Key Concepts

• A standard normal distribution has a mean of 0 and a standard deviation of 1.
• A normal distribution is bell-shaped and the total area under the normal distribution curve is 1.
• A z-score tells you how many standard deviations a value is above or below the mean.
• Transforming a value to its z-score allows you to apply the empirical rule.
• The empirical rule states that for a normal distribution: 68% of the distribution lies within one standard deviation of the mean, 95% of the distribution lies within two standard deviations of the mean and 99.7% of the distribution lies within three standard deviations of the mean.

Glossary

standard normal distribution: a continuous random variable (RV) X ~ N(0, 1); when X follows the standard normal distribution, it is often noted as Z ~ N(0, 1).

z-score: the linear transformation of the form [latex]z=\frac{x-M}{\sigma}[/latex]; if this transformation is applied to any normal distribution X ~ N(μ, σ) the result is the standard normal distribution Z ~ N(0,1). If this transformation is applied to any specific value x of the RV with mean μ and standard deviation σ, the result is called the z-score of x. The z-score allows us to compare data that are normally distributed but scaled differently.