Standard Normal Distribution ([latex]z[/latex] Distribution)

Let’s Summarize

• A continuous random variable is not limited to distinct values. We cannot display the probability distribution for a continuous random variable with a table or histogram. We use a density curve to assign probabilities to intervals of [latex]x[/latex]-values. We use the area under the density curve to find probabilities.
• We use a normal density curve to model the probability distribution for many variables, such as weight, shoe sizes, foot lengths, and other human physical characteristics. Normal curves are mathematical models. We use [latex]\mu[/latex] to represent the mean of a normal curve and [latex]\sigma[/latex] to represent the standard deviation of a normal curve. We use Greek letters to remind us that the normal curve is not a distribution of real data. It is a mathematical model based on a mathematical equation. We use this mathematical model to represent the perfect bell-shaped distribution.
• Recall: For a normal curve, the empirical rule for normal curves tells us that [latex]68\%[/latex] of the observations fall within [latex]1[/latex] standard deviation of the mean, [latex]95\%[/latex] within [latex]2[/latex] standard deviations of the mean, and [latex]99.7\%[/latex] within [latex]3[/latex] standard deviations of the mean.
• To compare [latex]x[/latex]-values from different distributions, we can standardize the values into a standard normal distribution. If we convert the [latex]x[/latex]-values into [latex]z[/latex]-scores, the distribution of [latex]z[/latex]-scores is also a normal density curve with a mean of [latex]0[/latex] and a standard deviation of [latex]1[/latex]. This curve is called the standard normal distribution. We can then use the standard normal curve to find probabilities for any normal distribution.