Mistake Master
The Test Says Something Differs. It Never Says What, or Which Way
You'll learnhow to carry out a chi-square test — adding one squared, scaled discrepancy per cell, reading the p-value from the upper tail with degrees of freedom that come from the table's shape, writing the conclusion in context, and the three conclusions a significant chi-square does not license.
The statistic adds up how far each cell landed from what the null predicted, squared so that overshoots and undershoots both count, and divided by the expected count so that a gap of 20 means more where 30 were expected than where 300 were. For the three-school table that total is 12.15 on 2 degrees of freedom, and the p-value is the area to its right — always the right, because only a large statistic disagrees with the null. At 0.0023 it clears any usual standard, and then the sentence has to be written carefully, because a chi-square result invites three conclusions it cannot support: which cell is responsible, what caused it, and, when the p-value is large, that the variables are independent. It cannot even say which direction the difference runs, and this page proves that rather than asserting it: swap the two response columns so school A runs low instead of high, and the statistic comes back identical to the last decimal. Twelve short steps; press Next for each one.