MathBench presents...

A different kind of variability

The answer is that what the standard deviation tells you how much the population varies. As you do more sampling, your standard deviation should stay approximately the same. There is variability in the population, and the standard deviation is measuring it.

But when we do more and more sampling, we are also getting closer and closer to figuring out the real average. Otherwise why do more sampling? What we need is a new number that tells us how close we are to the actual mean.

I won't explain why this works, but it is a well-established fact that if you divide the standard deviation by the square root of the sample size, you get a number called a standard error (SE), and that number tells you how close you are to the true mean.

sculpture thumbSE (STANDARD ERROR) = SD / Sqrt(n)

A rule of thumb: 95% of the time, the true average lies within 2 SE's of your sample average.

So, as I do more and more sampling, n gets bigger and the standard error gets smaller. That means I can narrow in on the true average.

Let's try some examples. Let's say I have measured 9 songs (in Statisticalese, I say n=9, where “n” means “number in sample”).

Fill in the blanks

click "?" for hints    key:wrong answer, right answer

n 9
average 250 seconds
standard deviation 30 seconds
standard error

So, we are 95% sure that true average is between

and

but if I meassure 100 songs:

Fill in the blanks

click "?" for hints    key:wrong answer, right answer

n 100
average 250 seconds
standard deviation 30 seconds
standard error

So, we are 95% sure that true average is between

and
   

So all that extra sampling paid off -- we can narrow down the range around the true average from 40 seconds to 12 seconds.