Statistics Basics: Population, Sampling, Data Types, and Frequency Tables

Statistics: collect, analyze, interpret data to predict future tendencies.
Key sequence: Collect $\rightarrow$ Analyze $\rightarrow$ Calculate/Graph $\rightarrow$ Present $\rightarrow$ Interpret $\rightarrow$ Predict.
Population: The entire group of interest.
Sample: A subset of the population from which data is collected.
Parameter: A numerical characteristic of the population (e.g., population mean $\mu$ ).
Statistic: A numerical value calculated from sample data, used to estimate a parameter (e.g., sample mean $\bar{x}$ ).
Variable: The quantity measured for each subject (what the data represents).
Relation: Population $\rightarrow$ Sample $\rightarrow$ Parameter $\rightarrow$ Statistic $\rightarrow$ Data (variable) $\rightarrow$ Interpretation.
Caveat: A sample statistic is an estimate of the population parameter; a larger, well-chosen sample generally provides a better estimate.

Qualitative (Categorical): Non-numeric data (e.g., hair color, blood type).
Quantitative (Numeric): Numeric data (e.g., number of classes, GPA).
- Discrete: Countable values, usually integers (e.g., number of classes).
- Continuous: Measured values that can have decimals (e.g., weight, height, GPA).
Nuance: Number of correct questions is discrete; quiz score (with partial credit) can be continuous.

Goal: Sample should represent the population; bias occurs when it doesn't.
Sampling Methods:
- Simple random sampling: Every member has an equal chance.
- Stratified sampling: Divide populations into subgroups (strata) and sample from each.
- Cluster sampling: Divide into clusters, sample some clusters, survey all (or sample) within selected clusters.
- Systematic sampling: Select members at regular intervals (e.g., every 10th).
- Convenience sampling: Sample those easiest to reach (prone to bias).
Takeaway: Clearly state population and sampling method to avoid biased, non-generalizable results.

Frequency tables: Organize data by showing how many times each value/category occurs.
- Ungrouped (raw): Lists exact data values and their frequencies.
- Grouped (binned): Data grouped into intervals (bins) with a defined width.
- Bin width (w): Difference between the start of consecutive bins (e.g., $w = 4$ for bins [1-4], [5-8]).
Tallying: Quick method to count occurrences.
Cumulative Frequency: Running total of frequencies.

Relative frequency: Proportion of observations for a data value $\text{Relative frequency} = \frac{f_i}{n}$ .
Expressed as a fraction, decimal, or percentage.
Percentage: Relative frequency multiplied by 100.
Cumulative relative frequency: Running total of relative frequencies, summing to 1.0 (100%).

Use frequency/cumulative frequency tables to answer questions like "how many work no more than 5 hours" or "what proportion work at least 5 hours?"
Identify relevant data values first, then apply frequencies.

Know key terms: population, sample, parameter, statistic, variable.
Distinguish data types: qualitative vs quantitative (discrete vs continuous).
Be comfortable with frequency tables, tallying, and grouped data (bins/width).
Compute and interpret frequencies, relative frequencies, cumulative counts, proportions, and percentages.
Understand sample statistics estimate population parameters, and good sampling improves estimates.
Use a calculator; focus on conceptual understanding.