Statistics and Data Analysis

Measures of Variation

• Variance: average of the squared differences from the mean
• Standard deviation: square root of the variance

Correlation

• Correlation: measures the strength and direction of a linear relationship between two variables
• Correlation coefficient (r): ranges from -1 to +1
  • Positive values indicate a positive relationship
  • Negative values indicate a negative relationship
  • Zero indicates no linear relationship

Hypothesis Testing

• Hypothesis testing: statistical method to make inferences about a population based on sample data
• Steps:
  • Formulate null and alternative hypotheses
  • Set a significance level (α)
  • Collect and analyze data using appropriate statistical tests
  • Compare the calculated test statistic with the critical value or p-value
  • Make a decision to either reject or fail to reject the null hypothesis based on the evidence

Errors in Hypothesis Testing

• Type I error: incorrectly rejecting a true null hypothesis (false positive)
  • Controlled by lowering the significance level (α)
• Type II error: incorrectly failing to reject a false null hypothesis (false negative)
  • Controlled by increasing sample size or improving test sensitivity (power)

Parametric and Non-Parametric Tests

• Parametric tests: assume data come from a specific distribution (e.g., normal distribution)
  • Examples: t-tests, ANOVA, Pearson correlation
• Non-parametric tests: do not make assumptions about data distribution
  • Examples: Mann-Whitney U test, Wilcoxon signed-rank test, Spearman correlation

Analysis of Variance (ANOVA)

• Assumptions:
  • Normality: residuals (errors) are normally distributed
  • Homogeneity of variances: variances of groups being compared are approximately equal
  • Independence: observations within each group are independent of each other

Non-Parametric Tests

• Advantages:
  • Robust to violations of assumptions (e.g., normality, homogeneity of variance)
  • Applicable to ordinal or non-normally distributed data
  • Easier to interpret without stringent requirements
• Disadvantages:
  • Less powerful than parametric tests with normally distributed data
  • May lose efficiency with larger sample sizes
  • Provide less precise estimates of population parameters

Chi-Square Test

• Chi-square test: determines if there is a significant association between categorical variables in a contingency table
• Example: assessing whether there is a relationship between gender and voting preference in a political survey

Factor Analysis

• Factor analysis: statistical technique that identifies patterns or underlying dimensions (factors) among a set of variables
• Role in psychological measurement: reduces data complexity by grouping variables based on shared variance, aiding in the development of more concise and interpretable measurement scales

Reliability

• Reliability: consistency and stability of measurement over time and across different conditions
• Types:
  • Test-retest reliability: consistency of scores across repeated administrations of the same test
  • Internal consistency reliability: consistency among items within a measurement scale (e.g., Cronbach's alpha)
  • Inter-rater reliability: consistency between different raters or observers scoring the same behavior or event

Effect Size

• Effect size: quantifies the strength of a relationship or the magnitude of an effect independent of sample size
• Importance: helps researchers assess the practical significance of findings, compare results across studies, and make informed decisions about the relevance of their research in practical settings

Research Designs

• Cross-sectional design: collects data from participants at a single point in time to compare different groups or assess relationships between variables
• Longitudinal design: follows the same group of participants over an extended period to study developmental changes, stability of behaviors, or the effects of interventions over time

Statistical Inference

• Statistical inference: using sample data to make generalizations or predictions about a population
• Applications: hypothesis testing, estimating population parameters, and drawing conclusions based on statistical evidence derived from sample data

Data Visualization

• Scatterplot: graphical representation of the relationship between two continuous variables
• Confidence interval: range of values calculated from sample data that is likely to include the true population parameter, with a specified level of confidence (e.g., 95% confidence)

Regression Analysis

• Assumptions:
  • Linearity: relationship between variables
  • Independence of errors
  • Homoscedasticity: constant variance of errors
  • Normality of residuals
• Applications: predicting outcomes based on predictor variables, identifying significant predictors of behavior or outcomes, and understanding relationships between variables

Meta-Analysis

• Meta-analysis: statistical technique for combining results from multiple studies to obtain an overall estimate of the effect size
• Purpose: provides a comprehensive summary of existing research, identifies patterns or inconsistencies across studies, and enhances the reliability and generalizability of findings

Factorial Design

• Factorial design: experimental design involving the simultaneous manipulation of two or more independent variables (factors)
• Use: allows researchers to study main effects of each factor and interactions between factors, providing insights into how different variables interact to influence outcomes