data.

Variables come in two kinds, and the first question about any variable is which kind it is.

A qualitative variable records a quality, a category the person or thing falls into. Sex, province, religion, marital status and the fuel used for cooking are qualitative. Their values are names, not amounts, even when they are coded as numbers on a form, such as 1 for Punjab and 2 for Sindh. Some books call a qualitative variable an attribute.

A quantitative variable records an amount, a value that is a number by its nature. Age, income, the number of rooms in a house, the yield of a plot and the price of a kilogram of sugar are quantitative.

Quantitative variables split again, and this split matters in every chapter on distributions. A discrete variable can take only separate values with gaps between them, usually whole numbers you get by counting. The number of children in a family, the number of rooms, and the number of defective bulbs in a box are discrete. A family can have 3 children or 4, never 3.6. A continuous variable can take any value in a range, at least in principle, and you get it by measuring. Height, weight, time, temperature and the yield of a plot are continuous. A child’s height can be 112.3 centimetres or 112.31, as finely as the ruler allows.

One test settles most cases. Ask whether the value is counted or measured. Counting gives a discrete variable, measuring a continuous one. Figure 1 sets the kinds out as a tree.

A variable that is continuous in nature is often recorded as if it were discrete. Age is continuous, since time runs smoothly, but the census records “age as of the last birthday”, a whole number.¹ Money is measured to the paisa, but income is usually recorded in whole rupees. The variable is still continuous. The recording has only been rounded, and the methods for continuous data still apply.

5. The four scales of measurement

Knowing that a variable is qualitative or quantitative is not enough. You also need to know what its numbers mean, because that decides what you may do with them. In 1946 the American psychologist S. S. Stevens set out the answer that statistics still uses. He defined measurement, “in the broadest sense”, as “the assignment of numerals to objects or events according to rules”, and he showed that different rules give four different scales.³ Each scale adds one more thing the numbers can tell you. Figure 2 draws them as a staircase.

On a nominal scale the numbers or names only label categories. You can say whether two things are the same or different, and nothing more. A cricketer’s shirt number, the code for a province, a person’s blood group and a CNIC number are nominal. Stevens’s own example was the numbers on football players’ shirts.³ It makes no sense to average shirt numbers or to say that province 4 is twice province 2.

On an ordinal scale the categories also have an order. You can say that one is greater or better than another, but not by how much. A student’s position in class, the grades A, B and C, the rank of a civil servant, and the five answers of a Likert scale from “strongly disagree” to “strongly agree” are ordinal. The gap between first and second in a class need not equal the gap between second and third.

On an interval scale the order is there and equal differences mean equal amounts, but the zero is chosen by convention and does not mean “none”. Temperature in degrees Celsius is the standard example. The rise from 10 °C to 20 °C is the same amount of heat as the rise from 20 °C to 30 °C, so differences make sense. But 0 °C is only the point where water freezes, not the absence of heat, so 20 °C is not “twice as hot” as 10 °C. Calendar years are interval too. The year 2000 is not twice as late as the year 1000.

On a ratio scale there is also a true zero, which means none of the thing at all, so ratios make sense as well. Height, weight, income, age, the number of children and the yield of a plot are on a ratio scale. A farm of 20 acres is twice the size of one of 10, and a family with no income has none at all.

The scales are cumulative, as Stevens put it. Everything that can be done with nominal data can be done with ordinal data, and so on up the staircase, with the ratio scale allowing everything.³ A CE-2025 MCQ turned on this. “Ranking scale also includes the properties of ________ scale”, with the options nominal, interval, ratio and all of these.⁷ A ranking (ordinal) scale has the nominal scale’s property of sorting into categories, and adds order. It does not have the interval scale’s equal steps. The answer is nominal.

6. How to classify a variable and name its scale

In the hall, a question may give you a list of variables and ask you to classify them, or an MCQ may ask for one. Use these steps in this order, and write down the answer to each.

1. Say exactly what is recorded. “Income” is not enough. Is it monthly household income in rupees, or an income group such as “below Rs 50,000”? The same idea recorded two ways can fall on two different scales.
2. Ask whether the values are categories or amounts. Categories make the variable qualitative, amounts make it quantitative. A number used only as a code, such as a roll number, is a category.
3. For a quantitative variable, ask whether it is counted or measured. Counted means discrete, measured means continuous.
4. Ask what the numbers allow. If they only say same or different, the scale is nominal. If they also give an order, it is ordinal. If equal differences are equal amounts, it is at least interval. If zero means none of the thing, so that “twice as much” makes sense, it is ratio.
5. Check with one test sentence. Try “A is twice B” and “the average of these is”. If “twice” makes sense, the scale is ratio. If only “the average” makes sense, it is interval. If only “A comes before B” makes sense, it is ordinal. If none does, it is nominal.

Step 4 is the one candidates rush. A qualitative variable is always nominal or ordinal. A quantitative variable is usually interval or ratio. Most quantitative variables a CSS question gives you, such as income, yield, weight, age and counts, are ratio, and temperature in Celsius and calendar years are the common interval examples.

7. Why the scale decides what you may work out

The scale is not a label for its own sake. It decides which summary of the data has a meaning, and a summary without meaning is a common way to mislead. Stevens’s paper lists the statistics each scale permits.³ For nominal data you may count the cases in each category and find the mode, the most common category. For ordinal data you may also use the median and percentiles, which depend only on order. For interval data you may use the mean and the standard deviation, which need equal differences. Only for ratio data may you use the coefficient of variation, which compares a spread to the mean and so needs a true zero.

Take temperature, and see what goes wrong when the scale is ignored. Suppose a winter day in Lahore is 15 °C and a summer day is 30 °C. In Celsius the second number is twice the first, and it is tempting to say the summer day is twice as hot. But Celsius has an arbitrary zero. On the Kelvin scale, whose zero is the true absence of heat, the two days are 288.15 K and 303.15 K. The ratio is $303.15/288.15=1.05$, so the summer day holds about 5 per cent more heat energy, not 100 per cent more. Figure 3 draws the two scales side by side.

The same caution applies to the Likert scales that surveys use. A five-point scale codes “strongly disagree” as 1 up to “strongly agree” as 5. These codes are ordinal. Averaging them to get “3.4” treats the step from “disagree” to “neutral” as equal to the step from “agree” to “strongly agree”, which the respondents never said. Researchers often do average them, and argue that with many questions the error is small. In an exam answer, call a Likert item ordinal, as the CE-2026 MCQ expected, and if you average one, say that you are treating it as interval.⁸

Here is the rule that follows. Before you compute a summary, name the scale and check that the summary is allowed on it. The mode is safe for any data, the median for ordinal and above, the mean for interval and above, and ratios only for ratio data.

8. Where data come from, primary and secondary

Data reach an inquiry in one of two ways. Primary data are collected first-hand by the investigator for the purpose at hand. A researcher who asks 300 farmers in Sahiwal about their wheat yield, and PBS when it takes the census, are collecting primary data. Secondary data were collected by someone else for another purpose and are used again. A student who takes the literacy rate from the census report, or a journalist who quotes the State Bank’s inflation figure, is using secondary data.

The distinction is about who collected the data and why, not about the data themselves. The census figures are primary data for PBS and secondary data for everyone who uses them afterwards.

Each kind has its strengths. Primary data fit the purpose exactly, since the investigator chose what to ask and how, and the investigator knows how accurate they are. But they cost time and money. Secondary data are cheap and quick, and often cover more people or more years than any one researcher could. But they may not fit the purpose, their definitions may differ from yours, and their accuracy is harder to judge.

So before using secondary data, ask four questions of it. Who collected it, and are they reliable? For what purpose, and does their definition match yours? When, and is it recent enough? How, and was the method sound? The census definition of a literate person, someone who “can read a newspaper and write a simple letter in any language with understanding and can make simple calculation”, is an example of what you must check before comparing literacy rates from two sources.¹ A survey that counted anyone who can sign their name would give a higher rate.

Secondary data come from published sources, such as the reports of PBS, the Pakistan Economic Survey, the State Bank’s bulletins and the reports of the UN and the World Bank. They also come from unpublished records, such as the registers of hospitals, schools, police stations and companies.

9. How primary data are collected, and choosing a method by the size of the inquiry

There are several standard ways of collecting primary data. Each suits a different kind of inquiry.

In direct personal investigation, the investigator goes to each person and collects the data in person. The data are accurate and the investigator can explain each question. But it is slow and costly, so it suits only a small inquiry in a small area.

In indirect oral investigation, the investigator asks third parties who know the facts, such as witnesses, neighbours or officials, instead of the people themselves. It is used when the people concerned cannot or will not answer, as in an inquiry into an accident or into drug addiction. Its accuracy depends on the honesty and knowledge of the witnesses.

In questionnaires filled by respondents, the questions are sent by post, by email or through a website, and each respondent fills them in. It is cheap and covers a wide area. But only people who can read and are willing will answer, and those who answer may differ from those who do not. The census’s self-enumeration portal of 2023 was a questionnaire of this kind.¹

In schedules filled by enumerators, trained enumerators carry the list of questions, called a schedule, from house to house, ask each question and record the answer. It suits a large inquiry, reaches people who cannot read, and gives uniform answers if the enumerators are well trained. It costs more than a mailed questionnaire. Pakistan’s censuses and most PBS surveys use this method.

Through local correspondents, agents in each area, such as a newspaper’s district reporters or an agriculture department’s field staff, send regular estimates. It is cheap and quick, but rough, since each correspondent judges in their own way.

In registration, the law requires an event to be recorded when it happens, such as a birth, a death, a marriage or the sale of a vehicle. Over time the register becomes a complete record. NADRA’s registration of births and deaths is the national example, and the chapter on demography discusses how complete it is.

Choosing by the size of the inquiry

CE-2024 asked which methods you would use “when the field of inquiry is (i) small, (ii) fairly large and (iii) very large”, with “due regard to accuracy, labor, cost and time”.⁶ The answer follows from the methods’ strengths. For a small field, a few villages or one factory, direct personal investigation is best, because it is the most accurate and the cost stays low when the field is small. For a fairly large field, a district or a province, use schedules filled by trained enumerators, perhaps on a sample rather than everyone, or questionnaires where the respondents can read. For a very large field, a whole country, use enumerators with schedules across the country for a census, or a well-designed sample survey that reaches every region at far lower cost, with registration and local correspondents for continuing records. The exercise on CE-2024’s question works this question through with you.

A census and a sample are the two great choices in every inquiry, whatever its method. A census studies every member and has no sampling error, but it is slow and costly, and its size makes its own mistakes, such as people missed or counted twice. A sample studies a part, chosen by chance, and is quicker, cheaper and often more accurate per person, since fewer, better-trained staff do the work. The chapters on sampling in Part-II teach how to choose a sample and how to measure the error it brings.

10. A worked case, the variables of the 2023 census

The census form of 2023 asked each household about its members and its house. Here are ten of the variables it recorded, as the census report defines them, classified by the steps for naming a variable’s scale.¹

Take three rows slowly, because they show the reasoning. Level of education looks like it could be a number, years of schooling, and if the census had recorded years it would be a ratio scale. But it recorded levels, and the step from primary to middle is not the same amount of learning as the step from intermediate to a bachelor’s degree. So it is ordinal. Age is measured, a continuous variable, but the census writes it as “age as of the last birthday”.¹ The scale is still ratio, since an age of zero means just born and a person of 40 has lived twice as long as one of 20. The number of persons in a household is counted, so it is discrete. Its average for Pakistan was 6.33 in 2023, a number no household can have, which is no problem. The mean of a discrete variable need not be one of its possible values.¹

Now see what the scales allow the census report to say. For province, sex and fuel it can only give counts and percentages, and the mode, such as “firewood was used by 52.72 per cent of households”. For education it can also give the median level. For age and household size it can give means and spreads, and say that the average household was 6.33 persons.¹

11. Your turn with help, CE-2024’s question on collecting data

Here is CE-2024 Q.2(a), for 10 marks.⁶

What methods you employed in the collection of statistical data when the field of inquiry is (i) small, (ii) fairly large and (iii) very large, if you are to pay due regard to accuracy, labor, cost and time.

Work it with me, one step at a time. Before you read each answer, write your own.

Step 1. What does the question ask you to balance? Write the four things it names.

Step 2. For a small field, say one village or one factory, which method and why?

Step 3. For a fairly large field, say a district or a province, which methods and why?

Step 4. For a very large field, a whole country, which methods and why?

Step 5. How would you end the answer?

12. Variables and inquiries to classify on your own

Try each before you open its answer.

1. Classify each variable as qualitative or quantitative, discrete or continuous where it applies, and name its scale. (a) The blood group of a donor. (b) The number of cars sold by a showroom in a month. (c) The time a student takes to finish a paper. (d) A student’s position in class. (e) The temperature in Jacobabad at noon, in °C. (f) A household’s monthly electricity bill in rupees.

2. A survey asks shopkeepers, “How satisfied are you with the new tax system? 1 Very unsatisfied, 2 Unsatisfied, 3 Neutral, 4 Satisfied, 5 Very satisfied.” A report says the average answer was 2.6. What scale are the answers on, and what should the report have said instead?

3. Is the CNIC number of a citizen a quantitative variable? Explain in two sentences.

4. A researcher uses the literacy rate from the 2023 census in a study of her own district. Is she using primary or secondary data, and what should she check before comparing it with a rate from an NGO’s survey?

5. A university wants to know how its 400 final-year students rate its teaching. Which method of collection would you use, and would you take a census or a sample? Give one reason for each choice.

13. The 2023 census, a count checked by a sample

Pakistan’s seventh census, of 2023, shows every idea of this chapter at work on one inquiry, the largest the country makes. Its own report sets out how it was done.¹

It began with a decision about method. A Census Advisory Committee of demographers recommended a digital census with every structure geo-tagged, a single questionnaire, and counting each person at their usual place of residence, the method PBS calls de jure, used in all of Pakistan’s earlier censuses.¹ The other method counts people wherever they are found on census night. The committee judged it too costly in money and staff for Pakistan. The CCI approved the plan, the work plan and the questionnaire at its 49th meeting on 13 January 2022.¹

The data were collected by two of the methods of primary collection at once. First, from 20 February to 10 March 2023, a self-enumeration web portal let households fill in the questionnaire themselves. About 2.6 million people did so, roughly one person in every 93.¹ Then came the main field operation. Houses and structures were listed from 1 to 10 March, and from 12 March about 121,000 trained enumerators, carrying 120,000 tablets, went from house to house, filling in the census form as a schedule. The field work was extended four times at the provinces’ request and ended on 30 April, and in some districts on 21 May, “mainly due to under-enumeration”.¹ Figure 4 lays the dates out.

A census tries to count everyone, but no census succeeds completely. Some people are missed, and some are counted twice. So the census was checked by a sample survey, the Post Enumeration Survey, held from 8 to 19 July 2023.¹ Its sample was 2,500 enumeration blocks in 48 districts, chosen “using stratified random sampling where strata within provinces were made on the basis of similar characteristics like growth rates, population etc.”, with a relative margin of error of 1.1 per cent.¹ To keep the check independent, its enumerators were given blocks they had not counted in the census. The final results were compiled “by factoring in the findings of the Post Enumeration Survey”, and the CCI approved them unanimously at its 50th meeting on 5 August 2023.¹

That sequence is the lesson. A census, the largest primary inquiry, was itself tested by a sample, because a careful sample of 2,500 blocks can measure the errors of a count of 241.5 million that no one can recount in full. Censuses have used samples in another way too. In 1981 and 1998, questions on disability were asked only on a “long form” given to a sample of households, which is why the 2017 census, which dropped the long form, had no figure on disability.¹

The results then became data for everyone else. The count of 241,499,431 people, the sex ratio of 106.12 males to every 100 females, the average household of 6.33 persons and the literacy rate of 60.65 per cent are primary data for PBS and secondary data for every ministry, researcher and student who uses them.¹ Under Article 51 of the Constitution, the count decides the provinces’ seats in the National Assembly.²

14. What students mix up

A statistic and statistics. In the plural, statistics are numerical facts about a group. In the singular, statistics is the science. Later in this subject, “a statistic” in the singular also has a third meaning, a number worked out from a sample, such as a sample mean. The chapter on descriptive and inferential statistics teaches it.

Qualitative and discrete. A discrete variable is still quantitative. It is counted, like the number of children. A qualitative variable is a category, like sex, even when it is coded 1 and 2.

Discrete and continuous by the recording. Age recorded in whole years looks discrete, but age is continuous. Decide by the nature of the variable, counted or measured, not by how many decimals were written.

Interval and ratio. The test is the zero. If zero means none of the thing, the scale is ratio. Temperature in Celsius and calendar years have conventional zeros and are interval.

Ordinal codes treated as amounts. A Likert code of 4 is not twice a code of 2. The numbers only keep the order.

Primary and secondary by the data. The same data are primary for the body that collected them and secondary for everyone else. What decides it is who collected them and for what purpose.

A census and a sample survey. A census covers every member of the population. A sample survey covers a part chosen to stand for the whole. “Survey” alone does not say which.

15. What FPSC has asked

• CE-2025 Q.2(b), 10 marks. “Define Statistics. Discuss the importance of study of statistics by giving examples. How it can help the extension of scientific knowledge?”⁷ The model answer at the end of the chapter covers it.
• CE-2024 Q.2(a), 10 marks. The methods of collecting data for a small, fairly large and very large field of inquiry, with regard to accuracy, labour, cost and time.⁶ The exercise on CE-2024’s question works it.
• Part-I. “A study based on complete enumeration is known as” (census survey) and “Statistical results are” (true on average) in CE-2024.⁶ “Statistics deals with” (aggregative facts) and “Ranking scale also includes the properties of ________ scale” (nominal) in CE-2025.⁷ “Which of the following best describes what statistics deals with?” (aggregative facts) and “A Likert scale is a” (ordinal variable) in CE-2026.⁸

16. Facts to carry into the hall

• Statistics, plural. Numerical facts about an aggregate, affected by many causes, collected systematically for a purpose and placed in relation to each other.
• Statistics, singular. The science of collecting, organising, presenting, analysing and interpreting data, and drawing conclusions under uncertainty.
• Origin of the word. German Statistik, the study of political facts and figures, from Latin status, state.⁴
• Limits. Deals with aggregates, not individuals. Only with what can be measured. True on average. Can be misused. Needs trained hands.
• Variables. Qualitative (categories) or quantitative (amounts). Quantitative are discrete (counted) or continuous (measured).
• Scales (Stevens, 1946). Nominal (same or different), ordinal (order), interval (equal differences, conventional zero), ratio (true zero). Cumulative.³
• Permitted summaries. Mode for nominal. Median for ordinal and above. Mean and standard deviation for interval and above. Coefficient of variation for ratio.³
• Primary data are first-hand, secondary data are someone else’s. Check who, why, when and how.
• Methods of collection. Direct personal investigation, indirect oral investigation, questionnaires by respondents, schedules by enumerators, local correspondents, registration.
• Census 2023. 241,499,431 people. Self-enumeration portal 20 February to 10 March 2023, about 2.6 million entries. Field enumeration from 12 March, ended 30 April (some districts 21 May), 120,929 enumerators trained, 120,000 tablets. Post Enumeration Survey 8 to 19 July 2023, 2,500 blocks in 48 districts by stratified random sampling. Approved by the CCI on 5 August 2023.¹

17. What loses marks on this topic

• Defining statistics in one sense only. Give both the plural and the singular sense, each with an example.
• Listing uses without examples. “Statistics is used in business” earns little. “A cement factory tests a sample of bags from each batch and judges the batch by it” earns the mark.
• Calling a coded category quantitative. A province coded 1 to 4 is nominal.
• Calling Celsius temperature ratio. It is interval, because its zero is conventional.
• Naming methods of collection without fitting them to the field. CE-2024 asked for a method for each size of field, judged on accuracy, labour, cost and time. Answer for each size, with reasons.
• Writing in everyday words. The 2018 examiner’s complaint applies here most of all, since this topic is mostly “explain”. Use the terms of this chapter exactly.⁵

18. A worked answer for CE-2025 Q.2(b)

Define Statistics. Discuss the importance of study of statistics by giving examples. How it can help the extension of scientific knowledge? (10)

Give it about eighteen minutes. Here is the answer laid out as it should stand in the answer book.

Definition. The word statistics is used in two senses. In the plural, statistics are numerical facts about a group, affected by many causes, collected systematically for a stated purpose and placed side by side for comparison, for example the population of each province in the 2023 census. In the singular, statistics is the science of collecting, organising, presenting, analysing and interpreting numerical data, and of drawing valid conclusions from them in the presence of uncertainty. Its two branches are descriptive statistics, which summarises data, and inferential statistics, which draws conclusions about a population from a sample.

Importance, with examples.

• Government and planning. Under Article 51(5) of the Constitution, National Assembly seats are allocated to provinces on the basis of the last census. Budgets, schools, hospitals and the price indices of PBS all rest on statistics.
• Economics. National income, inflation, unemployment and poverty are measured statistically, and policy is judged against them.
• Business and industry. Demand is forecast, and quality is controlled by testing samples, as a pharmaceutical firm tests tablets from each batch.
• Agriculture. Crop estimates guide imports and procurement, and field trials decide whether a new wheat variety truly yields more.
• Medicine and public health. New drugs are tested against control groups, and a nation’s health is tracked through birth, death and infant mortality rates.

Its limits, briefly. It studies aggregates, not individuals. Its results are true on average. It can be misused by those who choose data to suit a conclusion.

How it extends scientific knowledge. Science advances by testing ideas against evidence, and evidence always varies. Statistics supplies the rules for that test. It decides how large a sample is needed, designs the experiment so that its comparison is fair, measures the variation by chance, and states how much confidence the conclusion deserves. A new drug or seed is accepted only when a test shows that its effect is larger than chance could explain. In this way statistics turns observation into knowledge that other scientists can check and repeat.

Conclusion. Statistics is both the record of a society in numbers and the method by which it reasons from incomplete evidence. That is why every modern state, and every science that measures, depends on it.

What to remember from this chapter

• Statistics means numerical facts about a group (plural) and the science of collecting and reasoning from data (singular). The word comes from the Latin for “state”.
• It is used wherever decisions are made about many people or things, in government, economics, business, agriculture, medicine and science.
• It studies aggregates, not individuals, and its results are true on average.
• A variable is qualitative (categories) or quantitative (amounts), and quantitative is discrete (counted) or continuous (measured).
• The four scales are nominal, ordinal, interval and ratio. The scale decides which summaries mean anything.
• Primary data are collected first-hand, secondary data are someone else’s. The method of collection should fit the size of the inquiry.
• Pakistan’s 2023 census collected data by self-enumeration and by enumerators, and was checked by a stratified sample survey before the CCI approved it.

Check yourself

Answer each in your own words before you open the answer. The last one comes from the chapter on how to sit the paper.

1. Give the two meanings of the word statistics, with an example of each. (The section on statistics in two senses.)

2. Why is “Ali is 23 years old” not a statistic in the plural sense? (The section on statistics in two senses.)

3. Give three limits of statistics. (The section on what statistics cannot do.)

4. Is the number of rooms in a house discrete or continuous, and what is its scale? (The section on data and variables.)

5. Why is temperature in degrees Celsius an interval scale and not a ratio scale? (The section on the four scales.)

6. Which averages may you use for ordinal data, and why not the mean? (The section on why the scale decides.)

7. How did PBS check the 2023 census, and why was a sample good enough to check a count of the whole country? (The section on the 2023 census.)

8. For a fairly large inquiry, a province, which method of collecting data would you choose and why? (The section on collecting primary data.)

9. How many questions must you answer from each section of Part-II, and what happens to a third from one section? (The chapter on how to sit the paper.)