Suppose I roll a six-sided die 1000 times and write down the number of times each face comes up. How do I test whether the die is fair? Can I use a chi-squared test where the expected number of each face is 1000/6=167?

There also appears to be a multinomial test, but that seems less likely to be baked into stats packages and software.

Related question.

  • 2
    $\begingroup$ You can certainly use a chi-square (but don't round off the expected value; leave it at 1000/6). Some directly relevant posts: A, $\, $ B, $\, $ C, $\, $ D, ... (ctd) $\endgroup$
    – Glen_b
    Dec 26, 2013 at 5:41
  • $\begingroup$ (ctd)... and some potentially relevant discussion in E. Yes, the multinomial test should work, but the chi-square should do just about as well. $\endgroup$
    – Glen_b
    Dec 26, 2013 at 5:42
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    $\begingroup$ A multinomial exact test could need to consider ${1005 \choose 5} = 8,\!459,\!043,\!543,\!951$ cases. Even allowing for symmetrical possibilities, that leaves $12,\!193,\!703,\!764$ distinct cases, which is still rather large. There are further efficiencies possible, but this is still probably not the way to go. $\endgroup$
    – Henry
    Dec 26, 2013 at 9:38
  • $\begingroup$ Thanks to all who referred me to other questions, I wasn't finding them on my own. $\endgroup$
    – dfrankow
    Dec 26, 2013 at 22:52

1 Answer 1


Apply the chi-square test of goodness of fit with number of possible cases - 1 degrees of freedom and the null hypothesis being the discrete uniform distribution as you pointed out. This is a textbook example for that.


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