The Law of Large Numbers (section 7.2) tells us that the results of a chance process repeated over-and-over again independently are unpredictable in the short-run but have a regular and a predictable pattern in the long-run. Averages and proportions from a random sample tend to hone in on their corresponding population values. More stability is associated with more trials and more volatility is associated with fewer trials.

The result is not surprising. If we think about the Expected Value of a measurement as the long-run average value, then it makes sense that we should get closer to that long-run average as we get closer to the idealized "long run".

This lesson studies the random error that tells us how far off the sample is from the population. A random sample makes all possible samples of size n equally likely.

• n = 1: If n = 1 in Example 8.1, we would just pick one person randomly from the population and the possible proportions we might get are just 0 (if the person we pick doesn't have a Bachelor's degree) or 1 (if the person we pick has a degree). A histogram of the proportions we might get from all possible samples of size n = 1 would look like Figure 8.1A.
• n = 10: Similarly, if we use a sample of size 10, then the possible sample proportions we might get are 0 (if none of the ten have a degree) or 0.1 (if one of the ten has a degree) or 0.2 (if 2 of the ten have degrees), etc... A histogram of all of the different sample proportions we might get with n = 10 is given in Figure 8.1B.
• n = 100: Finally, a histogram of the sample proportions we might get with all possible samples of size 100 is given in Figure 8.1C.

With n = 1, the sample proportions go all the way from 0 to 1. With n = 10, the preponderance of sample proportions is between 0.2 and 0.6. With n = 100, the bulk of the proportions are between 0.33 and 0.43. As the sample size grows there is less variation in the proportions we might get - that's the Law of Large Numbers. But there is another pattern emerging in Figure 8.1. As the sample size grows, the histogram of the possible sample proportions takes a familiar shape - with large samples it looks like the normal curve!