Probability densities and spinners

Back to home page In our basic data analysis workshops, we use an idea from John Kruschke's Doing Bayesian Data Analysis: we use spinners to generate random values for continuous variables and introduce the concept of probability density.

We start off with simple spinners representing a uniform distribution over a range from, say, 0 to 0.5. We discuss the problems of attaching a probability to an exact value, which leads to probability of a range of values and hence probability density.

The spinner for a uniform (0, 0.5) distribution and the corresponding probability density plot are shown below:

For uniform distributions this quite intuitive. The probability that the spinner comes to rest between 0.125 and 0.25 (assuming it is actually fair!) is proportional to the area of that segment of the circle. If the total area of the circle = 1, probabilities equal the area of the relevant segments. And we can see that the same applies to the area under the curve in the plot.

Spinners work for other continuous distributions too. If we want to simulate draws from a population of squirrels with normally-distributed weights with mean 1000g and standard deviation 145g, we can make a spinner like the one shown at the top of the page.

The probability of drawing a squirrel weighing within 100g of the mean (ie, 900 to 1100g) is slightly more than 0.5. The probability of a result between 1150 and 1200g is proportional to the area of the dark red segment in the plot above.

But relating the spinner to the shape of the plot of the normal distribution is not intuitive. To help understanding the relationship I put together the animation below, showing the spinner unfolding and morphing into the usual bell-curve plot:

Your browser does not support the object tag.

You can download the animation as a .mp4 file here.

R code to produce the graphs is here. I used the ImageMagick program to produce the final video.

Comments: Please email comments to mike at mmeredith dot net

Updated 16 Sept 2013 by Mike Meredith