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

Combined SDs for fun and profit

So let's combine those standard deviations!

	n	average	difference	standard deviation (SD)	combined SD a/k/a "pooled variance"
Fish-2-Whale	8	267 g	94 g	44
Control	8	173 g		28

 

Note that the combined SD is actually a bit smaller than either SD on its own. That's the effect of sample size. What if we used 800 fish instead of 8 in each tank? The average and SD for each tank probably would not change much, but what about the combined SD?

	n	average	difference	standard deviation (SD)	combined SD a/k/a "pooled variance"
Fish-2-Whale	800	267 g	94 g	44
Control	800	173 g		28

Now the combined SD would be 1.85, which means we could detect much smaller shifts in growth rates.

Or, we could even try to be clever, and say, "well, it's the Fish-2-Whale tank that has the bigger variance, so let's add more fish to that tank, and fewer to the generic tank."

	n	average	difference	standard deviation (SD)	combined SD a/k/a "pooled variance"
Fish-2-Whale	1000	267 g	94 g	44
Control	600	173 g		28

Remember, average growth and SD still don't change, and the TOTAL number of fish we needed to take care of didn't change either -- but the pooled SD went down even a little more, from 1.85 to 1.81.