Lecture Notes on Measures of Dispersion

Section 3.2 Overview

• Focus on measures of dispersion to understand data spread.
• Follow-up to measures of central tendency from Section 3.1.

Distribution Basics

• Distribution Definition: Collection of data values forming a population and describing arrangement.
• Key Questions:
  1. Shape of distribution (bell-shaped, symmetric, skewed, etc.).
  2. Center of distribution (mean, median, mode).
  3. Spread of data values (dispersion).

Measures of Dispersion

• Objective:
  • Determine range, standard deviation, variance.
  • Use empirical rule and Chebyshev's inequality.

Range

• Simplest measure of spread: Maximum value minus minimum value.

Standard Deviation

• Measures spread around the mean.
• Symbol: Sigma (σ) for population, s for sample.
• Calculation:
  • Population: Divide by N (population size).
  • Sample: Divide by n - 1 (sample size adjusted).
• Importance: Indicates how much variation exists from the mean.
• Variance: Standard deviation squared.

Empirical Rule

• Applicable only to bell-shaped distributions.
• Key Percentages:
  • 68% within ±1 standard deviation.
  • 95% within ±2 standard deviations.
  • 99.7% within ±3 standard deviations.
• Helps in estimating data spread.

Chebyshev's Inequality

• Less precise, applies to any distribution shape.
• Formula: 1 - 1/k^2, where k is the number of standard deviations.
• Provides minimum percentage of data within k standard deviations.

Examples

• Comparisons of data sets using histograms.
• Understanding spread through real data examples (e.g., university IQ scores).

StatCrunch Usage Notes

• Critical: Choose correct standard deviation type (adjusted or unadjusted) based on whether dealing with a sample or a population.
• Symbols:
  • Capital N: Population size.
  • Little n: Sample size.
  • Mu (µ): Population mean.
  • x-bar (x̄): Sample mean.

Summary

• Understand measures of dispersion and their importance in statistical analysis.
• Difference in calculations between population and sample data critical to accurate analysis.
• Use empirical rule for bell-shaped distributions and Chebyshev's inequality for general use.