8: And another way: the standard error

Graphs using standard deviation (SD) tell you what a big population of fish would look like — whether their sizes would be all uniform, or somewhat variable in size, or very variable in size. Sometimes, though, you don’t really care what a population looks like, you just want to know, did a treatment (like Fish2Whale instead of other competing brands) make a difference on the average? In that case you measure a bunch of fish because you’re trying to get a really good estimate of the average effect, despite whatever variation might be present in the populations. In statistical language we say the focus is on estimating a population mean based on a sample mean.

two men moving fish tankLet’s say your company decides to go all out to prove that Fish2Whale really is better than the
competition. They convert a supply closet into an acquarium, hatch 400 fish, and tell you to do a
HUGE experiment. The whole idea of the HUGE experiment is to get a really accurate measurement of
the effect of Fish2Whale, despite the natural differences such as temperature, light, initial size
of fish, solar flares, and ESP phenomena. The return on their investment? Really small error bars.

But how do you get small error bars? Just using 400 fish WON’T give you a smaller SD. A huge population
will show just as much variation in size as a small population. Instead, you need to use a quantity called
the "standard error", or SE, which is the same as the standard deviation
DIVIDED BY the square root of the sample size.
Since you fed 100 fish with Fish2Whale, you get
to divide the standard deviation of each result by 10 (i.e., the square root of 100). Likewise with
each of the other 3 brands. So your reward for all that work is that your error bars are much smaller:

bar graph with SE

Why should you care about small error bars? Well, as a rule of thumb, if the SE error bars for the 2
treatments do not overlap, then you have shown that the treatment made a difference.
(This is not a conclusive statistical test, but simply a way to visualize what your results mean. Many statistical tests are actually based on the exact amount of overlap of the SE bars, but they can get quite technical. For now, we’ll just assume that no overlap = a true difference between the treatments.)

So, in order to
show that Fish2Whale really is better than the competitors, NOT ONLY does the mean growth need to be
higher, but (mean minus SE) for Fish2Whale must be bigger than (mean plus SE) for the other brands. In other words, the error bars shouldn’t overlap. It’s
a little easier to see on a graph:

If you turn on javascript, this becomes a rollover


No overlap means the 2 treatments really had different effects (on average). If there is overlap, then the two treatments did NOT have different effects (on average). The good news is, you already know how to make this kind of graph. Just use the SE instead of SD and you’re good.

fish in tank cat in tank