Finding the mean and standard deviation of a binomial random variable | AP Statistics | Khan Academy

We're told a company produces processing chips for cell phones at one of its large factories. Two percent of the chips produced are defective in some way. A quality check involves randomly selecting and testing 500 chips. What are the mean and standard deviation of the number of defective processing chips in these samples?

So, like always, try to pause this video and have a go at it on your own, and then we will work through it together.

All right, so let me define a random variable that we're going to find the mean and standard deviation of, and I'm going to make that random variable the number of defective processing chips in a 500 chip sample. So let's let X be equal to the number of defective chips in a 500 chip sample.

So, the first thing to recognize is that this will be a binomial variable. This is binomial. How do we know it's binomial? Well, it's made up of 500; it's a finite number of trials, right over here. The probability of getting a defective chip—you could do this as a probability of success. It's a little bit counterintuitive that a defective chip would be a success, but we're summing up the defective chips, so we would view the probability of a defect—or, I should say, defective chip—it is constant across these 500 trials.

And we will assume that they are independent of each other. 0.02. You might be saying, "Hey, well, are we replacing the chips before or after?" But we're assuming it's from a functionally infinite population. Or, if you want to make it feel better, you could say, "Well, maybe you are replacing the chips." They're not really telling us that right over here. So, we'll assume that each of these trials are independent of each other and that the probability of getting a defective chip stays constant here.

And so, this is a binomial random variable or binomial variable. We know the formulas for the mean and standard deviation of a binomial variable, so the mean— the mean of X, which is the same thing as the expected value of X, is going to be equal to the number of trials (n) times the probability of a success on each trial (p).

So what is this going to be? Well, this is going to be equal to—we have 500 trials, and then the probability of success on each of these trials is 0.02. So it's 500 times 0.02. And what is this going to be? 500 times two hundredths is going to be—it's going to be equal to 10. So that is your expected value of the number of defective processing chips or the mean.

Now, what about the standard deviation? So the standard deviation of our random variable X—well, that's just going to be equal to the square root of the variance of our random variable X. So I could just write it—I'm just writing it all the different ways that you might see it because sometimes the notation is the most confusing part in statistics.

And so this is going to be the square root of what? Well, the variance of a binomial variable is going to be equal to the number of trials times the probability of success in each trial times one minus the probability of success in each trial.

And so in this context, this is going to be equal to—you’re going to have the 500—500 times 0.02. 0.02 times 1 minus 0.02 is 0.98, so times 0.98. And all of this is under the radical sign. I didn't make that radical sign long enough.

And so what is this going to be? Well, let's see. 500 times 0.02, we already said that this is going to be 10. 10 times 0.98, this is going to be equal to the square root of 9.8. So it's going to be, I don't know, three point something. If we want, we can get a calculator out to feel a little bit better about this value.

So I'm going to take 9.8 and then take the square root of it, and I get 3.5. Round to the nearest hundredth: 3.13. So this is approximately 3.13 for the standard deviation. If I wanted the variance, it would be 9.8, but they ask for the standard deviation, so that's why we got that. All right, hopefully, you enjoyed that.