Significance

The measure of whether the results of research were due to chance. The more statistical significance assigned to an observation, the less likely the observation occurred by chance.

p-value

The way in which significance is reported statistically (i.e. p<.01 means that there is a less than 1% chance that the results of a study are due to random chance). Note that generally p-values need to be fairly low (.01 and .05 are common) in order for a study to make any strong claims based on the results.

Example:

A study had one group of students (Group A) study using notes they took in class; the other group (Group B) studied using notes they took after class using a recording of the lecture. Students in Group A scored higher on a test than Group B. The study reports a significance of p<.01 for the results.
This means that whatever the reason students who took notes in class did better on the test, there is only a 0 - 1% chance that the results are due to some random factor (such as Group A having smarter students than Group B).

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Basic Statistical Terms