It is not difficult to visualize uncertainty. Just think of flipping a series of coins where each toss is independent. The probability of tossing a head is 0.5, and -log(0.5) is 1 for base 2 logarithms. The probability of tossing h heads in a row is simply

Solving for h produces

Almost everything we do statistically has uncertainty, -logp, at the core. Statistical literature refers to uncertainty as negative log-likelihood.

First, springs are the machine of fit for a continuous response model (Farebrother, 1987). Suppose that you have n points and that you want to know the expected value (mean) of the points. Envision what happens when you lay the points out on a scale and connect them to a common junction with springs (see Connect Springs to Data Points). When you let go, the springs wiggle the junction point up and down and then bring it to rest at the mean. This is what must happen according to physics.

When the fit is constrained by hypotheses, you test the hypotheses by measuring this same spring energy. Suppose you have responses from four different treatments in an experiment, and you want to test if the means are significantly different. First, envision your data plotted in groups as shown here, but with springs connected to a separate mean for each treatment. Then exert pressure against the spring force to move the individual means to the common mean. Presto! The amount of energy that constrains the means to be the same is the test statistic you need. That energy is the main ingredient in the F-test for the hypothesis that tests whether the means are the same.

Effect of Pressure Cylinders in Partitions shows what the situation looks like for a single category such as the medium size cars (see the mosaic column from Carpoll.jmp labeled medium in A Mosaic Plot for Categorical Data). Suppose there are thirteen responses (cars). The first level (American) has six responses, the next has two, and the last has five responses. The response probabilities become 6⁄13, 2⁄13, and 5⁄13, respectively, as the pressure against the response partitions balances out to minimize the total energy.

As with springs for continuous data, you can divide your sample by some factor and fit separate sets of partitions. Then test that the response rates are the same across the groups by measuring how much additional energy you need to push the partitions to be equal. Imagine the pressure cylinders for car origin probabilities grouped by the size of the car. The energy required to force the partitions in each group to align horizontally tests whether the variables have the same probabilities. A Mosaic Plot for Categorical Data shows these partitions.