The Chi^2 test statistic can be less than or equal to 1. It happens to be zero e.g. when for all categories the observed count equals the expected count. This means a perfect match. It cannot be less than 0 because of the squaring of the differences between observed-expected.
The Chi square test can be equal to zero or more. It equals zero when expected/theoretical values are equal to the observed ones, in which case you accept the null hypothesis.
@Abdelmajid Amine: you almost never accept the null hypothesis. Even in the ideal case of your observations being exactly what you expect: it still may be by chance from a slightly different model than the null hypothesis one.
You are Wright Emmanuel. The exact formulation is 'not to reject' in stead of 'accept' the null hypothesis.
Is it true for Rietveld Refinement of X-ray diffraction data that chi square can be less than or equal to 1 ?
It is true for any chi-square, otherwise it cannot be a chi-square. But, if the degrees of freedom are « great », it is probably the sign that something's going wrong in your fit: overestimation of the residual variance or overfitting, for instance. Or you were especially lucky when performing the experiment...
as the denominator is always with power 2 you can not find it negative (less than 1)
I got it now. I think because the equation of chi square based on the summation of the power of the subtraction of the expected from the observed divided by expected.
So, you always have a larger value in the Numerator rather than denominator.
please check this link for the equation
https://en.wikipedia.org/wiki/Chi-squared_test
No, you don't. Chi-square *can* be below 1, and it happens. Especially for small degrees of freedom. To convince yourself, just try a chi square on the following table:
10 11
14 15
In fact, the numerator can be as small as you want, because it's differences between two values, hence the closest the values, the smallest the numerator.
Thank you. I think you have to understand the ASSUMPTIONS of Chi square test first. Once you get you will understand my previous answer.
please, reread toe assumptions of chi square test.
I think *you* have to understand the assumptions of a chi-square test... Under the null hypothesis, the expected and observed values should be the same, that is all your numerators should be 0, so the chi-square should be 0.
Did you tried the chi-square on the table I sent ? You would have observed a chi-square well below 1.
NB: just a guess to try to understand you... if by « positive » you mean « test significant »... Then use a correct wording, because it confuses everyone. In that case, for usual type I errors, yes indeed for a chi-square test to be significant, it mus be > 1. But that was not the question asked. And your « demonstration » is unclear.
thank you.
The first assumption of chi square is " No more than 20% of the expected counts are less than 5 ". this condition is cracked you your "mentioned 2x2 table"
Thank you.
Dear Hakan Tarık Meriç
The chi-squared is one of the most studied and well-known distributions in statistics. The following is from Wikipedia:
"The chi-square distribution (also chi-squared or χ2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables."
As Emmanuel Curis says, values between 0 and 1 are easily found with few degrees of freedom. For example, pchisq (1, 1) = 0.6826895, then more than two-thirds of the values in the distribution are less than 1. With 2 degrees of freedom, pchisq (1, 2) = 0.3934693, and for 3 df, pchisq (1, 3) = 0.198748.
It is true that for many degrees of freedom, the proportion decreases rapidly, but, as it is a continuous distribution, there are infinite values less than 1. Even zero is a value with zero probability, but it is possible to find when each one of standardized normal variables is zero.
I hope you are convinced from the existence of chi-square values less than 1 :)
Ahmed Khalifa Khalifa : you are confusing the value of khi-square distance (or test criterion), that can be computed for any table and requires no assumption, and the approximation of the exact distribution of this distance under the null hypothesis of independence, that is acceptable (in usual contexts) if all theoretical counts are > 5 (other conditions exist however, like the "less than 20 % of counts < 5 and no 0, in the theoretical table" you gave; in practice, that most often does not have a great importance, if you really need to use one rule and not the other, you probably should instead use an exact test...).
And if that still does not convince you, and neither does Jorge Ortiz Pinilla's answer ... Just replace the table I gave by almost ten times the table:
100 101
141 151
All theoretical counts are well above 5, and I let you the pleasure to discover the value of the khi-square criterion...
The Chi-square value can be less than 1 but it can not be negative. Read more
https://www.reneshbedre.com/blog/chi-square-test.html
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