Key Concepts in Statistical Analysis for PSYC 1010

Data

Numerical info (scores, measurements).

Variables

Characteristics that vary (e.g., gender, age).

Case

Entity from which data is collected (e.g., people, cities).

Descriptive Stats

Describe and summarize data (univariate or multivariate).

Inferential Stats

Generalize from sample to population; includes hypothesis testing and cause-effect relationships.

Discrete

Whole numbers, no fractions (e.g., yes/no, categories).

Continuous

Infinite fractional values, limited by instrument precision (e.g., age).

Independent Variable

The cause or predictor (e.g., number of drinks).

Dependent Variable

The effect or outcome (e.g., blood alcohol level).

Correlation

Relationship between variables without cause/effect.

Regression

Predict outcome (Y) from predictor(s) (X); can be single or multiple.

Population

Entire group (e.g., all Canadians 65+).

Sample

Subset of the population (e.g., 50 Guelph students).

Parameters

Describe population.

Statistics

Describe sample.

Random Sampling

Ideally unbiased but can be difficult and ethically complex.

Scientific Method

Steps: Observation → Question → Hypothesis → Experiment → Analyze → Conclusion → Replicate.

Deductive Reasoning

General → Specific.

Inductive Reasoning

Specific → General.

Quantitative Research

Numeric, statistical.

Qualitative Research

Descriptive, open-ended.

Nominal Scale

Categories (e.g., yes/no).

Ordinal Scale

Ranked order (e.g., income levels).

Interval Scale

Equal intervals, no true zero (e.g., temperature).

Ratio Scale

True zero (e.g., age, salary).

Mean

Average, affected by outliers.

Median

Middle value, unaffected by outliers.

Mode

Most frequent score.

Normal Distribution

Symmetrical bell curve; mean = median = mode.

Positive Skew

Few high scores.

Negative Skew

Few low scores.

Kurtosis

Leptokurtic: Tall and thin; Platykurtic: Flat; Mesokurtic: Normal.

Range

Difference between highest and lowest scores.

Variance

The average of squared deviations from the mean.

Standard Deviation

Square root of variance; measures spread of scores.

Average Deviation

Mean of all individual score deviations from the mean.

Coefficient of Variation

Standard deviation ÷ mean; useful for comparing variability across different units.

Range

Difference between the highest and lowest scores.

Normal Distribution

Symmetrical, bell-shaped curve.

68% Rule

68% of scores fall within ±1 SD of the mean.

Positive Skew

Distribution with few high scores.

Negative Skew

Distribution with few low scores.

Z-Scores

Standardized score showing how far a value is from the mean in SD units; useful for comparing across different distributions.

Sample Space

All possible outcomes.

Event

A specific outcome.

Probability Function

Ensures all event probabilities sum to 1.

Mutually Exclusive

P(A or B) = P(A) + P(B).

Not Mutually Exclusive

Events can overlap (e.g., ace of diamonds).

Bernoulli Distribution

Probability of success/failure with sampling with replacement.

Null Hypothesis (H₀)

No difference or effect.

Alternative Hypothesis (H₁)

Predicts a difference or effect.

Type I Error (α)

False positive - rejecting H₀ when it's true.

Type II Error (β)

False negative - failing to reject H₀ when it's false.

Power

Probability of correctly rejecting H₀ (1 - β).

Z-Test

Used when population standard deviation is known.

T-Test

Used when population parameters are unknown.

Independent T-Test

Used for two separate groups.

Paired T-Test

Used when one group is measured twice.

Degrees of Freedom (df)

One sample: N - 1; Independent t-test: N - 2.

Critical T-Value

Used to determine rejection region; If calculated t > critical t → reject H₀.

Confidence Intervals

Range around a sample mean where the population mean likely falls.

Point Estimate

Exact value from sample data.

Interval Estimate

Range around the point estimate with a confidence level (e.g., 95%).

ANOVA (Analysis of Variance)

A parametric test used when comparing more than 2 means.

Between-group variance

Variation due to differences between group means.

Within-group variance

Variation within each group.

One-factor ANOVA

Different participants in each group (e.g., comparing schools).

Repeated-measures ANOVA

Same participants measured multiple times (e.g., before/after study).

Two-factor / Three-factor ANOVA

Tests for interactions between two or more independent variables (e.g., school type × region × income level).

Bonferroni-Dunn correction

Divide α by the number of comparisons to keep overall error rate at 0.05.

Assumptions of ANOVA

Normal distribution, homogeneity of variance, independence of observations, interval or ratio data.

F-Ratio

F = MSbetween / MSwithin; F > 1 indicates more variability between groups than within, suggesting potential statistical significance.

Degrees of Freedom

df between = k - 1 (where k = number of groups); df within = N - k (where N = total sample size); df total = N - 1.

ANOVA Steps

1. Calculate sum of squares (SS): total, between, and within. 2. Calculate degrees of freedom (df). 3. Calculate mean square (MS = SS / df). 4. Calculate the F-ratio (MSbetween / MSwithin). 5. Compare Fobtained to Fcritical (use F-table). 6. If Fobtained > Fcritical, reject H₀.

Post-Hoc Tests

Used only if ANOVA is significant to determine which groups are significantly different; Tukey's HSD is a common post-hoc test.

Correlational Studies

Predictor variable (X-axis) is the independent variable; Criterion variable (Y-axis) is the dependent variable.

Correlation Analysis

Measures the strength and direction of the relationship between two variables (X and Y).

Scatterplots

Show how closely data points fit the regression line (line of best fit).

Outliers

Vertical outliers affect the relationship, while horizontal outliers are called leverage points.

PPMC (Pearson's r)

The correlation coefficient (r) ranges from -1 to 1, showing strength and direction (positive/negative) of the relationship.

Strength of r

0.00-0.25: Little to no correlation; 0.25-0.50: Fair correlation; 0.50-0.75: Moderate to good correlation; 0.75: Good to excellent correlation.

R² (coefficient of determination)

Indicates the percentage of variability in one variable explained by the other variable; e.g., R² = 0.49 means 49% of variability in Y is explained by X.

Limitations of PPMC

A high r-value doesn't prove causation; range and extreme data points can affect the results; assumes a linear relationship between variables.

Regression Analysis

Models the relationship between a dependent variable (Y) and an independent variable (X) using a straight line.

Linear Regression Equation

Y = bX + a (where b is the slope and a is the Y-intercept).

Slope

Indicates the direction of the relationship (positive or negative).

Y-intercept

Value of Y when X = 0.

Prediction in Regression

The line of best fit helps predict Y from X, and error is the difference between the predicted and actual values.

Coefficient of Determination (R²)

Measures the proportion of variability in Y explained by the variability in X; e.g., if R² = 0.495, 49.5% of the variance in Y is accounted for by X.

RCT (Randomized Controlled Trials)

Establish cause-and-effect relationships, the gold standard for drug trials.

Historical Data

Can show correlation but doesn't establish causality.

Statistical Testing

Null Hypothesis (H₀): No relationship or effect between variables.

Non-Directional Hypothesis

Open to the possibility of either a positive or negative relationship.

T-tests & ANOVA

Used to compare means between two or more groups; T-test compares two groups, ANOVA compares more than two groups.

Example: Covid-19 Case Study

Analyzed the relationship between vaccination rates and new case counts, showing statistical significance (p < 0.05).

Conclusion of Correlation Studies

In correlation studies, r shows the relationship, R² quantifies the explanation, and regression predicts outcomes.

Statistical Significance

Statistical significance (e.g., p-value < 0.05) confirms whether a relationship exists without concluding causality.