📌 Snapshot
- How to turn survey data into a decision: the seven project steps, a toothpaste case study and solved numericals.
- Why the objective comes first, and how it fixes the target group, the type of data and the tools you pick.
- The toothpaste study worked through: mean and standard deviation of income, mean spend, pie angles and percentages.
- The glossary terms CUET lifts into one line questions: sampling error, census, class mark, seasonality and more.
📖 Detailed Notes
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 8.
A factory, no data, one question: what would you advise?
X is a young entrepreneur who wants to open a toothpaste factory. X has money and enthusiasm, but no idea whether people want gel or plain paste, how much a family spends in a month, or what makes a family switch brands. You are the adviser. You cannot guess. You need numbers, and you need a method for getting them.
This chapter is that method. It adds no new formula. It shows how the tools you already know (collecting, organising and presenting data, averages, spread and correlation) fit together in one project, from a vague question to a clear recommendation. CUET asks about the order of the steps, the sample project's figures and the glossary terms at the end.
The big idea: every later step is decided by the first one
A project has seven steps, and they run in a fixed order. The first step quietly decides all the others, because the objective tells you who to ask, what to ask and which tool will answer it.
| Step | What you do | Tool or idea behind it |
|---|---|---|
| 1. Identify the problem | State clearly what you want to study | Examples: sale of a car, mobile phone, shoe polish or soap; a water or electricity problem of an area; consumer awareness about rights |
| 2. Choose the target group | Decide whose behaviour the question is about | Cars: middle and higher income groups. Soap: all rural and urban consumers. Safe drinking water: both urban and rural population |
| 3. Collect data | Primary, secondary or both, using the objective as the guide | Questionnaire or interview schedule; census or sample |
| 4. Organise and present | Tabulate, then draw suitable diagrams | Tables, bar diagrams, pie diagrams |
| 5. Analyse and interpret | Calculate the average, the variability and any relationship | Mean, standard deviation, correlation |
| 6. Conclude | Draw meaningful conclusions; predict future prospects and suggest growth or policy steps where possible | Reasoning from the results |
| 7. Bibliography | List every secondary source used | Magazines, newspapers, research reports |
Why the order cannot change. You cannot choose a target group before you know the objective. You cannot design a questionnaire before you know the target group. You cannot calculate an average before you have data. Each step uses the output of the one before, which is why CUET loves "which step comes first" and "which step comes after analysis" questions.
Why the bibliography lists only secondary sources. Data you collected yourself came from your own survey, so there is nothing to cite. A bibliography credits the magazines, newspapers and reports someone else produced.
Collecting data: the choices you make in Step 3
Your objective decides whether you need primary data (collected first hand), secondary data (already published) or both. Primary collection uses a questionnaire or an interview schedule, delivered by personal interview, post, phone or email. A questionnaire sent by post must carry a covering letter that explains the purpose of the inquiry, because a stranger will not answer questions without knowing why you are asking.
| Feature | Primary data | Secondary data |
|---|---|---|
| Source | Collected by you, first hand | Already published by someone else |
| Fits your question | Exactly, because you design the questions | Only as far as the published data happen to fit |
| When it is used | When no suitable data exist | When time, money and manpower are short and the information is easily available |
| Needs a bibliography entry | No | Yes |
You also choose how many to ask. A census takes observations on every individual in the population. A sample survey takes them on a representative subset. If you use a sample, the sampling method must suit the population. Random sampling gives every individual an equal chance of being picked, and a published table of two-digit random numbers (the last appendix of the chapter) lets you pick units without any personal bias.
| Feature | Census method | Sample survey method |
|---|---|---|
| Who is observed | Every individual in the population | A representative set (the sample) |
| Cost and time | High | Lower |
| Main risk | Cost and effort | Sampling error |
Two errors are easy to confuse.
| Error | Meaning | Example |
|---|---|---|
| Sampling error | The numerical difference between the estimate from the sample and the true value for the population | The sample mean is ₹104; the true mean is ₹110, so the error is ₹6 |
| Non-sampling error | Errors that arise in collection due to sampling bias, non-response or mistakes in data acquisition | Half the postal questionnaires never come back |
Beyond the textbook. For the exam: a sample is a representative subset chosen by random sampling, and its error is the gap from the true value. In reality: sampling error usually shrinks as the sample grows, but non-sampling error does not. A biased question or a low response rate stays just as bad in a bigger sample. Why it matters: this is why a careful 100-household survey can beat a careless 1,000-household one, and why the covering letter matters so much.
Pause and predict: you post 100 questionnaires and only 12 come back. Which kind of error is the biggest worry?
Check your answer: non-sampling error, through non-response. The 12 who replied may differ from the 88 who did not, and no calculation on 12 answers can repair that.
The toothpaste project: what X needs to know
The adviser decides to collect primary data through a questionnaire. Six things are needed: the average monthly spend on toothpaste, the brands in demand, customers' attitude to those brands, preferences about ingredients, the major media influences on demand, and how income relates to all of these. If a tried and tested questionnaire exists for a similar study, you may reuse it after suitable changes. Otherwise you design one yourself and make sure every required item is asked.
The questionnaire has fifteen questions. They cover the family's identity, ages and size, monthly income, urban or rural location, the bread-winner's occupation, whether toothpaste is used, essential qualities of a good toothpaste, the brand used, 100 gram packs used per month, satisfaction, willingness to try a new product, the features wanted (plain, gel, antiseptic, flavoured, carries protection, fluoride) and the main sources of information (cinema, exhibitions, internet, magazines, newspapers, radio, sales representatives, television). Each question feeds a business decision: the features asked about feed the product design, and the information sources feed the advertising plan.
The simplified report is based on 100 households, 67% urban and 33% rural. Here is everything the report finds.
| Item | Finding |
|---|---|
| Age of persons (500 persons) | Below 10: 74; 10 to 20: 56; 20 to 30: 91; 30 to 40: 146; 40 to 50: 93; above 50: 40. Most persons are in the 20 to 50 group |
| Family size (100 families) | 1 to 2 members: 20; 3 to 4: 40; 5 to 6: 30; above 6: 10. Most families have 3 to 6 members |
| Monthly family income (100 households) | 0 to 10,000: 20; 10,000 to 20,000: 40; 20,000 to 30,000: 30; 30,000 to 40,000: 10. Mean ₹18,000, standard deviation ₹9,000 |
| Monthly spend on toothpaste | Mean ₹104 per household per month, standard deviation ₹35.60 |
| Occupation of the main earner | Service 30; professional 5; manufacture 10; trader 40; any other 15 |
| Preferred brand (households) | Pepsodent 20; Colgate 18; Close-up 12; Cibaca 9; OralB 7; Sensodyne 7; Aquafresh 5; Meswak 5; Pearl 4; Anchor 4; Babool 3; Promise 3; any other 3 |
| Basis of selection | Standardised marking 50; quality 45; price 35; company's brand name 35; taste 20; advertisement 15; ingredients 10; tried a new product 10; persuaded by the dentist 5 |
| Ingredient preference | Antiseptic 80; gel 70; flavoured 50; plain 40; carries protection 30; fluoride 10 |
| Media influence (families) | Television 47; newspaper 30; cinema 25; magazine 20; radio 18; sales representative 15; exhibits and stall 10 |
The concluding note of the report: most users are urban, most persons surveyed are in the middle age groups, families have 3 to 6 members, monthly income is mostly between ₹10,000 and ₹30,000, the main occupations are service and trading, and about ₹104 a month is spent on toothpaste per household. Pepsodent, Colgate and Close-up are the most preferred brands, people prefer gel or antiseptic pastes, and television is the most popular medium for reaching people.
Why only some of these columns add to 100. Each household names one brand and one occupation, so those columns total 100. The ingredient column totals 280 and the media column 165, because a family could tick more than one option.
Worked examples: from raw table to decision
Worked example: mean and standard deviation of income. Using the income table, find the mean and standard deviation by the step deviation method, taking the assumed mean $A = 20{,}000$ and class width $h = 5{,}000$ (both in ₹).
| Step | What we do | Result |
|---|---|---|
| 1 | Write the class marks and $d' = (x - 20000)/5000$ | Marks 5,000; 15,000; 25,000; 35,000. $d' = -3, -1, 1, 3$ |
| 2 | Multiply by the frequencies 20, 40, 30, 10 to get $fd'$ | $-60, -40, 30, 30$, so $\sum fd' = -40$ |
| 3 | Square $d'$ and multiply by $f$ to get $fd'^2$ | $180, 40, 30, 90$, so $\sum fd'^2 = 340$ |
| 4 | Mean $= A + \dfrac{\sum fd'}{N} \times h$ | $20000 + (-0.4)(5000) = 18000$, i.e. ₹18,000 |
| 5 | SD $= h\sqrt{\dfrac{\sum fd'^2}{N} - \left(\dfrac{\sum fd'}{N}\right)^2}$ | $5000\sqrt{3.4 - 0.16} = 5000 \times 1.8 = 9000$, i.e. ₹9,000 |
Common wrong step. Dropping the square of the mean term in Step 5. That gives $5000\sqrt{3.4} \approx 9220$ (₹9,220), which is a plausible-looking number and not the answer. Another slip is forgetting to multiply by $h$, which leaves the SD as 1.8 rupees.
Worked example: mean spend on toothpaste. The spend table has class marks 20, 60, 100, 140, 180 with frequencies 5, 20, 40, 30, 5. Take $A = 100$ and $h = 40$.
| Step | What we do | Result |
|---|---|---|
| 1 | $d' = (x - 100)/40$ | $-2, -1, 0, 1, 2$ |
| 2 | $fd' = 5(-2),\ 20(-1),\ 0,\ 30,\ 10$ | $-10, -20, 0, 30, 10$, so $\sum fd' = 10$ |
| 3 | Mean $= 100 + \dfrac{10}{100} \times 40$ | ₹104 per household per month |
Common wrong step. Reporting the assumed mean (₹100) as the answer, or forgetting the factor $h$ and writing $100 + 0.1 = 100.1$. The correction $\dfrac{\sum fd'}{N}$ is in units of class widths, so it must be multiplied by 40.
Worked example: drawing the occupation pie. Convert the occupation counts (service 30, professional 5, manufacture 10, trader 40, any other 15) into angles.
| Step | What we do | Result |
|---|---|---|
| 1 | Total frequency | 30 + 5 + 10 + 40 + 15 = 100 |
| 2 | Angle $= \dfrac{\text{frequency}}{\text{total}} \times 360^\circ$, so each household is worth $3.6^\circ$ | Service $108^\circ$; professional $18^\circ$; manufacture $36^\circ$; trader $144^\circ$; other $54^\circ$ |
| 3 | Check | $108 + 18 + 36 + 144 + 54 = 360^\circ$ |
Common wrong step. Giving every slice $72^\circ$ because there are five categories. A pie divides the circle in proportion to the frequencies, not equally. The sum check at the end catches most slips.
Worked example: persons are not households. In the age table, what percentage of persons surveyed are aged 20 to 50?
| Step | What we do | Result |
|---|---|---|
| 1 | Add the classes 20 to 30, 30 to 40 and 40 to 50 | 91 + 146 + 93 = 330 persons |
| 2 | Divide by the total number of persons, which is 500 (not 100) | 330 / 500 |
| 3 | Convert to a percentage | 66% |
Common wrong step. Dividing by 100, the number of households, which gives 330%. A percentage can never exceed 100 for a part of a whole. The age table counts persons in 100 households, so the base is 500.
Worked example: why the media bars add to more than 100. Television influenced 47 families. Is that 47% of the families, and what does the total of all media bars tell you?
| Step | What we do | Result |
|---|---|---|
| 1 | The sample has 100 families, so 47 of 100 is 47% | 47% were influenced by television |
| 2 | Add all the media counts | 47 + 30 + 25 + 20 + 18 + 15 + 10 = 165 |
| 3 | Compare with 100 | 165 is more than 100, so families named more than one medium |
Common wrong step. Treating the bars as shares that must sum to 100% and rescaling them to fit. A bar diagram of multiple-answer data is not a partition, so a pie diagram would be the wrong picture for it.
Pause and predict: X wants to know whether richer households spend more on toothpaste. Which tool from earlier chapters answers that, and which step of the project is it part of?
Check your answer: correlation between income and monthly spend, at Step 5 (analysis and interpretation). The mean and standard deviation describe each variable separately, while correlation measures the relationship between the two.
From numbers to a conclusion
Step 6 is where the work pays off. For X, the report gives a direct recommendation: aim at urban families with moderate incomes, offer a gel or antiseptic paste with a recognised quality mark, price it against Pepsodent, Colgate and Close-up, and spend the advertising money on television first. Notice that every part of the advice traces back to a table in the report. A conclusion that does not follow from the data is opinion.
Beyond the textbook. For the exam: a random sample gives every individual an equal chance of selection, drawn from a random number table. In reality: large household surveys in India, such as those run by the National Sample Survey, usually pick villages and urban blocks first and households inside them, so that the sample covers rural and urban areas in proportion. Why it matters: the 67% urban and 33% rural split in the toothpaste study is a reminder that the target group decides what a good sample looks like.
Definitions to memorise
The glossary at the end of the chapter is a rich source of one line questions. The terms below are the ones most worth knowing word for word.
| Group | Term | Meaning |
|---|---|---|
| Data collection | Statistics | The method of collecting, organising, presenting and analysing data to draw meaningful conclusions; it also means the data themselves |
| Data collection | Population | All the individuals or units for whom information is sought |
| Data collection | Informant | The individual or unit from whom information is obtained |
| Data collection | Enumerator | The person who collects the data |
| Data collection | Questionnaire | A list of questions prepared by an investigator, answered by the respondent |
| Data collection | Structured questionnaire | Closed questions with alternative answers provided to choose from |
| Variables | Variable | A quantity measuring an attribute, which can take different values |
| Variables | Attribute | A qualitative characteristic that cannot be measured |
| Variables | Discrete variable | Takes only certain values and moves by finite jumps |
| Variables | Continuous variable | Can take any numerical value |
| Classification | Class interval | The difference between the upper and lower class limits |
| Classification | Class mark (class midpoint) | The middle value of a class, equal to (upper limit + lower limit) / 2 |
| Classification | Inclusive method | An item equal to the upper or the lower limit of a class stays in that class |
| Classification | Exclusive method | An item equal to a class limit is placed in the class above or below, not in this one |
| Classification | Frequency distribution | Shows how values are spread over classes with their class frequencies |
| Classification | Tally marking | Counting with strokes, grouped in fives |
| Time and shape | Time series | Data arranged in chronological order |
| Time and shape | Seasonality | Periodicity in data variation with a period of less than one year |
| Time and shape | Cyclicity | Periodicity in data variation with a period of more than one year |
| Time and shape | Univariate, bivariate | A frequency distribution of one variable, or of two variables |
| Time and shape | Weighted average | An average in which the data points carry different weights |
Other terms in the glossary include assumed mean (an approximate value used to simplify calculation), range (maximum minus minimum), decile (a value that divides the data into ten equal parts) and percentile (one of 99 values dividing the data into hundred equal parts).
Exam traps
| Trap | Why it tempts | The truth |
|---|---|---|
| Bibliography lists primary data | You collected the data, so it feels like a source | It lists secondary sources only: magazines, newspapers and research reports |
| Choosing the target group is the first step | The target group is about "who", which feels like the start | Identifying the problem comes first, then the target group |
| Mean spend ₹18,000 and SD ₹104 | The two pairs of figures look alike | Income: mean ₹18,000, SD ₹9,000. Toothpaste spend: mean ₹104, SD ₹35.60 |
| Newspaper is the top medium | It is a familiar, trusted medium | Television is first with 47 families, then newspaper with 30 |
| Seasonality means yearly repetition | Seasons come every year | Seasonality is under one year; cyclicity is over one year |
| Sampling error is the same as non-response | Both are "survey errors" | Sampling error is the sample estimate minus the true value; non-response belongs to non-sampling error |
| Any postal questionnaire can go out as it is | The questions already explain themselves | A postal questionnaire needs a covering letter stating the purpose |
| Class mark and class midpoint differ | Two names look like two ideas | They are the same value: (upper limit + lower limit) / 2 |
Formula sheet
| Quantity | Formula | Unit | When to use |
|---|---|---|---|
| Class mark | $\dfrac{\text{upper limit} + \text{lower limit}}{2}$ | Same as data | Representing a class in a grouped table |
| Class interval | $\text{upper limit} - \text{lower limit}$ | Same as data | Finding the class width $h$ |
| Step deviation | $d' = \dfrac{x - A}{h}$ | None | Simplifying mean and SD arithmetic |
| Mean (step deviation) | $\bar{x} = A + \dfrac{\sum fd'}{N} \times h$ | Same as data | Grouped data with equal class widths |
| Standard deviation | $\sigma = h\sqrt{\dfrac{\sum fd'^2}{N} - \left(\dfrac{\sum fd'}{N}\right)^2}$ | Same as data | Spread of grouped data |
| Pie angle | $\dfrac{\text{frequency}}{\text{total}} \times 360^\circ$ | Degrees | Drawing a pie diagram |
| Percentage share | $\dfrac{\text{part}}{\text{whole}} \times 100$ | Per cent | Report observations; check the right base |
| Range | $\text{maximum} - \text{minimum}$ | Same as data | Simplest measure of spread |
60-second revision
| Topic | Remember |
|---|---|
| Seven steps | Problem, target group, collect, organise and present, analyse and interpret, conclude, bibliography |
| Data type | The objective decides primary, secondary or both |
| Postal survey | Needs a covering letter |
| Secondary data | Used when time, money and manpower are short |
| Sample | 100 households; 67% urban, 33% rural |
| Income | Mean ₹18,000; SD ₹9,000 |
| Toothpaste spend | Mean ₹104; SD ₹35.60 |
| Top brands | Pepsodent 20, Colgate 18, Close-up 12 |
| Top medium | Television, 47 families |
| Sampling error | Sample estimate minus the true value |
| Non-sampling error | Bias, non-response, acquisition mistakes |
| Seasonality / cyclicity | Under one year / over one year |
Where this chapter connects
Every step of the project calls on an earlier chapter. Step 3 uses Collection of Data for primary and secondary sources and sampling. Step 4 draws on Organisation of Data for classes and frequencies and on Presentation of Data for bar, pie and histogram diagrams. Step 5 uses Measures of Central Tendency for the mean and Correlation for the link between income and spend. When you compare prices across years, Index Numbers takes over. The opening ideas of the subject are in Introduction.
🎯 Practice MCQs
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Q1. Which is the first step in developing a statistical project?
▸ Show answer & explanation
Answer: B
The objective must be clear before anything else. Choice of the target group is step 2, collection of data step 3 and conclusions step 6, so only identifying the problem comes first.
Q2. A questionnaire sent to respondents by post must be accompanied by:
▸ Show answer & explanation
Answer: A
A postal questionnaire needs a covering letter that explains why the inquiry is made. The bibliography belongs at the end of the project report, and the other items are not required.
Q3. In the sample toothpaste project, which pair is correct for the mean and standard deviation of monthly family income?
▸ Show answer & explanation
Answer: C
The step deviation table gives mean $20000 + (-0.4)(5000) = 18000$ and SD $5000\sqrt{3.4 - 0.16} = 9000$. Option A gives the pair for toothpaste spend, option B swaps the two values, and option D lists the assumed mean and the class width.
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Q4. Match List I with List II. List I: (i) Organisation and presentation (ii) Analysis and interpretation (iii) Collection of data (iv) Bibliography. List II: (p) Mean, standard deviation, correlation (q) Bar and pie diagrams (r) Questionnaire or interview schedule (s) Secondary sources used
▸ Show answer & explanation
Answer: A
Diagrams belong to organisation and presentation, the statistical measures to analysis, the questionnaire to collection, and the list of secondary sources to the bibliography.
Q5. Assertion (A): Secondary data are usually used when time, money and manpower are short and the information is easily available. Reason (R): Secondary data are always more accurate than primary data.
▸ Show answer & explanation
Answer: C
The assertion is true. The reason is false because accuracy depends on how the data were collected, and secondary data may not fit the objective exactly. They are used for convenience and cost, not because they are more accurate.
Q6. In the age distribution of 500 persons in the sample (below 10: 74; 10 to 20: 56; 20 to 30: 91; 30 to 40: 146; 40 to 50: 93; above 50: 40), what percentage of persons are aged 20 to 50?
▸ Show answer & explanation
Answer: C
The three classes add to 91 + 146 + 93 = 330 persons, and 330 / 500 × 100 = 66%. Option D divides by 100 households instead of 500 persons, and a part of a whole can never be 330%.
Q7. In the occupation data (service 30, professional 5, manufacture 10, trader 40, any other 15, out of 100 families), the angle of the pie slice for traders is:
▸ Show answer & explanation
Answer: D
Each family corresponds to 3.6°, so traders get 40 × 3.6° = 144°. A slice of 72° would result from dividing the circle equally among five categories, and 40° and 100° ignore the scale of 360°.
Q8. The numerical difference between the estimate from a sample and the corresponding true value for the population is called:
▸ Show answer & explanation
Answer: C
This is the definition of sampling error. Non-sampling error arises from bias, non-response or mistakes in data acquisition. The class interval is the gap between class limits and the range is maximum minus minimum.
Q9. Which of the following are sources of non-sampling error? (i) Sampling bias (ii) Non-response (iii) Mistakes in data acquisition (iv) Using a sample instead of a census
▸ Show answer & explanation
Answer: C
Non-sampling error comes from sampling bias, non-response and errors in acquiring data. Using a sample instead of a census is a method choice that leads to sampling error, not a non-sampling error.
Q10. The spend table has class marks 20, 60, 100, 140 and 180 with frequencies 5, 20, 40, 30 and 5. Using the assumed mean 100 and class width 40, the mean spend per household is:
▸ Show answer & explanation
Answer: C
The values of $d'$ are -2, -1, 0, 1, 2, so $\sum fd' = -10 - 20 + 0 + 30 + 10 = 10$. Mean $= 100 + (10/100)(40) = 104$. Option A forgets to multiply by the class width and option B subtracts instead of adding.
Q11. In the sample project, 47 families were influenced by television, 30 by newspaper, 25 by cinema, 20 by magazine, 18 by radio, 15 by sales representatives and 10 by exhibits. The total of 165 for 100 families shows that:
▸ Show answer & explanation
Answer: B
The questionnaire allowed more than one option to be ticked, so a family can appear in several bars. The sample size stays 100, and nothing here suggests an error or secondary data.
Q12. A sales figure repeats its pattern every three years. This kind of periodic variation in a time series is called:
▸ Show answer & explanation
Answer: B
Periodicity with a period of more than one year is cyclicity, and a three year pattern qualifies. Seasonality has a period of less than one year, and the other two options are not types of variation.
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