Changes

\hat{p}=\frac{\sum\limits_{i=1}^Mx_i}{\sum\limits_{i=1}^MN_i}

\hat{p}=\frac{\sum\limits_{i=1}^Mx_i}{\sum\limits_{i=1}^MN_i}

</math>

</math>

is the fraction of trials that are successful in the all experiments. The chi-square test is, on average, at a minimum when all success probabilities are equal (<math>p_i=p</math> for <math>i=1,2,\ldots,M</math>) and will, on average, increase whenever the data follows any other hypothesis (<math>\sum\limits_{i=1}^M

is the fraction of trials that are successful in all experiments. The chi-square test is, on average, at a minimum when all success probabilities are equal (<math>p_i=p</math> for <math>i=1,2,\ldots,M</math>) and will, on average, increase whenever the data follows any other hypothesis (<math>\sum\limits_{i=1}^M

\left(p_i-\bar{p}\right)^2>0</math> where <math>\bar{p}</math> is the mean of the success probabilities). Thus the chi-square test is an effective hypothesis test for the data and hypotheses from Blount et al.

\left(p_i-\bar{p}\right)^2>0</math> where <math>\bar{p}</math> is the mean of the success probabilities). Thus the chi-square test is an effective hypothesis test for the data and hypotheses from Blount et al.