Basics of Statistics

1. Populations vs. Samples

  • Population: Entire group of interest (e.g., all SU undergraduate students, all cars in a parking lot).
  • Sample: Subset of the population used for analysis.
  • Key criteria for a good sample:
    • Random: Every experimental unit has an equal chance of being selected.
    • Representative: Reflects the population’s diversity (e.g., includes all genders, majors).

Example: To study SU students’ sleep habits, randomly select 200 students across all majors/years.

2. Variables & Data Types

Variables describe experimental units (e.g., age, major, car color).

Numerical (Quantitative) Data

  • Expressed as numbers with units.
  • Examples: Height (cm), exam scores, years of education.

Categorical (Qualitative) Data

  • Non-numeric labels or categories.
    • Nominal: No inherent order (e.g., gender, car color).
    • Ordinal: Ordered categories (e.g., product ratings: Poor, Fair, Good).

3. Summarizing Data

Numerical Data

  • Mean: Average value:
    \(\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i\)
  • Variance: Spread around the mean:
    \(s^2 = \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2\)

Categorical Data

  • Use frequency counts or percentages (e.g., 30% Biology majors, 70% Engineering).

4. Visualizing Data

Choose graphs based on data type:

  • Numerical Data:

    • Histograms: Show distribution shape.
    • Boxplots: Highlight median, quartiles, outliers.

  • Categorical Data:

    • Bar charts: Compare category frequencies.
    • Pie charts: Show proportions (use sparingly!).