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Page 1: Statistical Analysis and Design

Key Statistical Terms

• Mean: The average value; calculated as ( x = \frac{\sum x}{n} )

• Median: The middle value of a data set when ordered.

• Range: Difference between the maximum and minimum values ( \text{Range} = \text{max}(X) - \text{min}(X) )

Variance and Standard Deviation

• Variance: Measure of data spread; calculated by ( s^2 = \frac{\sum(x - \bar{x})^2}{n - 1} )

• Standard Deviation (SD): Square root of variance ( s = \sqrt{s^2} )

Z-score and Standard Error

• Z-score: Represents the number of standard deviations a data point is from the mean ( z = \frac{x - \bar{x}}{s_x} )

• Standard Error (SE): Measures the accuracy of a sample mean estimate as ( SE = \frac{SD}{\sqrt{n}} )

Confidence Intervals

• 95% Confidence Interval: Range within which we expect the true population mean to lie, calculated as:

  • Upper bound = ( \bar{x} + (1.96 \times SE) )

  • Lower bound = ( \bar{x} - (1.96 \times SE) )

Hypothesis Testing

Chi-Square Tests

• One-Sample Chi-Square: Assesses differences between observed and expected frequencies:

  • ( \chi^2 = \frac{\sum (O - E)^2}{E} )

• Degrees of Freedom (df): Calculated as ( df = k - 1 )

T-tests

• One-Sample t-test: Compares sample mean to a known value:

  • ( t = \frac{\bar{x} - \mu}{\frac{SD}{\sqrt{N}}} )

• Within-Subjects t-test: Analyzes repeated measures:

  • ( t = \frac{D̄}{SD_D \sqrt{N}} )

• Between-Subjects t-test: Compares means of two different groups:

  • ( t = \frac{\bar{X_1} - \bar{X_2}}{s_p \sqrt{\frac{1}{N_1} + \frac{1}{N_2}}} )

Effect Size

• Cohen's d: A measure of effect size calculated as ( d = \frac{2t}{\sqrt{df}} )

Page 2: Critical Values

Critical Values for Chi-Square and t-test

Probability Concepts

• Probability Calculation: ( P = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}} )

Page 3: Research Planning and Design

Constructs and Reliability

• Psychological Construct: Abstract concepts in psychology (e.g., intelligence).

• Operationalization: Converting constructs into measurable variables.

• Reliability: Consistency of a measure. Types include:

  • Test-Retest Reliability: Stability over time.

  • Interrater Reliability: Consistency across different raters.

  • Internal Reliability: Consistency of results across items in a test.

Validity Types

• Validity: Degree to which a test measures what it claims to measure. Types:

  • Face Validity: Appears to measure the intended construct.

  • Content Validity: Covers the entire content area of the construct.

  • Criterion Validity: Correlates with outcome criteria.

  • Convergent Validity: Correlation with similar constructs.

  • Discriminant Validity: Lack of correlation with dissimilar constructs.

Experimental Design

• Causal Relationships: Establishing cause and effect.

• Experimental Designs:

  • Within-Subjects Design: Participants serve as their own control.

  • Between-Subjects Design: Different participants in each group.

• Extraneous Variables: Variables that may affect results; must be controlled.

• Random Assignment: Randomly assigning participants to groups to reduce bias.

Non-Experimental Design

• Correlational Design: Examines relationships between variables without causation.

• Longitudinal Design: Studying the same group over time.

• Quasi-Experiments: Lacks random assignment yet compares groups.

• Observational Research: Involves observing subjects in their natural environment.

Survey Design

• Survey Instruments: Tools for gathering responses; includes questionnaires and interviews.

• Question Types:

  • Closed Questions: Fixed responses.

  • Open Questions: Free-form responses.

• Potential Biases:

  • Leading Questions: Influence responses.

  • Double-Barrelled Questions: Two queries in one.

Sampling Techniques

• Population vs. Sample: The entire group vs. a subset.

• Probability Sampling: Random selection methods.

• Non-Probability Sampling: Non-random selection methods, like convenience sampling.