# A Chi squared test for data with error

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1. Jun 16, 2016

Hi everyone.

I am totally new to statistics so my question may or may not be simple!
I know that for the data fitting we can do a chi squared test like:
$$\chi^2 = \Sigma \frac{(f_{data}-f_{model})^2}{(error_{data})^2}$$

So I have been doing this for a while, but now I have some data with different error, let's say like:
$$f_i = 2 ^{+0.9}_{-0.1}$$
How should I do the chi squared test for this?! What should I consider as the error? 0.9 ? 0.1?

2. Jun 16, 2016

### EnumaElish

Notation in (2) is unfamiliar. Do you mean range = [1.9, 2.9]?

3. Jun 17, 2016

yes, it means it can go from 1.9 to 2.9.
But until now, I have used Chi squared test only for normal distributions, which are for instance:
$$f_i = 2_{0.1}^{0.1}$$
i.e. error in both sides are the same.

4. Jun 17, 2016

### chiro

The chi-square test you are thinking of is regression based and I'm wondering why you can't transform the variance if it isn't in some standard form.

Usually doing transformations on random variables to get evaluate a test statistic is common and the most used one is standardizing a Normal distribution where you have Z = (X - mu)/sigma.

A similar transformation can be done to get it in the normal chi-square form and inferences based on this transformation can be made.

5. Jun 17, 2016

### EnumaElish

I'm still uncomfortable with the notation. What is the significance of "2"? Is it the mean or the median or the mode? In the case of [1.9, 2.9] isn't it possible to re-center the distribution so as to make it symmetric?

6. Jun 20, 2016

I think it means that the mean is 2. and as the distribution is normal, the $$\mu^2=0.1$$
However if the distribution is not normal, then $$\mu^2$$ would be different from left and right side of the mean.

7. Jun 20, 2016

Dear Chiro,

So you mean I can change the shape of the distribution to a nomal one?
but is it a right thing to do?
I mean if there are observational points, then doesn't this change the data completely?!

8. Jun 20, 2016

### EnumaElish

No it does not change the data. Suppose there is a test for determining if a sample is from a normal distribution. Suppose there isn't a test for determining if a sample is from a lognormal distribution. If the data are suspected to be lognormal, what are we going to do? Well, we can "log the data" so as to turn them into data distributed normally. Then apply the normality test. That's possible because the "log" of lognormal is normal. Chiro is suggesting a similar transformation.