Chapter 4: The Normal Distribution

STAT 218 - Week 3, Lecture 2

January 23th, 2024

Notations

We will use some notations for important parameters and statistics as follows:

Important

Population Mean: \(\mu\)
Sample Mean: \(\bar{y}\)


Population Standard Deviation: \(\sigma\)
Sample Standard Deviation: \(s\)

Introduction to Normal Curves - I

  • We can think of normal curves as a smooth approximation to a histogram based on a sample of \(Y\) values.
  • We can describe the population distribution of a quantitative variable \(Y\) by
    • calculating its mean \(\mu\) and its standard deviation \(\sigma\) AND
    • using a density curve
      • density curve: relative frequencies as areas under the curve.
  • Let’s have a look at Example 4.1.2 in our book.

Introduction to Normal Curves - II

  • There is no one single normal curve but many normal curves
    • each normal curve has its own mean \(\mu\) and its standard deviation \(\sigma\)

Normal Curves, Continuous Variables and Probability

  • If a numeric variable has a continuous distribution, we can find probabilities by using the density curve for that variable.
    • For that continuous variable, the probability would be equivalent to a specific area under the density curve.
      • The area under a normal curve is always equals to 1.
        • Why?

Let’s Meet with Standard Normal Distribution

Standardization Formula

\(Z\) = (\(Y\) - \(\mu\)) / \(\sigma\)

Let’s See Examples

  • Please refer Example 4.1.2 and 4.3.1 from our course textbook.

Review 2 - Quiz

See the ungraded Review 2 Quiz on Canvas (Module Week 3).

References

Samuels, M. L., Witmer, J. A., & Schaffner, A. A. (2003). Statistics for the life sciences (Vol. 4). Upper Saddle River, NJ: Prentice Hall.