Quick Reference: Stats Basics — Variables, Data Types, Notation

Descriptive vs Inferential Statistics

• Descriptive: describes and summarizes data; examples include graphs, histograms, scatter plots, pie charts; purpose is to organize and communicate findings.
• Inferential: generalizes from a sample to a population using probability to assess uncertainty; moves from the specific to the general.

Population vs Sample; Parameters vs Statistics

• Population: all possible observations; a parameter is a population characteristic (constant).
• Sample: subset of the population; a statistic is a sample-based estimate of a population parameter.
• Notation:
  • Population: mean $\mu$, standard deviation $\sigma$, etc. (Greek letters)
  • Sample: mean $\bar{x}$, standard deviation $s$, sample size $n$ (Roman letters)
• Key idea: parameters are fixed constants (in theory) for the population; statistics are estimates from the sample.

Notation and Core Concepts

• The average of the population: $\mu$
• The average of the sample: $\bar{x}$
• Population standard deviation: $\sigma$; Sample standard deviation: $s$
• Sample size: $n$ (population size often $N$)
• These distinctions matter for calculations and interpretations.

Data Types and Variables

• Variables: the numbers collected that can vary; they describe observations.
• Independent variable vs dependent variable: independent is the condition you manipulate; dependent is the data you measure.
• Discrete vs Continuous:
  • Discrete: integers only (no decimals). Examples: number of students.
  • Continuous: decimals allowed. Examples: time, blood pressure.
• Implication: variable type guides which graphs and analyses are appropriate.

Qualitative / Categorical Data

• Qualitative (categorical) data fall into categories, not inherently numeric.
• Nominal: categories with no natural order (e.g., car brands: Honda, Toyota, Ford).
• Ordinal: categories with a natural order but unequal intervals (e.g., Freshman, Sophomore, Senior; first/second/third place).
• Note: Averages of nominal/ordinal categories are generally not meaningful; use appropriate summaries.

The Research Process (high-level)

• Steps: background literature, design experiment, collect data, analyze data, interpret and present results.
• Emphasize clear communication of results; this is as important as the analysis itself.

Historical Context Highlights (brief)

• Statistics used in war decisions; probability informs cost-benefit analyses.
• John Snow’s cholera map was an early example of linking data to causal interpretation.
• Early population data (e.g., Domesday Book) illustrate long-standing use of counting for decisions.