Wiki Page, Math 13 - Perez Sandbox : Module 12 - Wrap Up Probability Distributions for Continuous Quantitative Variables-2

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2026-01-17T21:15:15-08:00

Let’s Summarize

A continuous random variable is not limited to distinct values. It is a measurement such as foot length. We cannot display the probability distribution for a continuous random variable with a table or histogram. We use a probability density curve to assign probabilities to intervals of x-values. We use the area under the curve to find probabilities.

We use a normal probability 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 to represent the mean of a normal curve and 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.

For a normal curve, the Empirical Rule tells us that there is a 68% chance that observations fall within 1 standard deviation of the mean, 95% within 2 standard deviations of the mean, and 99.7% within 3 standard deviations of the mean.

To compare x-values from different distributions, we standardize the values by finding a z-score.

A z-score measures how far X is from the mean in standard deviations. In other words, the z-score is the number of standard deviations X is from the mean of the distribution. For example, Z = 1 means the x-value is 1 standard deviation above the mean.

If we convert the x-values into z-scores, the distribution of z-scores is also a normal probability density curve. This curve is called the standard normal distribution. We use an applet with the standard normal curve to find probabilities for any normal distribution.

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