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Normal Distribution Overview

Oct 1, 2025

Overview

This lesson introduces the normal distribution (bell-shaped curve), discusses its characteristics, the empirical rule, and gives practical examples of its application.

Introduction to the Normal Distribution

  • The normal distribution is a bell-shaped curve representing how data values are spread in many real-world scenarios.
  • Common examples include heights, IQ scores, SAT scores, life span of light bulbs, and popcorn kernels popping.
  • The normal distribution is also called the Gaussian distribution.

Characteristics of the Normal Distribution

  • The mean, median, and mode are all equal and located at the center of the curve.
  • The distribution is symmetrical about the mean.
  • The shape is determined by the mean (μ) and standard deviation (σ).

The Empirical Rule (68-95-99.7 Rule)

  • About 68% of data falls within one standard deviation (μ ± 1σ) of the mean.
  • About 95% falls within two standard deviations (μ ± 2σ).
  • About 99.7% falls within three standard deviations (μ ± 3σ).
  • Due to symmetry, 34% lies between the mean and one standard deviation (both above and below).
  • 13.5% of data lies between one and two standard deviations (both above and below).
  • 2.35% lies between two and three standard deviations (both above and below).

Applying the Empirical Rule: Examples

  • Call center example: With mean = 63 calls and σ = 3, 68% of receptionists answer between 60 (μ-1σ) and 66 (μ+1σ) calls daily.
  • Midterm example: With mean = 85% and σ = 6%
    • 91% is one standard deviation above the mean; 84% score below (50% + 34%), 16% score above.
    • 73% is two standard deviations below the mean; 97.5% score above, 2.5% score below.

Key Terms & Definitions

  • Normal Distribution — A symmetric, bell-shaped data distribution defined by its mean and standard deviation.
  • Mean (μ) — The average value, center of the distribution.
  • Standard Deviation (σ) — Measures the spread or dispersion of the data.
  • Empirical Rule (68-95-99.7 Rule) — States percentages of data within 1, 2, and 3 standard deviations of the mean.

Action Items / Next Steps

  • Review the empirical rule and practice applying it to different normal distribution scenarios.
  • Prepare for homework or further reading on more advanced normal distribution problems.