# How to determine if a dichotomous variable is randomly distributed or is predicted by other instances

I have a product with a measurement of X features. Each feature can either be a PASS or a FAIL. Which statistical test can I use to tell if the failures are randomly distributed across all features?

Example: You have a sample of 684 cars with 4 tires each. Each car can have either 0, 1, 2, 3, or 4 flat tires. Are the flats randomly distributed across all tires, or does the presence of one flat predict the presence of another?

Sample dataset:

Num of flats / QTY observed
0 flats / 645
1 flat / 27
2 flats / 4
3 flats / 7
4 flats / 1

My best guess is to estimate the "expected values" using a normal distribution based on the average and standard deviation of the original dataset. However, I can't run a successful chi-square with an expected value of zero, and I don't know if it's at all appropriate to estimate a normal distribution based on the original dataset.

My calculated "expected values' are: 645, 58, 0, 0, 0

• You can see from outer space that these are not randomly (uniformly) distributed, 0 flats is far more prevalent. So what exactly is your question? – user2974951 Jan 15 at 9:00
• Welcome to cross valiated. – cbeleites supports Monica Jan 15 at 10:48
• Hi user2974951, my question is whether the flats are evenly distributed across ALL TIRES. So it would look more like (0 = good tire, 1 = flat): Car 1: 0000 Car 2: 0010 Car 3: 0000 Car 4: 1011 My questions are whether the 1's are evenly distributed over all possible locations, or if they are likely to appear in multiple per car. Imagine a real-world condition where cars that drive on bad roads have a higher incidence of flats overall. However the data would have taken up too much space if I had presented it like that. – Amanda Jan 15 at 17:25
• Can you give a link to your data then? Some kind of poisson model should do. – kjetil b halvorsen Jan 15 at 20:37