# How Random Are You?

Here’s a nice little activity for a Statistics class. I was taken through it at an Open University tutorial earlier today.

1. Ask students to draw a 4 inch (or 10cm) square. (That part will likely take the most time out of the whole activity…)
2. They should then randomly mark 16 points in the square.
3. Then the square should be divided into 4 x 4 smaller squares:
4. Then count the number of points inside each smaller square.

For the example I have drawn above, the data is: 1, 0, 2, 2, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 1, 1. (Sanity check: 16 pieces of data and they sum to 16 which is the number of points plotted.)

## What do we mean by random?

In this instance we would expect ‘random’ points to be distributed according to a Poisson process, and thus that the mean and variance for our data set should be equal.

Firstly note that every student will have a mean of 1. (16 points in total, divided by 16 squares.)

Calculate variance using your favourite formula. Mine is to find  $S_{xx}=\sum x^2 - n\bar{x}^2$ and then divide by $n$. For my data, $\sum x^2 = 22$ giving $\sigma^2=\frac38$.

## How random are you?

The closer in value your mean and variance are (thus, the closer your variance is to 1) the better you have done at plotting the points randomly. If your variance is much smaller than 1 then you have separated the points too much and they thus have more underlying order than intended. If your variance is much greater than 1 then you have clustered the points too tightly in some area(s).

Be reassured that most people will have a variance less than 1!

(Mathematical note: this is effectively the ‘index of dispersion’, $D=\sigma^2/\mu$. You might also prefer to divide by $n-1$ when finding variance. Feel free!)

(Another mathematical note: if you check the Poisson probabilities, then you would expect approximately six squares with 0 points, six squares with 1 point, three squares with 2 points and one square with 3 points.)