Foundations: Data and Statistics

Data

In statistics, data are collections of observations or experimental results. Data may be gathered by observing existing conditions (observational studies) or by conducting experiments where conditions are controlled.

• Observation: The researcher observes behavior or conditions without intervention. Example: Observing traffic patterns.

• Experiment: The researcher sets up a condition or treatment to study its effect. Example: Testing whether a vaccine is effective.

• Big Data: Refers to the study and analysis of very large datasets, often across populations or categories, to inform decisions.

Statistics is the process of planning a study, collecting and representing data, analyzing it, and communicating results. It includes:

• Experimental or study design

• Data collection

• Data representation (e.g., graphs)

• Data analysis (e.g., summaries, hypothesis tests)

• Presentation and interpretation of results

Key idea: Statistics is not just calculation; explaining what results mean is essential.

Population, Sample, Parameter, and Statistic

Population vs. Sample

Understanding the difference between population and sample is fundamental:

• Population: The entire group the study aims to describe. Example: All women in the United States.

• Sample: A subset of the population that is actually observed or measured.

Census vs. Survey

• Census: Attempts to study the entire population.

• Survey/Sample Study: Studies only a sample from the population.

Parameter vs. Statistic

• Parameter: A numerical summary describing a population.

• Statistic: A numerical summary describing a sample.

Example: If the average height for the population is 63.7 inches (parameter), and a sample of 200 women has an average height of 63.0 inches (statistic), the difference is expected due to sampling variability.

• Reasons a statistic may differ from its parameter:

  • Sample size is not large enough

  • Ordinary random sample-to-sample fluctuation

  • The parameter itself may be incorrect

Types of Data

Quantitative vs. Categorical Data

Data can be classified as quantitative or categorical:

• Quantitative: Numeric values with meaningful interpretation (e.g., age, weight, height).

• Categorical (Qualitative): Labels or groups used to classify observations (e.g., yes/no, gender, favorite color).

• Important exception: Numeric-looking values (e.g., Group 1, Group 2) are categorical if arithmetic operations are not meaningful.

Discrete vs. Continuous Quantitative Data

• Discrete: Separate, countable outcomes. Examples: Die result, coin outcome, integers.

• Continuous: Values that can vary across a continuum and can be measured with increasing precision. Examples: Length, height, blood volume.

• Memory trick: Discrete = counted; Continuous = measured.

Discrete data can be infinite if countable (e.g., all integers). Continuous data are uncountable (e.g., all numbers in the interval 0 to 1).

Test: If infinite, ask if you could build a counting system to reach every value. If yes, still discrete; if no, continuous.

Designing a Statistical Study

Stages of Study Design

A statistical study is organized into three broad stages: prepare, analyze, and conclude.

Step 1 - Prepare

• Define the purpose of the study

• Specify the population

• Choose a sample and sample size

• Determine participant selection and sampling method

• Consider voluntary participation and possible bias

• Define variables and measurement units clearly

Best practice: Collect detailed information first; you can categorize later. Units must be stated clearly.

Step 2 - Analyze

• Create graphs or visual representations

• Compute descriptive statistics (e.g., mean)

• Identify unusual observations or outliers

• Apply inferential procedures (e.g., confidence intervals, hypothesis tests)

Outlier: A value far from other observations. Investigate plausibility, decide whether to retain or omit, and report handling transparently.

Step 3 - Draw Conclusions

• Interpret findings beyond calculations

• Assess bias, publishability, and practical importance

Sampling Methods

Sampling Methods Overview

Sampling methods describe how observations are selected from the population.

Stratified vs. Cluster: Stratified = some from every subgroup; Cluster = everyone from selected groups.

Snowball sampling: A subtype of convenience sampling, useful for hard-to-reach populations.

Cluster sampling: Chosen for practicality or when the cluster is the study target. Tradeoff: depth on one group vs. breadth across population.

Voluntary Response and Bias

Voluntary participation can produce a biased sample, as respondents may differ systematically from non-respondents.

• Example: Engaged students are more likely to answer an optional survey.

• Statistically, mandatory participation is better, but ethically, most surveys must be voluntary.

• Report voluntary design and potential bias; mitigate by reaching opposing groups if possible.

Sources of Error

Error Types

Core distinction: Sampling error can occur even with proper random sampling. Nonsampling error arises from other problems.

Nonsampling Error Sources

• Respondent side: Forgetting or lying about weight; height is more reliably reported.

• Researcher side: Leading questions, political bias, poor design.

Statistical vs. Practical Significance

Concepts

• Statistical significance: Is the observed difference unlikely to be explained by random variation alone?

• Practical significance: Is the effect large enough to be meaningful or useful in the real world?

A result can be statistically significant without being practically important, especially with a very large sample.

Exam-worthy idea: Statistically significant does not automatically mean important or useful.

Misuse: Large samples can produce statistically significant results for trivial effects. Practical significance is subjective.

Example: A vaccine reduces infection from 20% to 15%. Statistically significant, but practical importance depends on context.

Self-Check Questions

• Is "random positive integer" discrete or continuous, and why?

• What kind of significance does a 0.3% improvement with p < 0.001 on 2 million participants have?

• Name the sampling method for: every 12th name (systematic), men and women separately (stratified), one dorm and all residents (cluster).

• Sampling error, nonsampling error, or nonrandom sampling error: sample mean differs from population mean with no design flaws?

• Why is "underweight/normal/overweight" a worse survey item than "weight in pounds"?

• Give one reason a statistic can differ from its parameter even when nothing was done wrong.

Additional info: Simple random sampling was inferred as standard for introductory statistics, though not explicitly defined in the lecture transcript.