05 - The Normal Distribution

Normal Distribution Overview

  • Definition: A normal distribution is a continuous probability distribution characterized by its bell-shaped and symmetric curve.

  • Key Properties:

    • Described by two parameters: mean (center) and standard deviation (spread).

    • The normal curve is symmetric around the mean.

Rule of Thumb for Normal Distributions

  • Standard Deviations:

    • 1 Standard Deviation (68%): Approximately 68% of data lies within one standard deviation from the mean.

    • 2 Standard Deviations (95%): Approximately 95% of data lies within two standard deviations from the mean.

    • 3 Standard Deviations (99.7%): Approximately 99.7% of data lies within three standard deviations from the mean.

  • Common Interpretations:

    • 1 SD is considered "close"

    • 2 SDs is considered "far"

    • 3 SDs is considered "very far"

Visual Representation

  • Bell Curve Illustrations are useful for visualizing data distributions and understanding the data within standard deviation ranges.

Examples Using Normal Distribution

Example 1: Temperature Distribution

  • Given:

    • Mean = 72, Standard Deviation = 11

  • Temperatures Breakdown:

    • 68% fall between 61 and 83 degrees

    • 95% fall between 50 and 94 degrees

    • 99.7% fall between 39 and 105 degrees

Example 2: IQ Scores in Middle School

  • Given:

    • Mean = 103, Standard Deviation = 10

  • Calculations:

    • Percentage with IQ between 93 and 113: 68% (1 SD).

    • Percentage below IQ of 123: 97.5% (2 SDs).

    • Percentage with IQ between 83 and 93: 13.5% (between 2 SDs and 1 SD).

Percentiles

  • Definition: Percentiles divide data into 100 equal parts.

    • 80th percentile: 80% of data falls below this value.

  • SAT score example: Scoring in the 70th percentile means 70% scored lower, 30% scored higher.

Calculating Proportions using Calculator

  • Transition to more complex calculations not strictly within 1, 2, or 3 SDs.

  • Using the Normal CDF Function

    • Example: Scores in a national test with mean = 506, SD = 81

      • Proportion of scores below 574: Use normal CDF input with limits, mean, and SD.

      • Area calculations involve continuous inputs, typically using values close to negative/positive infinity:

        • Negative infinity can be substituted with a value like -10^99.

Inverse Norm Function

  • For finding values corresponding to specific percentiles. Example of finding the 30th percentile in a normal distribution uses inverse norm where area to left is input value.

Practice Problems

  1. Hummingbird Wing Beats

  • Average = 55, SD = 4

  • Proportions:

    • Between 47 and 63: 95%

    • Between 51 and 67: 83.85%

    • Less than 59: 84%

  1. Diastolic Blood Pressure

  • Given Mean = 82, SD = 4

  • Proportions:

    • Less than 80: 30.85%

    • Higher than 89: 4.01%

    • 85th Percentile: 86.14

Conclusion

  • Understanding of normal distributions is crucial for statistical calculations and further explorations like hypothesis testing.