Data Basics Types of Variables Numerical (Quantitative) Continuous: Can take any value within a range (e.g., height, weight). Discrete: Can only take

Data Basics

Types of Variables

• Numerical (Quantitative)

  • Continuous: Can take any value within a range (e.g., height, weight).

  • Discrete: Can only take distinct values (e.g., number of students in a class).

• Categorical

  • Regular: No inherent order (e.g., colors).

  • Ordinal: Meaningful order but no consistent difference between categories (e.g., Likert scale responses).

Relationships Among Variables

• Associated (Dependent) Variables: Show some connection.

• Independent Variables: No connection between variables.

• Scatterplots & Correlations: Used to analyze positive, negative, or no associations between variables.

Observational Studies & Sampling Strategies

Populations & Samples

• Population: Entire group of interest (described by parameters).

• Sample: Subset of the population used for the study (described by statistics).

Sampling Bias / Poor Sampling Designs

• Non-response Bias: When only a small fraction of sampled individuals respond.

• Voluntary Response Bias: When individuals self-select, often those with strong opinions.

• Convenience Sampling: Sampling individuals that are easily accessible.

Types of Sampling

• Simple Random Sample: Every individual has an equal chance of selection.

• Stratified Sampling: Population divided into subgroups (strata) with random samples from each.

• Cluster Sampling: Randomly selecting entire groups (clusters) rather than individuals.

• Multistage Sampling: Sampling in stages using different methods.

Experiments

Principles of Experimental Design

1. Control: Compare treatment and control groups.

2. Randomize: Randomly assign subjects to treatments.

3. Replicate: Ensure enough samples or replicate the study.

4. Block: Group subjects based on known variables that affect the response.

Other Key Concepts

• Placebo Effect: Psychological response to a non-active treatment.

• Blinding: Prevents bias by concealing treatment assignment.

Ethics in Data Collection

• Institutional Review Boards (IRB): Protects study subjects.

• Informed Consent: Participants must be fully informed and provide consent.

• Confidentiality vs. Anonymity:

  • Confidentiality: Identities are separated from the data.

  • Anonymity: Identities are not collected.

Examining Variables

• Distribution of a Variable: Describes values a variable takes and their frequency.

• Graphical Tools:

  • Categorical Variables: Pie charts, bar graphs.

  • Quantitative Variables: Histograms, stemplots, boxplots.

Summarizing Data

Categorical Variables

• Frequency Distribution Table: Shows counts & percentages.

• Bar Plot: Represents frequencies or proportions; can be stacked/side-by-side.

• Contingency Table: Summarizes data for two categorical variables.

Quantitative Variables

• Measures of Center:

  • Mean: Average of the dataset.

  • Median: The middle value of ranked data.

• Measures of Spread:

  • Variance: Average squared deviation from the mean.

  • Standard Deviation: Square root of variance.

  • IQR (Interquartile Range): Q3 - Q1.

• Stemplots & Histograms:

  • Show data distribution and density.

  • Shape Descriptions: Unimodal, bimodal, symmetric, skewed.

  • Bin Width: Affects visualization of data distribution.

• Box Plots (Box-and-Whisker Plots):

  • Visualizes quartiles, IQR, and outliers.

  • Useful for comparing multiple groups.

• Robust Statistics:

  • For skewed distributions: Median & IQR are better measures than mean & standard deviation.

Correlation & Line of Best Fit

• Scatterplots: Visualize relationships between two numerical variables.

• Correlation Coefficient (r):

  • Measures strength & direction of linear relationships (-1 to +1).

• Least Squares Regression Line:

  • Minimizes squared residuals; used for prediction.

• Residuals:

  • Difference between observed & predicted values.

  • Residual Plots: Assess model fit.

• R² (Coefficient of Determination):

  • Proportion of variability explained by the model.

• Slope & Intercept:

  • Slope: Expected change in response variable per unit change in explanatory variable.

  • Intercept: Expected value when explanatory variable = 0.

• Prediction & Extrapolation:

  • Predicting within data range is reliable.

  • Extrapolation (beyond the data range) is often unreliable.