Collecting data and the statistical cycle

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Apuntes de la lección

The Statistical Cycle

The statistical cycle is a four-stage process: pose a question, collect data, process and represent data, and interpret results.
The cycle is iterative: interpreting results often leads to new questions, starting the cycle again.
Each stage must be planned carefully to ensure the data collected can actually answer the original question.
A clear, focused problem definition is essential before any data is collected.

Primary and Secondary Data

Primary data is collected first-hand by the person or team carrying out the investigation.
Secondary data is data that has already been collected by someone else, such as from a website, book, or government report.
Primary data is more reliable for your specific question but can be time-consuming and expensive to collect.
Secondary data is quicker and cheaper to obtain but may not perfectly match your question or may be outdated.

Populations and Samples

A population is the entire group of people or items that you want to understand.
A sample is a subset of the population, selected to represent the whole.
Collecting data from the whole population is called a census; it is often impossible, too costly, or too time-consuming.
Sampling provides insights when measuring the entire population is infeasible.
A good sample should be representative of the population, meaning it reflects its characteristics accurately.

Random Sampling

Random sampling gives every member of the population an equal chance of being selected.
Random sampling helps to avoid bias and produce a representative sample.
Methods include using random number generators, lottery methods, or picking names from a hat.
A large sample size alone does not guarantee accuracy if the sampling method is biased.

Sources of Bias

Bias occurs when a sample is not representative of the population.
Bias can arise from the sampling method, the way questions are asked, or who chooses to respond.
A famous example is the 1936 Literary Digest poll, which predicted a Republican win but was biased because names were taken from magazine subscription lists and telephone directories, which over-represented Republicans.
Even a very large sample can be deeply flawed if it is biased.
To reduce bias, use random sampling and ensure questions are neutral.

Designing a Questionnaire

Questions should be clear, concise, and easy to understand.
Avoid leading questions that suggest a particular answer, e.g. 'Don't you agree that...?'
Avoid biased questions that make one answer seem more acceptable, e.g. 'How much do you enjoy our excellent service?'
Avoid double-barrelled questions that ask about two things at once, e.g. 'Do you like the food and the staff?'
Provide balanced response options and avoid overlapping categories.
Pilot the questionnaire on a small group to check for problems before full data collection.

Processing and Representing Data

After collection, data must be processed: cleaned, organised, and summarised.
Common representations include tables, bar charts, pie charts, and averages.
The choice of representation should suit the type of data and the question being asked.
Processing may involve calculating statistics such as the mean, median, mode, or range.

Interpreting Results and Drawing Conclusions

Interpretation involves analysing the processed data to answer the original question.
Conclusions should be based on evidence and should acknowledge any limitations or possible bias.
It is important to distinguish between correlation and causation; a relationship does not prove cause.
Results from a sample are estimates for the population and may have a margin of error.
For example, sample counts in Singapore elections give an indicative result with a 4% margin of error at a 95% confidence interval, but are not official results.

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Preguntas de práctica

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1.Which of the following best describes a population in statistics?
Easy
- AThe complete set of all people or items with the characteristics being studied
- BA small group selected from a larger group
- CThe number of people living in a country
- DThe data collected from a sample
2.Which of the following is an advantage of using a sample instead of a census?
Easy
- AIt is cheaper and faster to collect data
- BIt gives exact information about every individual in the population
- CIt eliminates all bias
- DIt requires no sampling method
3.A sample should be representative of the population to allow valid conclusions.
Easy
True or false?
4.A researcher wants to know the average height of students in a school. She selects every 10th student from an alphabetical list of all students. What type of sampling method is this?
Medium
- ASystematic sampling
- BSimple random sampling
- CStratified sampling
- DQuota sampling
5.Which of the following are potential sources of bias in a sample? (select all that apply)
Medium
- AUsing a convenience sample
- BUsing a random number generator to select participants
- CNon-response bias
- DUndercoverage of certain groups
- EUsing a large sample size
6.In the 1936 Literary Digest poll, more than two million people responded, but the prediction was wrong. What was the main reason for the error?
Medium
- AThe sample was biased because it was drawn from magazine subscribers and telephone directories
- BThe sample size was too small
- CThe poll was conducted too early
- DThe respondents lied about their voting intention
7.Match each term with its correct definition.
Medium
- Population
- Sample
- Census
- Bias
- The complete set of all items or individuals of interest
- A subset of the population selected for study
- Data collected from every member of the population
- A systematic error that makes results unrepresentative
8.Put the stages of the statistical cycle in the correct order.
Medium
- Pose a question
- Collect data
- Process and represent data
- Interpret results

Más preguntas

9.Which of the following are stages of the data handling cycle? (select all that apply)
Medium
- Apose a question
- Bcollect data
- Cprocess and represent data
- Dinterpret data
- Epublish results in a newspaper
10.Match each type of data with its correct description.
Easy
- Primary data
- Secondary data
- Population
- Sample
- Data collected by the person who will use it
- Data that has already been collected by someone else
- The whole group that is being studied
- A subset of the population selected for study
11.Put the stages of the data handling cycle in the correct order.
Medium
- Interpret data
- Pose a question
- Process and represent data
- Collect data
12.Which of the following are advantages of using a sample rather than a census? (select all that apply)
Medium
- ALower cost
- BFaster data collection
- CIt is always more accurate
- DIt can be used when measuring the whole population is impossible
- EIt eliminates all bias
13.A researcher wants to estimate the average height of students in a school. She selects every 10th student from an alphabetical list of all students. What type of sampling is this?
Medium
- ASimple random sampling
- BSystematic sampling
- CStratified sampling
- DQuota sampling
14.A census collects data from every member of the population.
Easy
True or false?
15.A factory produces a batch of 800 light bulbs. The quality manager checks 50 bulbs from the batch. What is the population in this investigation?
Easy
- AThe 50 bulbs checked
- BAll 800 bulbs in the batch
- CAll light bulbs made by the factory that day
- DThe quality manager
16.A supermarket manager wants to investigate how long customers wait in the checkout queue at different times of day. Which of the following best describes the population in this investigation?
Medium
- AThe checkout staff on duty
- BThe length of the checkout queue at various times
- CAll customers who use the supermarket
- DThe tills in the supermarket
17.A researcher wants to study the electrical conductivity of copper. She takes repeated measurements from several pieces of copper wire. What is the population in this investigation?
Medium
- AThe pieces of copper wire she tested
- BAll copper wire in the world
- CThe overall behaviour of electrical conductivity of copper over infinitely many measurements
- DThe researcher's measuring instrument
18.A hospital wants to test a new 'quit smoking' programme on 100 patients to predict how well it would work nationwide. In this study, what is the sample?
Medium
- AAll people in the country who might use the programme
- BThe 100 patients in the test group
- CThe doctors running the programme
- DThe success rate of the programme
19.Which of the following is a reason for using a sample rather than a census?
Easy
- ASampling always gives exact results
- BSampling is cheaper and faster than a census
- CSampling avoids the need to define a population
- DSampling removes all possible bias
20.A statistician wants to estimate the average weight of all apples in an orchard. She picks 40 apples and weighs them. Which term describes the 40 apples?
Medium
- APopulation
- BCensus
- CSample
- DSuperpopulation
21.Which of the following are reasons why sampling is often preferred to a census? (select all that apply)
Medium
- ALower cost
- BFaster data collection
- CIt is sometimes impossible to measure the whole population
- DIt always gives a perfectly accurate result
- EIt removes the need for a sampling frame
22.Which of the following statements about populations and samples are correct? (select all that apply)
Medium
- AA population includes all people or items with the characteristics one wishes to understand.
- BA sample is a subset of a population.
- CA sample is always larger than the population.
- DA representative sample should reflect the whole population.
23.A sample is always representative of the population, no matter how it is selected.
Easy
True or false?
24.In the 1936 Literary Digest poll, the sample was biased because the names were obtained from magazine subscription lists and telephone directories.
Medium
True or false?
25.A supermarket manager wants to investigate how long customers wait in the checkout queue at different times of day. She records the waiting times of 200 customers on a Saturday morning. What is the population in this investigation?
Medium
- AAll customers who use the supermarket's checkouts
- BThe 200 customers whose waiting times were recorded
- CAll customers who shop on a Saturday morning
- DThe total time all customers spend waiting in queues
26.A researcher wants to study the electrical conductivity of copper. She takes repeated measurements from several pieces of copper wire. What is the sample in this study?
Medium
- AThe repeated measurements she takes
- BAll pieces of copper wire in the world
- CThe average conductivity of copper
- DThe pieces of wire she did not measure
27.A factory produces a batch of 2000 items. A quality inspector checks a sample of 80 items. What fraction of the population is in the sample? Give your answer as a fraction in simplest form.
Hard
- A\frac{1}{25}
- B\frac{1}{20}
- C\frac{1}{40}
- D\frac{2}{5}
28.What is the name of the subset of individuals selected from a statistical population?
Easy
- AA census
- BA sample
- CA superpopulation
- DA parameter
29.What is the term for a sample in which every member of the population has an equal chance of being selected?
Easy
- AA biased sample
- BA quota sample
- CA random sample
- DA convenience sample
30.Which of the following best describes primary data?
Medium
- AData collected by someone else and used by you
- BData collected by you for the first time for your own investigation
- CData that has already been summarised in a table
- DData that has been published in a textbook
31.Which of the following best describes secondary data?
Medium
- AData collected by someone else, which you then use
- BData collected directly from the population by you
- CData collected using a random sample
- DData that has not yet been processed
32.A questionnaire asks: 'Do you agree that the new school timetable is a great improvement?' What is the main problem with this question?
Medium
- AIt is a leading question
- BIt is too long
- CIt asks for a numerical answer
- DIt uses a random sample
33.A factory produces a batch of 4000 items. A quality inspector checks a sample of 200 items. What percentage of the population is in the sample?
Medium
- A2%
- B5%
- C10%
- D20%
34.A population has 3600 members. A sample of 180 members is taken. What fraction of the population is in the sample? Give your answer in simplest form.
Medium
- A1/10
- B1/20
- C1/18
- D1/40
35.Which of the following is a source of bias in a sample?
Easy
- ASelecting names at random from a complete list
- BSelecting names from a telephone directory only
- CSelecting every 10th name from an alphabetical list
- DSelecting names using a random number generator
36.In the 1936 Literary Digest poll, more than two million people responded but the prediction was wrong. What was the main reason for the error?
Medium
- AThe sample was too small
- BThe sample was biased because names came from magazine subscription lists and telephone directories
- CThe respondents were chosen at random
- DThe poll was carried out too late
37.Which of the following are types of data? (select all that apply)
Medium
- APrimary data
- BSecondary data
- CRandom data
- DBiased data
- EQualitative data
38.What is the population in this investigation?
Medium
- AThe 200 customers recorded
- BAll customers who use the supermarket
- CThe checkout staff
- DThe manager
39.Put the stages of the data handling cycle in the correct order, starting with the first stage.
Medium
- Pose a question
- Collect the data
- Process and represent the data
- Interpret the results and draw conclusions
40.Match each term about collecting data with its correct description.
Medium
- Primary data
- Secondary data
- Population
- Sample
- Data collected first-hand by the person carrying out the investigation
- Data that already exists, having been collected by someone else
- The whole group of people or items being studied
- A subset of individuals selected from the population
41.What term describes a sample in which every member of the population has an equal chance of being selected?
Medium
42.Complete the sentence about the statistical cycle.
Easy
The statistical cycle starts with posing a ____ and ends with interpreting the results.
43.A school has 1200 students. A sample of 60 students is selected. What percentage of the population is in the sample?
Medium
44.Complete the sentence about sampling.
Medium
A sample in which every member of the population has an equal chance of being selected is called a ____ sample.
45.Complete the sentence about the 1936 Literary Digest poll.
Medium
The 1936 Literary Digest poll predicted a Republican win, but the prediction was wrong because the sample was ____.
46.A school has 1500 students. A sample of 90 students is selected. What percentage of the population is in the sample?
Medium
47.A factory produces a batch of 2000 items. A quality inspector checks a sample of 80 items. What fraction of the population is in the sample? Give your answer as a decimal.
Medium
48.In the 1936 Literary Digest poll, what was the main reason the prediction was wrong?
Medium
49.A school has 800 students. A sample of 40 students is selected. What percentage of the population is in the sample?
Medium
50.A factory produces a batch of 5000 items. A quality inspector checks a sample of 250 items. What percentage of the population is in the sample?
Medium
51.A population has 2400 members. A sample of 120 members is taken. What fraction of the population is in the sample? Give your answer in simplest form.
Medium
52.A hospital tests a new 'quit smoking' programme on a sample of 150 patients. This sample is 5% of the population of patients who could potentially use the programme. How many patients are in the population?
Medium
53.Explain why a sample is often used instead of a census when collecting data.
Hard
54.Explain why the 1936 Literary Digest poll, despite having more than two million responses, gave a wrong prediction.
Hard
55.A school wants to find out how many hours per week students spend on homework. The headteacher decides to survey 100 students chosen at random from the school register. Explain why choosing the students at random is important.
Medium
56.A school has 1200 students. A sample of 60 students is selected. What percentage of the population is in the sample?
Medium
- A6 %
- B4 %
- C5 %
- D7 %
57.A school has 1500 students. A sample of 90 students is selected. What percentage of the population is in the sample?
Medium
- A6 %
- B5 %
- C7 %
- D8 %
58.A factory produces a batch of 2000 items. A quality inspector checks a sample of 80 items. What fraction of the population is in the sample? Give your answer as a decimal.
Medium
- A2.04
- B-0.96
- C1.04
- D0.04
59.A school has 800 students. A sample of 40 students is selected. What percentage of the population is in the sample?
Medium
- A5 %
- B7 %
- C4 %
- D6 %
60.A factory produces a batch of 5000 items. A quality inspector checks a sample of 250 items. What percentage of the population is in the sample?
Medium
- A5 %
- B4 %
- C7 %
- D6 %
61.A population has 2400 members. A sample of 120 members is taken. What fraction of the population is in the sample? Give your answer in simplest form.
Medium
- A1.05
- B2.05
- C-0.95
- D0.05
62.A hospital tests a new 'quit smoking' programme on a sample of 150 patients. This sample is 5% of the population of patients who could potentially use the programme. How many patients are in the population?
Medium
- A3600
- B3300
- C3000
- D2700

Apuntes de repaso

## Collecting Data and the Statistical Cycle **The statistical cycle** is a four-stage process: 1. **Pose a question** – be clear about what you want to find out. 2. **Collect data** – gather information from a population or sample. 3. **Process and represent** – organise the data (tables, charts, averages). 4. **Interpret** – draw conclusions and answer the original question. ### Primary vs secondary data - **Primary data** – collected first-hand for your own question (e.g. a survey you run). - **Secondary data** – already collected by someone else (e.g. government statistics, past records). ### Populations and samples - **Population** – *all* people or items with the characteristics you wish to understand. - **Sample** – a subset of the population, selected to estimate characteristics of the whole population. - A sample should be **representative** of the population. - Sampling is often cheaper and faster than a **census** (measuring every member of the population). Sometimes a census is impossible (e.g. testing every item in a batch would destroy it). ### Random sampling - **Random sampling** gives every member of the population a known chance of being chosen. - It helps produce an **unbiased** sample. - Example: drawing names from a hat, or using a random number generator. ### Bias and questionnaires - **Bias** occurs when the sample or the questions systematically favour certain answers. - **Leading questions** suggest a particular answer (e.g. “Don’t you agree that…?”). - **Biased samples** can arise from using a poor sampling frame (e.g. only asking people in a gym about exercise). - The 1936 Literary Digest poll predicted a Republican win because its sample came from magazine subscribers and telephone directories — heavily biased towards Republicans. The sample was huge but deeply flawed. ### Designing a good questionnaire - Ask clear, neutral questions. - Avoid leading or loaded wording. - Give balanced response options. - Keep it short and relevant. ### Drawing conclusions - Check whether the sample is representative. - Consider possible sources of bias. - Conclusions should be justified by the data and the way it was collected. **Key idea:** A large sample does not fix a biased sample. Representativeness matters more than size.

Population: All people or items with the characteristics one wishes to understand.
Sample: A subset of individuals selected from a population, used to estimate characteristics of the whole population.
Census: Recording data from the entire population.
Primary data: Data collected first-hand for the specific question being investigated.
Secondary data: Data that has already been collected by someone else.
Random sampling: Selecting a sample in such a way that every member of the population has a known chance of being chosen.
Bias: A systematic distortion that makes a sample or result unrepresentative of the population.
Leading question: A question that suggests or encourages a particular answer.
Representative sample: A sample that reflects the characteristics of the population.
Statistical cycle: The process of posing a question, collecting data, processing and representing it, and interpreting the results.

Tarjetas

What are the four stages of the statistical cycle?

Pose a question, collect data, process and represent the data, then interpret the results.

What is primary data?

Data that you collect yourself, first-hand, for the specific question you are investigating.

What is secondary data?

Data that has already been collected by someone else, such as from a website, book or government report.

Give one advantage of using secondary data.

It is often quicker and cheaper to obtain than collecting primary data yourself.

Give one disadvantage of using secondary data.

You cannot be sure how it was collected, so it may be biased, out of date or not quite fit your question.

What is a population in statistics?

The whole group of people or items that you want to find out about.

What is a sample?

A subset of individuals selected from a population, used to estimate characteristics of the whole population.

Why do statisticians usually take a sample instead of a census?

Sampling costs less and collects data faster; measuring the whole population is often impossible.

What is a census?

A survey that records data from every member of the population.

What makes a sample representative?

It reflects the whole population well, so the characteristics of the sample are similar to those of the population.

What is random sampling?

Selecting a sample in a way that gives every member of the population an equal chance of being chosen.

Why is random sampling used?

It helps produce an unbiased, representative sample and avoids the researcher choosing who takes part.

What is bias in sampling?

A systematic error that makes the sample unrepresentative of the population.

What is a leading question?

A question worded in a way that suggests or encourages a particular answer.

Give an example of a leading question.

For example: 'Don't you agree that the new school rules are unfair?' — it pushes the respondent towards agreeing.

What is a biased question?

A question that unfairly favours one answer or one group, so the data collected will not be neutral.

Give two tips for designing a good questionnaire.

Use clear, simple language and avoid leading or biased questions; keep questions relevant to the investigation.

What is a common problem with asking people questions face-to-face?

People may give the answer they think the interviewer wants, which can bias the results.

Why can a very large sample still be biased?

If the method of selecting the sample is biased, a large sample just repeats the same bias — size does not fix a flawed selection.

What was the main flaw in the 1936 Literary Digest election prediction?

Its sample was drawn from magazine subscription lists and telephone directories, which were heavily biased towards Republicans.

What does it mean to interpret data?

To explain what the data shows, draw conclusions and relate the findings back to the original question.

Why should you define the population before collecting data?

A clear definition of the population tells you exactly who or what your sample should represent.

What is a superpopulation?

A larger, sometimes hypothetical group from which the observed population is treated as a sample.

Give an example of a superpopulation.

If a quit-smoking programme is tested on 100 patients, the superpopulation is everybody in the country who could be given the treatment.

What is acceptance sampling used for?

To decide whether a batch or production lot of material meets the required specifications.

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