Introduction to Data Science Version 3

Imagine a gum ball jar full of gumballs of two different colors, red and blue. The jar was filled from a source that provided 100 red gum balls and 100 blue gum balls, but when these were poured into the jar they got all mixed up. If you drew eight gumballs from the jar at random, what colors would you get? If things worked out perfectly, which they never do, you would get four red and four blue. This is half and half, the same ratio of red and blue that is in the jar as a whole. Of course, it rarely works out this way, does it? Instead of getting four red and four blue you might get three red and five blue or any other mix you can think of. In fact, it would be possible, though perhaps not likely, to get eight red gumballs. The basic situation, though, is that we really don’t know what mix of red and blue we will get with one draw of eight gumballs. That’s uncertainty for you, the forces of randomness affecting our sample of eight gumballs in unpredictable ways.

Here’s an interesting idea, though, that is no help at all in predicting what will happen in any one sample, but is great at showing what will occur in the long run. Pull eight gumballs from the jar, count the number of red ones and then throw them back. We do not have to count the number of blue because 8 - #red = #blue. Mix up the jar again and then draw eight more gumballs and count the number of red. Keeping doing this many times. Here’s an example of what you might get:

Notice that the left column is just counting up the number of sample draws we have done. The right column is the interesting one because it is the count of the number of red gumballs in each particular sample draw. In this example, things are all over the place. In sample draw 4 we only have two red gumballs, but in sample draw 3 we have 6 red gumballs. But the most interesting part of this example is that if you average the number of red gumballs over all of the draws, the average comes out to exactly four red gumballs per draw, which is what we would expect in a jar that is half and half. Now this is a contrived example and we won’t always get such a perfect result so quickly, but if you did four thousand draws instead of four, you would get pretty close to the perfect result.

This process of repeatedly drawing a subset from a “population” is called “sampling,” and the end result of doing lots of sampling is a sampling distribution. Note that we are using the word population in the previous sentence in its statistical sense to refer to the totality of units from which a sample can be drawn. It is just a coincidence that our dataset contains the number of people in each state and that this value is also referred to as “population.” Next we will get R to help us draw lots of samples from our U.S. state dataset.