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## Binomial mean and standard deviation formulas

# Finding the mean and standard deviation of a binomial random variable

## Video transcript

- [Instructor] 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? 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 gonna find the mean
and standard deviation of, and I'm gonna make that random variable the number of defective processing chips in a 500-chip sample. Let's let x be equal to the
number of defective chips ... in 500-chip sample. 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, a 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, 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. So this is a binomial random variable, or binomial variable,
and we know the formulas for the mean and standard deviation of a binomial variable. 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, times 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 2 hundredths is going to be, it's going to be equal to 10. That is your expected value of the number of defective processing
chips, or the mean. Now, what about the standard deviation? 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. 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. So in this context, this
is going to be equal to, you're gonna have the 500, 500, times 0.02, 0.02, times one minus 0.02, is .98, so times 0.98. All of this is under the radical sign. I didn't make that
radical sign long enough. 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. 3-point-something. If we want, we can get a calculator out to feel a little bit
better about this value. I'm gonna take 9.8, and then take the square root of it, and I get, if I 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.