4 Normal Distribution

Chapter 5: Normal Probability Distributions

Introduction to Normal Distributions

  • Normal distributions are represented by a bell-shaped curve.

  • The curve is a probability density function (pdf) of a continuous random variable X.

Properties of Normal Distributions

  • A continuous random variable can take any value in an interval on the number line.

  • Normal distribution characteristics:

    • Mean, median, and mode are equal.

    • Curve is symmetric around the mean.

    • Total area under the curve = 1.

    • Approaches but never touches the x-axis.

Normal Curve Features

  • The graph is bell-shaped and symmetric about the mean.

  • Inflection points occur at μ - σ and μ + σ, marking the curve's upward and downward transition.

Mean and Standard Deviation

  • A normal distribution can have any mean (μ) and positive standard deviation (σ).

  • The mean indicates the curve’s center, while the standard deviation describes its spread.

Standard Normal Distribution

  • Standard normal distribution has a mean of 0 and a standard deviation of 1.

  • Z-score transformation allows any x-value to be expressed as a z-score to find probabilities using the standard normal table.

Finding Areas Under the Standard Normal Curve

  1. To find the area to the left of a z-score, use the standard normal table.

  2. For the area to the right, subtract the left area from 1.

  3. For areas between two z-scores, determine the corresponding areas and subtract.

Guidelines for Area Calculation

  • When calculating probabilities:

    • Draw the standard normal curve.

    • Shade the required area.

  • Use corrections for continuity when appropriate.

Empirical Rule

  • The Empirical Rule states:

    • Approximately 68% of data falls within 1 standard deviation from the mean.

    • Approximately 95% falls within 2 standard deviations.

    • Approximately 99.70% falls within 3 standard deviations.

Skewness

  • Skewness types:

    • Positive skew: Mean > Median > Mode.

    • Negative skew: Mode > Median > Mean.

    • Zero skew: Mean = Median = Mode.

Sampling Distributions and Central Limit Theorem

  • A sampling distribution is formed by taking samples from a population.

  • Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution as sample size increases (n ≥ 30).

Finding Probabilities Using the Central Limit Theorem

  • Calculate mean and standard error of sample means to determine probabilities based on the normal distribution.

Correction for Continuity in Binomial Approximation

  • When using normal distribution to approximate a binomial distribution, apply continuity correction.

  • Example method: Convert a discrete interval to a continuous range by adjusting endpoints.

Conclusion

  • Understanding normal distributions and their properties is crucial for calculating probabilities and interpreting statistical data.