Data and its collection Revision Notes

In statistics, a population refers to the complete set of individuals, items, or data points that share a common characteristic and are the focus of a study. For example, if a researcher wants to study the average height of all students in a school, the population is every student enrolled in that school. The population can be finite or infinite depending on the context, but it always represents the entire group about which conclusions are to be drawn.

• Population includes every member relevant to the research question.
• It forms the basis for any statistical analysis or inference.

A sample is a smaller subset selected from the population. It is used when it is impractical or impossible to collect data from the entire population. For instance, instead of measuring the height of every student in a large school, a researcher might measure only 100 students chosen carefully to represent the whole school. The sample should ideally reflect the characteristics of the population to allow valid conclusions.

• Samples are used to save time, cost, and effort.
• The quality of conclusions depends on how well the sample represents the population.

A census involves collecting data from every member of the population. This method provides complete information without sampling error but can be very time-consuming and expensive. For example, a government census aims to gather data about every resident in a country. While a census gives the most accurate data, it is often only feasible for small populations or when precise data is essential.

• Census = data collection from the entire population.
• No sampling error but can be costly and slow.
• Used when complete accuracy is required or population size is manageable.

A representative sample accurately reflects the characteristics of the population from which it is drawn. This means the proportions of key features (such as age, gender, or income level) in the sample closely match those in the population. For example, if 60% of a population are female, a representative sample should also have about 60% females. Representative samples allow statisticians to make reliable inferences about the population without surveying everyone.

• Ensures sample results can be generalized to the population.
• Reduces bias and improves accuracy of conclusions.
• Achieved through careful sampling methods.

Deciding whether to conduct a census or use a sample depends on factors like population size, resources, and required accuracy. A census is ideal for small populations or when complete data is necessary. Sampling is preferred for large populations or when time and cost constraints exist. When sampling, ensuring the sample is representative is crucial to avoid misleading results.

• Census: complete data, costly, time-consuming.
• Sample: less costly, quicker, but must be representative.
• Trade-off between accuracy and practicality.

Sampling is the process of selecting a subset of individuals from a larger population to collect data and make inferences about the whole population. Choosing an appropriate sampling method is crucial to obtain representative and unbiased results. The main sampling methods are simple random sampling, stratified sampling, systematic sampling, and quota sampling. Each method has distinct characteristics and requirements, especially regarding the use of a sampling frame.

• Sampling helps to study populations efficiently without surveying everyone.
• Different methods suit different types of populations and research goals.

Simple random sampling involves selecting individuals so that every member of the population has an equal chance of being chosen. To do this fairly, a sampling frame—a complete list of all individuals in the population—is essential. The sampling frame ensures that the selection process can be random and unbiased.

For example, if a school has 200 students and you want a random sample of 20, you need a list of all 200 students (the sampling frame). Then, you could use a random number generator to pick 20 unique student numbers from this list.

• Sampling frame: a list of all population members.
• Necessary for random selection to avoid bias.
• Example: Using a student register to select participants randomly.

Stratified sampling divides the population into distinct groups or strata based on a characteristic (e.g., gender, age group). Then, a random sample is taken from each stratum proportionally or equally. This method requires a sampling frame that includes information about which stratum each individual belongs to.

For instance, if a population has 60% females and 40% males, and you want a sample of 50, you might randomly select 30 females and 20 males from the sampling frame that lists all individuals and their gender.

• Population divided into strata based on characteristics.
• Sampling frame must identify strata membership.
• Ensures representation from each group.

Systematic sampling selects individuals at regular intervals from an ordered list of the population, which is the sampling frame. For example, if you want a sample of 25 from 500 individuals, you calculate the sampling interval k = 500 ÷ 25 = 20. Then, you randomly select a starting point between 1 and 20 and pick every 20th individual thereafter.

This method depends on having a complete and ordered sampling frame to avoid bias and ensure coverage across the population.

• Select every k-th individual from an ordered list.
• Sampling frame must be complete and ordered.
• Example: Choosing every 20th name from a list of 500.

Quota sampling involves selecting participants non-randomly to fill predetermined quotas based on characteristics such as age, gender, or occupation. Researchers select individuals who meet the criteria until the quota is met, often using convenience sampling methods.

Unlike other methods, quota sampling does not require a sampling frame because selection is based on meeting quotas rather than random selection from a complete list. However, this can introduce bias since not all individuals have an equal chance of selection.

For example, a researcher may decide to interview 30 males and 30 females at a shopping centre by approaching people until the quotas are filled.

• Participants chosen to meet quotas for specific groups.
• No complete list of population needed.
• Selection is non-random and may introduce bias.

Sampling bias occurs when the process used to select a sample causes certain members of the population to be more likely included than others. This leads to a sample that does not accurately represent the entire population, which can distort survey results and conclusions.

For example, if a survey about students' travel methods only includes those who arrive at school early, it may miss the travel habits of students who arrive later, resulting in biased data. Sampling bias reduces the reliability and validity of survey findings.

• Sampling bias means the sample is not representative of the population.
• It causes survey results to be misleading or inaccurate.
• Bias arises from the sampling method, not from random chance.

Certain sampling techniques inherently favour some individuals over others. For instance, convenience sampling, where participants are chosen based on ease of access, often leads to bias because it excludes those who are less accessible.

Selecting the first 10 individuals arriving at school is a form of convenience sampling. Early arrivals might share characteristics such as living nearby or having similar schedules, which are not representative of the whole student body.

Other biased methods include voluntary response sampling, where only those who choose to participate are included, potentially skewing results.

• Convenience sampling can exclude parts of the population.
• Time-based selection (e.g., first arrivals) can favour certain groups.
• Biased samples do not reflect population diversity.

Imagine conducting a survey to find out how students travel to school. If the survey selects the first 10 students who arrive each morning, the sample may be biased.

Those arriving early might live closer to school and walk, while students arriving later might use buses or cars. Thus, the sample overrepresents walkers and underrepresents other travel methods.

This bias means the survey results will not accurately reflect the travel habits of all students.

• First 10 arrivals likely share similar travel modes.
• Sample excludes late arrivals who may travel differently.
• Results will overestimate some travel methods and underestimate others.

To reduce bias, sampling methods should aim for randomness or stratification to ensure all groups are fairly represented.

Simple random sampling involves selecting individuals so that every member of the population has an equal chance of being chosen.

Stratified sampling divides the population into groups (strata) and samples proportionally from each, ensuring representation of key subgroups, such as year groups or travel modes in the school example.

• Random sampling gives equal selection chance to all.
• Stratified sampling ensures representation of subgroups.
• Avoid convenience or time-based sampling to reduce bias.

When reviewing survey results, consider how the sample was chosen. If the sampling method is biased, the results may not be reliable or generalisable to the whole population.

For example, a transport survey using the first arrivals sample should be treated cautiously, as it likely misrepresents the true travel patterns.

Understanding sampling bias helps in critically assessing data quality and making informed decisions based on survey findings.

• Check sampling method for potential bias.
• Question representativeness of the sample.
• Use knowledge of bias to interpret survey results carefully.

Sampling is a fundamental process in statistics used to select a subset of individuals from a larger population for study. A well-chosen sample allows us to make inferences about the entire population without surveying everyone, which is often impractical.

A random number table is a tool containing sequences of digits arranged randomly. It helps select samples impartially, avoiding bias. By using random numbers, each member of the population can be given an equal or defined chance of selection, depending on the sampling method.

This note focuses on three sampling methods—simple random, systematic, and stratified sampling—and how to use a random number table to implement them effectively. It also covers how to assess if the sample represents the population or its subgroups (strata).

• Sampling saves time and resources compared to a census.
• Random number tables ensure unbiased selection.
• Different sampling methods suit different situations.

Simple random sampling means every individual in the population has an equal chance of being selected. To use a random number table for this:

1. Assign a unique number to each member of the population, usually starting from 01, 02, 03, and so on.

2. Decide the sample size needed (e.g., 5 individuals). 3. Choose a starting point in the random number table at random (e.g., row 3, column 5). 4. Read numbers sequentially from the table, ignoring any numbers outside the population range or duplicates, until the sample size is reached. 5. Select the individuals corresponding to these numbers for the sample.

• Ensures fairness and equal chance for all.
• Avoids selection bias.
• Requires a complete and numbered sampling frame.

Systematic sampling selects members at regular intervals from an ordered list. To use a random number table:

1. Number the population from 1 to N (total population size).

2. Calculate the sampling interval k = N ÷ n, where n is the desired sample size. Round down if necessary. 3. Use the random number table to select a random starting number between 1 and k. 4. Select every kth individual after the starting point until the sample size is reached. For example, if N = 100 and n = 10, then k = 10. If the random start is 4, select individuals numbered 4, 14, 24,..., 94.

• Simpler and quicker than simple random sampling.
• Requires an ordered list of the population.
• Risk of bias if there is a hidden pattern matching the interval.

Stratified sampling divides the population into distinct subgroups called strata (e.g., gender, age groups, regions). Samples are then taken from each stratum proportionally or equally to ensure representation.

To use a random number table for stratified sampling:

1. Divide the population into strata and number individuals within each stratum separately. 2. Determine the sample size for each stratum, often proportional to its size in the population. 3. For each stratum, use the random number table to select the required number of individuals as in simple random sampling. 4. Combine the selected individuals from all strata to form the full sample.

• Ensures all subgroups are represented.
• Reduces sampling error compared to simple random sampling.
• Requires knowledge of strata and their sizes.

A sample is representative if it accurately reflects the characteristics of the population or specific strata. To determine representativeness:

Check if the sample includes members from all relevant strata in correct proportions (for stratified sampling).

For simple random and systematic samples, verify that no subgroup is systematically excluded or underrepresented due to the selection method or sampling frame limitations. Consider the sample size; very small samples may not capture population diversity well. Look for potential biases introduced by the sampling method or practical constraints (e.g., non-response). If a sample is not representative, conclusions drawn may be misleading or invalid.

• Representative samples reduce bias and improve inference accuracy.
• Stratified sampling helps ensure representativeness of subgroups.
• Always evaluate sampling method and sample composition critically.

Surveys are a common method for collecting data from a group of people. The type of questions asked in a survey greatly influences the quality and type of data collected. Two main types of questions are open and closed questions. Understanding their roles helps in designing surveys that meet specific research goals.

Open questions allow respondents to answer in their own words without restrictions. This flexibility encourages detailed, descriptive responses that can reveal opinions, feelings, or explanations.

For example, an open question might be: "What do you like most about your school?" This invites a variety of answers, providing rich qualitative data.

Advantages of open questions include the ability to gather in-depth information and unexpected insights that closed questions might miss. They are useful when the researcher wants to explore ideas or understand reasons behind behaviours.

• Encourage detailed, descriptive answers.
• Provide qualitative data rich in detail.
• Useful for exploring new topics or opinions.

Closed questions restrict respondents to a set of predefined answers, such as 'Yes' or 'No', multiple choice options, or rating scales.

For example, a closed question might be: "Do you attend school regularly? (Yes/No)" This limits answers to two options, making responses easy to categorise and analyse.

Advantages of closed questions include straightforward data analysis, as responses can be quickly counted and compared. They are efficient for collecting quantitative data and useful when specific information is required.

• Provide fixed response options.
• Enable easy and quick quantitative analysis.
• Efficient for collecting specific information.

Open questions can be time-consuming to answer and analyse because responses vary widely and require qualitative coding or interpretation. This can introduce subjectivity and make summarising results challenging.

Closed questions may limit respondents’ answers, potentially missing important information or nuances. They can also lead to biased results if the provided options do not cover all possible answers or if respondents feel forced to choose an option that does not fully represent their view.

• Open questions require more time and effort to analyse.
• Closed questions may restrict responses and miss nuances.
• Poorly designed closed questions can bias results.

The choice depends on the survey’s purpose. If the goal is to gather detailed opinions or explanations, open questions are appropriate. If the aim is to collect data that can be easily quantified and compared, closed questions are better.

Sometimes, surveys combine both types to balance depth and ease of analysis. For example, a survey might use closed questions to gather demographic data and open questions to explore attitudes.

• Match question type to survey objectives.
• Use open questions for exploratory or qualitative data.
• Use closed questions for quantitative or specific data.
• Combine both types when appropriate.

Data in statistics represents information collected for analysis. Correctly categorising data is essential because it influences how data can be summarised, visualised, and analysed. The main categories are qualitative and quantitative data, with quantitative data further divided into discrete and continuous types.

• Data classification affects choice of graphs and statistical tests.
• Misclassification can lead to incorrect conclusions.

Qualitative data, also known as categorical data, describes qualities or characteristics that cannot be measured numerically. Instead, it categorises items based on attributes or labels.

Examples include eye colour (blue, brown, green), types of transport (bus, car, bicycle), or favourite music genres (rock, jazz, classical).

Qualitative data can be nominal, where categories have no natural order, or ordinal, where categories have a meaningful order (e.g., satisfaction ratings: low, medium, high).

• Describes categories or qualities.
• Cannot be meaningfully added or averaged.
• Examples: gender, nationality, brand names.

Quantitative data represents numerical values that measure quantities. This data type allows arithmetic operations such as addition, subtraction, and averaging.

Quantitative data is further divided into discrete and continuous types based on the nature of the values it can take.

• Expressed numerically and measurable.
• Enables calculation of statistics like mean and standard deviation.

Discrete data is a subset of quantitative data that can only take specific, separate values, often whole numbers. These values are countable and there are gaps between possible values.

For example, the number of students in a classroom can be 20 or 21 but not 20.5. Another example is the number of cars passing a checkpoint in an hour.

Discrete data often arises from counting objects or events.

• Countable values with no intermediate values.
• Examples: number of pets, number of goals scored.
• Usually integers.

Continuous data is quantitative data that can take any value within a given range, including fractions and decimals. It results from measuring rather than counting.

Examples include height (e.g., 1.75 metres), temperature (e.g., 23.4°C), or time taken to complete a race (e.g., 12.56 seconds).

Continuous data can be recorded to varying degrees of precision depending on the measuring instrument.

• Measurable values with infinite possibilities within a range.
• Examples: weight, distance, time.
• Can be rounded or approximated depending on precision.