Interactive Main and Variance

This article tries to explain multiple concepts from statistics using a small Javascript illustration of the correlation of two variables.

The normal distribution in one dimension is described by the probability distribution

$$ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} $$

where \( \mu \) is the mean and \( \sigma \) is the standard deviation. While the mean is easily understood, the standard deviation measures how spread out the distribution is. A large \( \sigma \) implies that numbers further from the mean are more likely.

The Interactive Experiment

You can add points to the ‘canvas’ below by clicking with the mouse. After adding a point the mean and variation is recomputed and displayed. In the image you can see the mean displayed as a red dot. Around the mean there is two ellipsis showing the ‘iso-bar’ for (\sigma) and (2\sigma) - something I’ll explain in more details below.

The numerical values of the mean are shown in the first table next to the canvas. The second table shows the computed covariance.

Computations

The mean can be computed as

$$\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i,$$ and $$\bar{y} = \frac{1}{n} \sum_{i=1}^n y_i.$$

However, we use an updating formula.

The elements of the covariance matrix are computed using the means via the following formulas

$$ C_{11} = \frac{1}{n} \sum_{i=1}^n (x_i-\bar{x})^2 $$

$$ C_{22} = \frac{1}{n} \sum_{i=1}^n (y_i-\bar{x})^2$$

$$C_{12} = C_{21} = \frac{1}{n} \sum_{i=1}^n (x_i-\bar{x})(y_i-\bar{y}). $$

However, for illustration purposes I need something that corresponds to the standard deviation. Thus, I need to compute the square root of \(C\). For an 2 by 2 matrix as \(C\) there exists an easy formula, see e.g. this Wikipedia article and square root(s) of 2 by 2 matrices. If we define the square root of the determinant of \( C \) as \( s = \sqrt{C_{11}C_{22}-C_{12}^2} \). Observe that \( C \) is positive definite which implies that the determinant is positive and that \( s \) is a positive real number. Furthermore, define \(t = \sqrt{C_{11}+C_{22}+2s}\). Then we have

$$ S = \frac{1}{t} \left[ \begin{array}{cc} C_{11}+s & C_{12} \ C_{12} & C_{22}+s \end{array}\right] $$

The ‘standard deviation’ matrix is used to draw the ellipses. Essentially, the matrix is used to transform a unit circle. It is also used for the ‘add points’ functionality where random two numbers are taken from \( N(0,1) \) to generate a random point \( [x , y ]\) and then transformed by multiplicatilon by \( S \).