Join Eddie Davila for an in-depth discussion in this video Discrete mean, part of Statistics Foundations: Probability.
- [Instructor] Discrete random variables. They are experimental results, often characterized by whole numbers. They cannot be decimals. Suppose we monitor the number of drinks ordered per customer at a particular Starbucks location between ten AM and 11 AM. The number of drinks ordered per customer would be discrete. I mean, you couldn't order half a drink or 30% of a drink. Here's the data for the number of drinks ordered per customer at a particular Starbucks between ten AM and 11 AM. In that one hour, there were 40 different paying customers. Note, we're only counting the number of drinks in their order. 22 customers ordered only one drink. 10 customers placed an order for two drinks in their purchases. We even had one customer that ordered zero drinks during their purchase. Perhaps they only bought a pastry. As you can see, during this one hour, each customer ordered between zero drinks and five drinks. Using these discrete drink totals, we can find the relative frequencies. 0.025, or 2.5% of customers, ordered zero drinks; 0.55, or 55% of customers, ordered a single drink; and so on. To find the mean of this discrete probability distribution, we will use a weighted mean. To do this, multiply zero drinks times the relative frequency, 0.025. Multiply one drink by 0.55, 2 drinks by the relative frequency of 0.25, and so on. Those products are your weights. When you add those weights, you get 1.68 drinks. This is the mean of this probability distribution. Our average customer between ten AM and 11 AM ordered 1.68 drinks. Next, let's find the standard deviation for this probability distribution. This is so lengthy, so let's use a table. The first column is the number of drinks ordered. The second column is the mean we just calculated, 1.68. Our third column is the difference between the drinks ordered and the mean. We then square those values in column four, but we're not done yet. Let's carry over those squared values to column C in a new table below. This table also has a number of drinks ordered in column A, the relative frequencies in column B. In column D are the products of multiplying the squared values from column C and the relative frequencies from column B. The sum of those values is 1.07 drinks squared. This is our variance, but since working with drinks squared is tricky, we'll take the square root of 1.07, which is 1.03 drinks. Therefore, our standard deviation for this discrete probability is 1.03 drinks. This is a nice opportunity to put together a few things we've covered throughout our statistics course. The average customer in Starbucks between ten AM and 11 AM ordered 1.68 drinks in their drink purchase. The standard deviation for this probability distribution is 1.03 drinks. So one standard deviation from the mean is our mean, 1.68, plus one times our standard deviation of 1.03. This gives us 2.71. We can then find that formula to find the value for one standard deviation from the mean in the negative direction. So our calculated range, one standard deviation away from the mean, is 0.65 to 2.71. But since drink orders are discrete, this range really accounts for drink orders of one and two. We didn't get all the way down to zero, so we're at one. We didn't get all the way to three drinks, so we're at two. How about two standard deviations from the mean? Okay, and we can copy our formula here and here. So now, our range for two standard deviations from the mean goes from negative 0.38 to 3.74 drinks. Obviously, we can't order negative drinks, so our lower limit, our discrete range, is going to be zero, and we're not quite at four drinks, so we're going to go to three drinks. And then finally we'll do the same thing for three standard deviations. Once again, we can't go lower than four drinks, so this is going to be zero, three standard deviations lower, and three standard deviations in the positive direction, not quite five drinks, it will be four drinks. So if we look at one standard deviation from the mean is from one drink to two drinks, so according to our table, this accounts for 80% of our customers. If we go to two standard deviations, two standard deviations would go from zero drinks to three drinks. This accounts for 92.5% of all customer orders. And if we decide to go for three standard deviations, from zero drinks to four drinks, that would account for 97.5% of all our drinks. And you might remember that three standard deviations should actually capture 99.7% of our data points. So according to our calculations, you might classify that order of five drinks as an outlier. And just like that, you're becoming a real statistician.
Eddie explains that probability is used to make decisions about future outcomes and to understand past outcomes. He covers permutations, combinations, and percentiles, and goes into how to describe and calculate them. Eddie introduces multiple event probabilities and discusses when to add and subtract probabilities. He describes probability trees, Bayes’ Theorem, binomials, and so much more. You can learn to understand your data, prove theories, and save valuable resources—all by understanding the numbers.