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.