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Comparing Two Means :: What's wrong with just comparing averages in groups?
Let's say you want to know which professors have a higher average salary those in New York State or those in California. Both have their share of top-tier schools, have similar populations and costs of living.  Since you don't have time to ask every professor how much they make, you instead ask 10 schools from each state the average salary. Click sample on the graphic below to simulate sampling 10 professors from each state.


Click the Sample button several times to see how the samples and difference between states change. You can hover over any graduation cap to see the average salary from a particular school. They are sorted from lowest to highest.

This sampling exercise demonstrates that samples from a population will vary, often substantially. If you need an estimate of a difference, depending on the actual difference between the two groups chance fluctuations could be misleading you. We need a way to take into account chance when we compare groups. 

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