Student: When I was looking at the
normal distribution and the
skewed distribution activities, I noticed that the
histograms never looked much like the curves. When I made the bin size very big, the heights looked good
but it looked like a row of large blocks. When I made the bin size small, the heights were
very blocky because they only had a few values.

Mentor: Very good.

Student: So what do the curves really mean? Are they just guesses?

Mentor: In a way it is as if you took an average of all the graphs you made, with all the different
bin sizes. But there is a better mathematical way to look at it. Do you remember
experimental and
theoretical probability?

Student: Yes. Theoretical probability is when you add up the favorable outcomes and divide them by the
total outcomes. Experimental probability is when you try an experiment a lot of times, and
divide the successes by the trials.

Mentor: And when you did something a lot of times, those two numbers got very close to each other,
didn't they?

Student: Yes.

Mentor: And if you did it an infinite number of times, which is to say if you did it forever, you'd
actually get back the theoretical probability. You'll learn more about
infinity when you study fractals, but for now what's important about it is that big numbers that are
not infinite act more and more like infinity as they get closer to it. So we can believe that
the theoretical probability of something is the same as if you ran the experiment an infinite
number of times.

Student: So if I flipped a coin, and kept track of the number of heads, forever, I'd really get half
heads.

Mentor: Yes. And if you flipped two coins at a time, and kept track of how many heads you got each
time, you might get a histogram like this:

Student: That looks like some of the histograms I had when I set the bin size very large.

Mentor: Yes. It wouldn't make sense to change the bin size on that graph, but what if we flipped more
than two coins? Then there would be more possible values. If we flipped twenty coins each
time, the theoretical graph looks like this:

Student: That looks more like the normal curve than a lot of my histograms.

Mentor: Remember this is infinite trials. But you are seeing something that is important. As you have
more and more coins per flip, you have more and more possibilities. When you have infinite
trials, you get back an exact theoretical height, and as you get out toward infinite
possibilities, you get more and more points with intermediate heights. When you have infinite
possibilities and infinite trials, you get a smooth curve. Now how do you think this might be
different between statistics and probability?

Student: Well, in statistics you can't always repeat the same experiment twice, much less infinite
times.

Mentor: There's one more thing. In statistics you have as close to infinite possibilities as your
measurement allows, partly because of what you just said. You might be measuring the heights
of different plants, which could have an infinite number of random variables controlling how
tall they grew. But the same idea still works. If you perform an infinite number of trials of
an experiment with infinite possibilities, then the result will be a smooth theoretical curve.