Data Collection in Statistics

Introduction to the Practice of Statistics

Statistics is the science of collecting, analyzing, interpreting, and presenting data. It is essential for understanding variability in data and making informed decisions based on evidence rather than anecdote. The field of statistics provides tools to describe and understand sources of variability in populations and samples.

• Statistics: The science of learning from data, including collection, analysis, interpretation, and presentation.

• Data: Information collected from observations, counts, measurements, or responses.

• Purpose: To describe variability and make inferences about populations based on samples.

Statistics is a process involving several steps:

1. Identify the research objective.

2. Collect data needed to answer the research question.

3. Describe the data.

4. Perform inference (draw conclusions about the population).

Populations, Samples, and Individuals

In statistics, it is often impractical to study an entire population, so we use samples to make inferences about the population.

• Population: The entire group of individuals or items of interest.

• Sample: A subset of the population selected for study.

• Individual: A single member of the population or sample.

• Parameter: A numerical summary describing a characteristic of a population.

• Statistic: A numerical summary describing a characteristic of a sample.

Example: Suppose the proportion of all students on your campus who have a job is 0.849 (parameter). A sample of 250 students is taken, and 0.864 of them have a job (statistic).

Types of Variables

Qualitative (Categorical) vs. Quantitative Variables

Variables are characteristics or properties that can take on different values. They are classified as either qualitative or quantitative.

• Qualitative (Categorical) Variables: Describe qualities or categories (e.g., education level, phone type).

• Quantitative Variables: Represent numerical values (e.g., age, temperature, income).

Examples:

• Education level – Qualitative

• Today's high temperature – Quantitative

• Number of vending machines – Quantitative

• Whether a student is prepared for class – Qualitative

Discrete vs. Continuous Variables

Quantitative variables can be further classified as discrete or continuous:

• Discrete Variable: Takes on countable values (e.g., number of students in a classroom).

• Continuous Variable: Can take on any value within a range (e.g., time exercised in a day, temperature).

Examples:

• Income (in dollars) – Quantitative, usually discrete

• Grade earned in Algebra (percentage) – Quantitative, continuous

• Number of newly admitted students – Quantitative, discrete

Levels of Measurement

Variables can also be classified by their level of measurement, which determines the type of statistical analysis that is appropriate.

• Nominal: Categories with no inherent order (e.g., phone type, internet provider).

• Ordinal: Categories with a meaningful order but no consistent difference between values (e.g., education level, response to survey questions).

• Interval: Ordered categories with meaningful differences, but no true zero (e.g., temperature in Celsius).

• Ratio: Ordered categories with meaningful differences and a true zero (e.g., number of hours exercised, age in years).

Examples:

• Internet provider – Nominal

• Age (Young, Middle, Old) – Ordinal

• Age (in years) – Ratio

• Temperature (Celsius) – Interval

Observational Studies Versus Designed Experiments

Types of Statistical Studies

Statistical studies are generally classified as either observational studies or designed experiments.

• Observational Study: Researchers observe and measure characteristics without influencing the subjects.

• Designed Experiment: Researchers apply a treatment and observe its effects on the subjects.

Explanatory Variable: The variable that is manipulated or categorized to determine its effect.

Response Variable: The outcome measured in the study.

Example: In a study where children are randomly assigned to different types of music instruction and their IQ is measured, the type of music instruction is the explanatory variable and IQ is the response variable.

• Observational studies can only show association, not causation, due to potential lurking variables.

• Designed experiments can help establish causation but may have confounding variables.

Other Types of Data Collection

• Census: Collecting data from every individual in the population.

• Web Scraping (Data Mining): Using automated tools to collect large amounts of data from the internet.

Simple Random Sampling

Random Sampling Methods

To ensure valid results, individuals should be selected randomly from the population.

• Random Sampling: Every individual has an equal chance of being selected.

• Simple Random Sample: Every possible sample of a given size has an equal chance of being chosen.

Steps for Obtaining a Simple Random Sample:

1. Obtain a list (frame) of all individuals in the population and number them 1 to N.

2. Use a random number generator or table to select n individuals for the sample.

Example: To select 5 members of the House of Representatives (435 total) for a luncheon, number each member and randomly select 5 numbers.

Bias in Sampling

Types of Bias

If a sample does not accurately represent the population, it is said to be biased. There are three main sources of bias:

• Sampling Bias: When the sampling method systematically favors certain outcomes.

• Nonresponse Bias: When individuals selected for the sample do not respond.

• Response Bias: When respondents give inaccurate answers (intentionally or unintentionally).

Two types of errors in sampling:

• Nonsampling Errors: Errors not related to the act of selecting a sample (e.g., data entry errors, poorly worded questions).

• Sampling Error: The natural variation that occurs by chance because a sample, not the entire population, is surveyed.

Example: In the 1936 Literary Digest poll, sampling bias occurred because the sample was not representative of the general population (it included mainly magazine readers, car owners, and telephone users).

Example: A company surveys customers about satisfaction. If only highly satisfied customers respond, nonresponse bias may occur.

Additional info: Understanding sources of bias is crucial for interpreting the results of statistical studies and for designing better surveys and experiments.