Table of Contents

• Testing differences
• 1: How to convince your boss
• 2: Looking at the data
• 3: Is the average different?
• 4: How big is big?
• 5: How to combine SDs
• 6: Combining SDs for fun and profit
• t-calc and t-crit
• 7: Getting to t-calc
• 8: A formula for t-calc
• 9: Degrees of freedom
• 10: The bottom line (finally)
• Putting it all together
• 11: Give it a try
• 12: Once more for luck
• 13: The big picture

A formula for t_calc

Let's generalize what we had on the last page -- after all, you want to be able to do t-tests with your own data, not just with my data. I will call the treatments 1 and 2, rather than Fish-2-Whale and generic.

Here are the pieces: the difference between the averages is:

| avg₁ - avg₂ |

and the combined SD is

sqrt(SD₁/n₁ + SD₂/n₂)

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 Fish-2-Whale 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 -- "x-bar" is statisticalese for "the average value", and "s" means SD. It's the same formula!