A formula for t_{calc}
Let's generalize what we had on the last page  after all, you want to be able to do ttests with your own data, not just with my data. I will call the treatments 1 and 2, rather than Fish2Whale and generic.
Here are the pieces: the difference between the averages is:
 avg_{1}  avg_{2} 
and the combined SD is
sqrt(SD_{1}/n_{1} + SD_{2}/n_{2})
So the whole formula for t_{calc} is:


But even more important ... what does this formula mean?
You can think of the top part of the fraction as the difference created by the treatment ... in our case, how much difference did Fish2Whale make?
And you can think of the bottom part of the fraction as the amount of random variation present in the experiment  variation that is not due the treatment itself, but just random "slop".
So the ratio says "how important was my treatment compared to random slop?" The bigger the ratio, the more important your treatment.
In textbooks, you may see this written as
Don't despair  "xbar" is statisticalese for "the average value", and "s" means SD. It's the same formula!
photo credits: slop
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