Wiki Page, Math 13 - Perez Sandbox : Module 12 - Continuous Probability Distributions (1 of 2)

Wiki Page
2026-01-17T21:15:11-08:00

Transition to Continuous Quantitative Variables

As we discussed previously, shoe size is a discrete quantitative variable because we can list all possible shoe sizes. Foot length is not a discrete quantitative variable because foot lengths are not limited to distinct, separate values. Foot lengths can be any value over a continuous range of possibilities. In other words, foot length, unlike shoe size, can be measured as precisely as we want to measure it. For example, we can measure foot length to the nearest inch, the nearest half inch, the nearest quarter of an inch, the nearest tenth of an inch, etc. Therefore, foot length is a continuous quantitative variable.

What happens to the probability histogram when we measure foot length with more precision? When we increase the precision of the measurement, we will have a larger number of bins in our histogram. This makes sense because each bin contains measurements that fall within a smaller interval of values. For example, if we measure foot lengths in inches, one bin will contain measurements from 6-inches up to 7-inches. But if we measure foot lengths to the nearest half-inch, then we now have two bins: one bin with lengths from 6 up to 6.5-inches and the next bin with lengths from 6.5 up to 7-inches.

You can use the following applet to see what happens to the probability histogram as the width of intervals decrease. Change the interval width by clicking on 0.5 in., 0.25 in., or 0.1 in.

At the bottom of the applet is an option to add a curve. This curve comes from a mathematical equation adjusted to fit the shape of the probability histogram.

Check “Show curve” and click through the different bin widths.

Notice that as the width of the intervals gets smaller, the probability histogram gets closer to this curve. More specifically, the area in the histogram’s rectangles more closely approximates the area under the curve.

If we continue to reduce the size of the intervals, the curve becomes a better and better way to estimate the probability histogram.

We'll use smooth curves like this one to represent the probability distributions of continuous quantitative variables. This idea is discussed in more detail on the next page.

Mark this page as done, then click the "Next" or > button to continue.

Content by the Open Learning Initiative and licensed under CC BY.