# 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, its 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.

photo credits: fish tank | fish farm