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Understanding Normal Distribution Basics

Oct 19, 2024

Normal Distribution and the 68-95-99.7 Rule

Introduction to Normal Distribution

  • Normal distribution refers to data from a population or sample.
  • Parameter vs. Statistic:
    • Parameter: Describes data from a population (e.g., population mean (μ), population standard deviation (σ)).
    • Statistic: Describes data from a sample (e.g., sample mean (xÌ„), sample standard deviation (s)).

Characteristics of Normal Distribution

  • Bell-shaped density curve.
  • Data clusters around the central value (population mean μ).
  • Examples of normally distributed variables: height, weight, exam scores.

Population Mean (μ)

  • Characterizes Position:
    • Increasing μ moves the curve to the right.
    • Decreasing μ moves the curve to the left.

Population Standard Deviation (σ)

  • Characterizes Spread:
    • Larger σ = More spread out (flatter curve).
    • Smaller σ = Less spread out (taller curve).
  • Total area under the curve = 1 (or 100%).

Key Properties of the Normal Distribution

  • Unimodal: Single peak.
  • Symmetric: Equal halves when cut at the mean.
  • Completely characterized by μ and σ.

Notation for Normal Distribution

  • For a variable x:
    • Notation: x ~ N(μ, σ)
    • Indicates x follows a normal distribution with mean μ and standard deviation σ.

The 68-95-99.7 Rule

  • Describes the distribution of data in a normal distribution.
    • 68% of data lies within 1 standard deviation of the mean.
    • 95% of data lies within 2 standard deviations of the mean.
    • 99.7% of data lies within 3 standard deviations of the mean.

Example of Application

  • Heights of university students:
    • Mean height = 5.5 feet, σ = 0.5 feet.
    • Intervals:
      • 68% between 5 and 6 feet.
      • 95% between 4.5 and 6.5 feet.
      • 99.7% between 4 and 7 feet.

Extending the Rule

  • The normal distribution extends infinitely and never touches the x-axis.
  • Areas beyond 3 standard deviations become very small.

Practice Questions

  1. Area between 70 and 90:

    • μ = 70 (center), σ = 10.
    • Area = 95% within 2 standard deviations (70 to 90 and 50 to 70).
    • Area from 70 to 90 = 47.5% (half of 95%).

  2. Area between -2 and 1:

    • μ = 0, σ = 1.
    • Area from 0 to 1 = 34% (half of 68%).
    • Area from 0 to -2 = 47.5% (half of 95%).
    • Total area between -2 and 1 = 81.5%.

Conclusion

  • Understanding normal distribution is essential in statistics.
  • The 68-95-99.7 rule provides a powerful tool for approximating areas under the curve.

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