Study Guide for Stats PDF

Key Statistical Concepts 1. Statistical Models ○ Definition: Tools used by scientists to test hypotheses. These models predict an outcome variable based on one or more predictor variables. ○ Types: Vary depending on the research question. For example, linear regre...

Key Statistical Concepts 1. Statistical Models ○ Definition: Tools used by scientists to test hypotheses. These models predict an outcome variable based on one or more predictor variables. ○ Types: Vary depending on the research question. For example, linear regression for continuous outcomes or logistic regression for categorical outcomes. 2. Populations and Samples ○ Population: The entire group of interest in a study. ○ Sample: A subset of the population used to draw conclusions. Larger and random samples better represent the population. 3. Parameters ○ Definition: Values that describe characteristics of a population (e.g., population mean or variance). ○ Use: Researchers estimate these using sample data. 4. Estimation ○ Definition: The process of inferring population parameters from sample data. ○ Method: The method of least squares is a common technique, minimizing the difference between observed and predicted values. 5. Standard Error ○ Definition: Reflects how much a sample statistic (like the mean) might differ from the population parameter. ○ Importance: Indicates the reliability of the sample in representing the population. 6. Confidence Intervals ○ Definition: A range of values where the population parameter is likely to fall (e.g., 95% confidence interval). ○ Misconception: It is not an interval where we are 95% confident the parameter lies, but rather where 95% of similar intervals from repeated samples would contain the true parameter. 7. Null Hypothesis Significance Testing (NHST) ○ Null Hypothesis: No effect or difference exists in the population. ○ Alternative Hypothesis: An effect or difference does exist. ○ Key Terms: Alpha Level: Probability of a Type I error (rejecting a true null hypothesis, usually set at 0.05). Beta Level: Probability of a Type II error (failing to reject a false null hypothesis). Statistical Power: Likelihood of correctly rejecting the null hypothesis when false (aim: 0.80 or higher). Statistical Techniques  1. t-Test ○ Purpose: Compares the means of two groups. ○ Types: Independent t-test: For separate groups. Dependent t-test: For repeated measures or paired data. 2. Analysis of Variance (ANOVA) ○ Purpose: Compares means across three or more groups. ○ Extensions: Planned Contrasts: Test specific hypotheses. Post Hoc Tests: Explore all group differences after finding significant results. 3. Analysis of Covariance (ANCOVA) ○ Purpose: Compares group means while controlling for covariates. 4. Factorial Designs ○ Definition: Experiments with two or more independent variables. ○ Key Effects: Main Effects: Impact of one independent variable. Interaction Effects: Combined impact of variables on the outcome. 5. Exploratory Factor Analysis (EFA) ○ Purpose: Identifies clusters of related variables (factors). ○ Applications: Questionnaire design, data reduction. 6. Reliability Analysis ○ Purpose: Measures consistency of a test (e.g., Cronbach’s alpha). 7. Chi-Square Test ○ Purpose: Examines relationships between two categorical variables. 8. Loglinear Analysis ○ Purpose: Explores relationships among more than two categorical variables. 9. Logistic Regression ○ Purpose: Predicts categorical outcomes based on predictors. ○ Types: Binary (two outcomes) and multinomial (multiple outcomes). Graphing Data 1. Common Types of Graphs: ○ Histograms: Show frequency distributions. ○ Boxplots: Highlight data spread, medians, and outliers. ○ Bar Charts: Represent group means, often with error bars. ○ Line Charts: Illustrate trends over time. ○ Scatterplots: Show relationships between two continuous variables, often with a regression line. General Considerations 1. Assumptions ○ Statistical methods often require assumptions (e.g., normality of data). Results may not be valid if assumptions are violated. 2. Effect Sizes ○ Measure the magnitude of an effect, providing insight into practical significance beyond p-values. 3. Meta-Analysis ○ Combines results from multiple studies to estimate an overall effect size. 4. Bayesian Statistics ○ Focuses on updating prior beliefs with new data, an alternative to NHST. 5. Open Science ○ Practices like pre-registering studies and sharing data for transparency and reproducibility. Reporting Results Include details like: ○ Type of analysis. ○ Sample size. ○ Assumption checks. ○ Effect sizes and confidence intervals. ○ Statistical software used. Pearson's Chi-Square Test (χ²) Purpose: Tests whether there is a significant association between two categorical variables (e.g., gender and voting preference). When to Use: When you have counts (frequencies) in categories (e.g., Yes/No, Male/Female). How It Works: Compares observed frequencies (actual data) with expected frequencies (what you'd expect if there were no relationship). Reporting: Include: ○ Test statistic (χ²) ○ Degrees of freedom (df) ○ Sample size (N) ○ p-value (indicates significance). Example: "The analysis revealed a significant association between gender and opinion, χ²(1, N = 250) = 12.50, p <.001." Key Point: If p <.05, the variables are significantly associated. 2. Fisher's Exact Test Purpose: Like the chi-square test but used when sample sizes are very small or expected counts in some cells are low (< 5). When to Use: Small samples or contingency tables with low expected frequencies. Reporting:  Fisher's test does not calculate a test statistic (like χ²), so report the p-value directly. Example: "Fisher's exact test revealed a significant association between treatment group and outcome, p =.025." 3. t-Tests Purpose: Compare means between groups to determine if differences are statistically significant. Types: 1. One-Sample t-Test ○ Compares the mean of a single sample to a known population mean. ○ Example: Compare your class's average score to the national average. 2. Independent Samples t-Test ○ Compares means between two independent groups. ○ Example: Compare test scores between males and females. 3. Paired Samples t-Test ○ Compares means of the same group at two different times (e.g., pre- and post-test scores). Reporting: Include: ○ Test statistic (t) ○ Degrees of freedom (df) ○ Sample mean (M) and standard deviation (SD) ○ Confidence interval (CI) ○ p-value ○ Effect size (e.g., Cohen's d for independent/paired samples). Example (Independent Samples): "An independent samples t-test revealed a significant difference in anxiety scores between the control group (M = 10.50, SD = 2.30) and the experimental group (M = 8.20, SD = 1.80), t(48) = -3.25, p =.002, 95% CI [-3.80, -0.80], d = 1.20." 4. ANOVA (Analysis of Variance) Purpose: Tests whether means across more than two groups are significantly different. When to Use: When you have one independent variable with more than two levels (e.g., three diets and their effects on weight loss). How It Works: Compares variance between groups (due to the treatment) and within groups (random error). Reporting: Include: ○ F-statistic (F) ○ Degrees of freedom (df) ○ p-value ○ Effect size (e.g., eta-squared, η²). Example: "A one-way ANOVA revealed a significant effect of treatment on depression scores, F(2, 57) = 5.20, p =.008." Key Points: Post hoc tests (e.g., Tukey's HSD) are needed to determine which groups differ if the result is significant. 5. Regression Analysis Purpose: Examines relationships between variables and predicts an outcome (dependent variable) from one or more predictors (independent variables). When to Use: Predict scores (e.g., predicting GPA based on study hours). How It Works:  Fits a line through data points to explain how one variable changes based on another. Reporting: Include: ○ R² (proportion of variance explained by predictors) ○ F-statistic and p-value (test significance of the model) ○ Regression coefficients (b) and their p-values. Example: "The regression model significantly predicted academic performance, R² =.25, F(2, 120) = 15.60, p <.001. Study hours (b = 0.50, p <.001) had a significant positive effect." 6. Correlation Coefficients Purpose: Measure the strength and direction of the relationship between two variables. Types: Pearson's r: Linear relationships between continuous variables. Spearman's ρ: Monotonic relationships (continuous/ordinal data). Kendall's τ: Similar to Spearman's but better for small samples or tied ranks. Reporting: Include: ○ Coefficient value (r, ρ, or τ) ○ Confidence interval ○ p-value. Example: "Creativity was significantly related to competition success, τ = −0.30, 95% CI [−0.47, −0.12], p =.001." 7. Reliability Analysis Purpose: Evaluates the consistency of a measure (e.g., survey items). When to Use: When assessing internal consistency of scales (e.g., using Cronbach’s α). Reporting: Report the α value. Example: "The fear of maths subscale had high reliability, Cronbach’s α = 0.82." Key Statistical Symbols and Their Meanings Symbol Meaning M Mean (average) SD Standard Deviation (spread of scores) SE Standard Error (variability of the mean) df Degrees of Freedom (flexibility in data) p p-value (probability of null hypothesis) χ² Chi-Square Statistic t t-statistic (difference of means) F F-statistic (variance ratio) R² Proportion of variance explained b Regression coefficient (slope) β Standardized regression coefficient r Pearson correlation coefficient η² Effect size in ANOVA α Type I error probability