The Chi-Squared Test is a way to see if a model is valid or invalid. By using the test I can see if I have a “fair die,” “fair coin,” or other model that fits well enough into my observed data.

The formula is:

X^{2} = Σ (O – E)^{2}/E

So what does that mean? Well it reads the Chi-Square = Sum of all observed minus the expected values squared divided by the expected value. Let me make sense of that with an example:

Let’s say we roll a die 60 times and we get:

1 – 6

2 – 4

3 – 8

4 – 5

5 – 12

6 – 25

We got a lot of 6s so we want to check to see if the “die is fair.” We would expect that each value would come up 10 times, that is our expected value.

We start out by making a null hypothesis H_{O} that the die is fair.

We then want to pick our level of significance, α, let’s say 0.05 (95% confidence).

We find our critical value by looking at a Chi-Square table, going down to 5 degrees of freedom, going across to 0.05, and we see our critical value is:

11.07.

If our calculation for our X^{2} value, sometimes denoted as p, is less that 11.07 then we accept our null hypothesis, otherwise we reject it.

O.k. so let’s calculate our X^{2} value.

X^{2} = (6 – 10)^{2}/10 + (4 – 10)^{2}/10 + (8 – 10)^{2}/10 + (5 – 10)^{2}/10 + (12 – 10)^{2}/10 + (25 – 10)^{2}/10 = 16/10 + 36/10 + 4/10 + 25/10 + 4/10 + 225/10 = 310/10 = 31.

So we have to reject our null hypothesis. Something is fishy with that die …

If you like this blog and want to learn more about the Chi-Square Test and Statistics I encourage you to check out my courses, “How to Master Integrated Math 3, Unit 1 in 7 Simple Lessons,” coming Sept. 12th or “How to Master High School Statistics in 10 Simple Lessons,” coming soon! Wow!

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