Sometimes we are interested in the sum of the n draws rather than the average. Now, we get the average by taking the sum and dividing by n, and so we get the sum back by taking n times the average. So, it turns out that the formulas for the expected value and the standard error of the sum can simply be obtained by multiplying the formulas for the sample mean by n. The important thing to keep in mind here is that the standard error for the sum actually increases at a rate of square root n. So, while the standard error of the average goes down, the standard error of the sum goes up. Another important statistic that comes up all the time are percentages. We already looked at approval ratings for US presidents. That's a question that was introduced by George Gallup in the late 1930s, and it's asked regularly every week or so. That poll tries to figure out what percentage of likely voters approve of the way that the US President is handling his job. Looking closely at this, we see that the percentage of likely voters is actually an average. It's helpful to look at this in somewhat detail because it comes up all the time. It's called a framework for counting and classifying. In this example, the population consists of all likely voters which are about 140 million adults. Each of these likely voters falls into one of two categories, either they approve of the President's handling of the job, or they don't. What do we do now is we put the label '1' on each likely voter who approves and '0' on each who doesn't approve. The reason we do that is because then the number of likely voters who approve equals the sum of all 140 million labels. To see why that is, let's look at a simple example with five voters. Suppose the first voter approves, so we give him a label '1'. The second voter doesn't, so he gets a label '0'. The third voter doesn't, gets a label '0'. The next voter approves, gets a label '1', and the last voter doesn't, and gets a label '0'. So, if we look at the sum of the labels, then that will be 2, and indeed 2 of the 5 voters approved of the President's handling of his job. So by putting 0s and 1s on the labels, the sum of the labels simply counts how many approve, and likewise one sees that the percentage of likely voters who approve is simply the percentage of 1s among all the labels. So, if we introduce labels, then sampling n likely voters means that the number of voters in the sample who are approving is simply the sum of the n draws. Likewise, the percentage of voters approving is the percentage of 1s, and that's then simply the sample average. And we can multiply by 100% just to convert the whole thing into percentages. So, the bottom line is that percentages are simply averages after introducing 0 and 1 labels. That means we can simply use the formulas for averages and we find that the expected value of the percentage of 1s is Mu times 100%, and the standard error for the percentage of 1s is sigma over square root n times 100%, where mu is the population average and sigma is the standard deviation of the population of 0s and 1s. So now, we have a number of formulas for the average, for the sum and for the percentages. But really, they all look quite similar. In the end, it comes down to sigma over square root n. But remember that all of these formulas are actually for sampling with replacement, but polls are simple random samples. So, those are samples without replacement. Now, we already saw that in the case of sampling without replacement, where the sample size is much smaller than the size of the population, these two things are roughly the same, so all of these formulas are still approximately true. In fact, it turns out the formulas for the expected values are exactly true even for sampling without replacement and for the standard error it's only approximately true and they are explicit formulas to correct for sampling without replacement. All of these formulas are also true if we don't draw from a population but we simulate data. Simulating data means we generate data according to a probability histogram, for example, by using a computer. Now, what are mu and sigma in that case? Remember, when we draw from a population, mu is simply the population average and sigma is the population standard deviation. If we simulate a random variable X that has K possible outcomes, X1 to XK, then the formulas for mu and sigma square are given here. These formulas actually also apply to the case of sampling from a population, because if we sample from a population with K possible subjects, then the probability of picking one subject is simply 1 over K and the whole thing becomes the population mean. Likewise for sigma square, where this formula simply turns into the formula for the population standard deviation squared. In the case where the random variable X has infinitely many possible outcomes such as when X follows the normal curve, they are analogous formulas, but we don't need these for the rest of the course.
