Bivariate Normal Model:

A Classroom Project

M. W. MAXFIELD & B. C. LYON

Some recycled punch cards, 17 toothpicks, about 20 photocopies, a base of styrofoam or cardboard, and some co-operative labour, and your classroom can display a most educational bivariate normal distribution model. If one picture, in 2-space, is worth a thousand words, then how many words is a 3-dimensional model worth, ten-to-the 45?

I. Overview As you can see in the photograph, we do not actually construct the 3-dimensional bivariate normal surface, but only suggest it to the eye by 17 parallel plane sections, each cutting the surface in a familiar plane normal curve. A bivariate normal distribution applies to two variables, x and y, such as height and number of minutes to learn a sorting task. Recall that in a plane if y is subject to a normal distribution, then the area under the normal curve from, say, y = - 0.2 to y = 1.1, equals the probability, or relative frequency, of y measuring between -0.2 and 1.1. Analogously, in 3-space, the volume under the bivariate normal surface bounded, say, by the rectangular border x = 1.2, x = 1.4, y = -0.2, and y = 1.1, equals the probability of measurement pairs x,y lying in this region. We will show how to make the model illustrate two measurements x and y that are completely independent of each other, or measurements that have some degree of correlation.

As the photograph shows, our model is cut off when the brim of the sombrero reaches an arbitrary height above the base, in this case 0.01. Like the tails of the normal distribution curve for one variable, the brim of the bivariate normal surface is asymptotic to its base, a very wide brim indeed.

2. Symmetry

We have split our pattern in two so as to meet page-size constraints. The full pattern is obtained by symmetry.

3. Plane Sections Using the symmetric pattern as the master copy, make 17 photocopies on paper. (Our larger model was made of heavy gauge acetate film, imprinted by Thermofax, but paper is easier to handle.)

Choose four distinctive colours, A, B, C, and D. Using a straightedge, apply colour A to the left vertical dotted line on every copy. Similarly, colour all the right vertical dotted lines with colour B, upper horizontal lines with colour C, and lower horizontal lines with colour D. Later these colours will help pupils discover features of the bivariate normal distribution.

4. Slopes and Deviations Cut one copy of the largest normal section shown in the pattern, and cut two copies of each of the eight smaller sections, cutting on the solid black border. You will notice that the sections differ in width, each cut off when the tail drops to 0.01 units. The base of each section is the zero line, or axis, not the 0.01 line.

As you cut out a normal segment, you will actually feel the change in concavity at the points of inflection, one standard deviation from the mean.

Glue or tape a flat toothpick to the back of the vertical axis of each section, point down, about half the toothpick extending below the baseline. Affix a stiff backing to the paper section, over the toothpick. Recycled punch computer cards work well, even though they will not cover all of the larger sections. Let the card follow the base axis. Then trim away protruding portions of the card.

5. Base and Mounting As a base choose a flat piece of styrofoam or a cardboard carton, at least 50cm long, preferably longer. If the base is wide as well as long, so as to represent a plane, so much the better, but long packing strips of styrofoam will do.

Our model is quite flexible in its ability to represent different combinations of the five parameters of a bivariate normal distribution of x and y, two means µx and µy, two variances and and the correlation coefficient r relating the two variables. The assembling of the model depends on whether the variables x and y are independent or correlated.

First, suppose the variables x and y are independent. In this case there is no probabilistic tie between x and y. In this case, letting each plane section represent a conditional distribution of y, for a fixed x value, we note that a unit has been set for y, namely, the standard deviation , the horizontal distance between the mean and the point of inflection. Fortunately, we need not start over for different values of the y standard deviation, however, since everything is relative. We simply take the unit as measured and adjust the x standard deviation accordingly. If  is equal to then our model will have circular horizontal cross sections. If  is a multiple k of the y standard deviation, we use k times the fixed  measurement on each plane section as .

Once you have determined, divide it by 4 and use the resulting amount as the common distance between plane sections in the model. Lay a straightedge along the base and draw the x axis. Starting from the centre, punch small holes with a nail or skewer along the straightedge at 0, ±025 , ±0.5 , ±075, ±, ± 1.25,± 1.5, + l.75 and ±2. Make the holes narrower than the toothpicks, so the toothpicks will be held perpendicular to the base.

Insert the toothpicks, with the tallest section at 0, and the shorter sections ranging down on each side.

The model is complete.

Before we stop to analyze the properties of the model, let us return to the case of correlated variables. The fifth parameter of the bivariate normal distribution is the correlation coefficient r If r is positive, then large x-measurements tend to go with large y-measurements. If r is negative, then large x’s and small y’s tend to go together.

When r does not equal 0 it is necessary to rotate the axes in the base, through an angle having tangent /. This can be done by labelling. Take the y-unit on the 17 plane sections to represent , and use an x standard deviation of , by letting be expressed in multiples of . Divide the multiple by 4 to obtain the common space to be left between adjacent sections.

Whether you try to represent an actual combination of parameter values or you simply content yourself with spacing the sections equidistantly along an axis without measuring, you will be able to observe several things. First, there is a pronounced optical illusion, the centre curves appearing to be closer together than the outer curves. The tips of the colour A lie on a normal curve, as do the tips of colour B. Colours C and D, along with the truncation points, show that horizontal cross sections of the surface are elliptical. The ellipse is a circle when x and y are independent and have a common variance.

It is harder to demonstrate with your model, but the persistent normal character of the bivariate normal surface goes beyond vertical sections parallel to an axis in the base. Choose any line in the base plane and imagine a vertical plane through the line. It will intersect the surface in a normal curve.

The 17 individual sections illustrate conditional distributions, here arbitrarily taken to be distributions of y when x takes on one of the 17 fixed values from -2 to + 2. The normal curves through the colour A tips and the colour B tips illustrate conditional distributions of x for fixed values of y.

Louisiana Tech. University

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