Overview
This lecture introduces the concept of variability in statistics, explains measures such as range, variance, and standard deviation, and distinguishes between sample and population calculations.
Introduction to Variability
- Variability measures how much scores differ within a distribution.
- Understanding variability is critical for interpreting central tendency and for later statistical concepts.
Measures of Variability
- Range is the difference between the highest and lowest scores (a single value).
- Standard deviation and variance are preferred measures, especially for interval or ratio data with a normal distribution.
- Standard deviation shows the average deviation from the mean.
Examples of Variability
- Distributions can have identical means but different spreads; less variability makes the mean a better predictor.
- A normal distribution shows that about 68% of scores fall within one standard deviation of the mean.
Calculating Sample Variance and Standard Deviation
- Sample variance: average squared deviations of scores from the sample mean.
- Defining formula: sum all squared deviations, divide by the number of cases (n).
- Sample standard deviation: square root of the sample variance.
- Computational formulas are used for easier calculation with large samples.
Population Variance and Standard Deviation
- Population variance: sum of squared deviations from the population mean, divided by number of cases (N).
- Population standard deviation: square root of the population variance.
Estimating Population Parameters from a Sample
- Sample variance underestimates population variance, so use n-1 (degrees of freedom) in the denominator when estimating.
- Use lowercase s to represent estimated population variance and standard deviation from a sample.
Degrees of Freedom
- Degrees of freedom (df) is usually n-1 and adjusts for bias in population estimates.
- Used frequently in statistical formulas to provide unbiased estimators.
Descriptive vs. Inferential Statistics
- Descriptive statistics describe known data (use n in formulas).
- Inferential statistics estimate population parameters from samples (use n-1 in formulas).
Accounting for Variability
- Variance can be explained by variables in experimental or correlational studies.
- The amount of variance explained is important in interpreting statistical relationships.
Key Terms & Definitions
- Variability ā how much scores differ in a distribution.
- Range ā highest score minus lowest score (single value).
- Variance (sĀ² or ĻĀ²) ā average of squared deviations from the mean.
- Standard deviation (s or Ļ) ā square root of the variance; average deviation from the mean.
- Sample ā subset of a population used for analysis.
- Population ā complete set of individuals or scores being studied.
- Degrees of freedom (df) ā number of values free to vary, usually n-1 for samples.
- Descriptive statistics ā statistics that summarize or describe data.
- Inferential statistics ā statistics used to make inferences about a population from a sample.
Action Items / Next Steps
- Review and practice computing variance and standard deviation using both defining and computational formulas.
- Understand when to use n vs. n-1 in calculations.
- Prepare for problem-solving exercises in class.