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## Statistics and probability

### Course: Statistics and probability > Unit 9

Lesson 6: Binomial mean and standard deviation formulas- Mean and variance of Bernoulli distribution example
- Bernoulli distribution mean and variance formulas
- Expected value of a binomial variable
- Variance of a binomial variable
- Finding the mean and standard deviation of a binomial random variable
- Mean and standard deviation of a binomial random variable

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

For a binomal random variable, the mean is n times p (np), where n is the sample size and p is the probability of success. The standard deviation is the square root of np(1-p). We can use them to make predictions in a binomial setting. In this example, we look at how many defective chips we expect, on average, in a sample.

## Want to join the conversation?

- At3:16why don't we use sample standard deviation and sample mean. I don't think we should use population standard deviation and population mean in samples.(8 votes)
- Those rules (sample st. dev or population st. dev) do not apply to random variables, only to data sets.(13 votes)

- do the equations here for mean and SD, only apply if we've established that it is a binomial?(3 votes)
- Yes, only for binomial random variables.(3 votes)

- Why is the n under the radical for the standard deviation? Isn't n technically a constant that we should be multiplying the standard deviation by and therefore it should stay outside? This way, we're also square rooting n, which I don't understand. As I understand it, we're taking the Bernoulli distribution and expanding it by n, and usually when we do this we just multiple both the mean and std deviation by n, but in this case we multiplied the mean by n and the standard deviation by the square root of n...(1 vote)
- A binomial random variable with n trials and success probability p, is the sum of n
**independent**Bernoulli random variables each with success probability p.

One property of variance is that the variance of the sum of**independent**random variables is the sum of their variances. So the**variances**, not the standard deviations, add. Therefore, the**variance**is proportional to n, and so the standard deviation is proportional to sqrt(n).(4 votes)

- So the variance and standard deviation of the "defective parts" and the "non-defective parts" are identical values. Yes?(2 votes)
- Yes. It makes sense to. If defective parts vary by 10 the non defective parts will also vary by 10 because they are directly correlated.(1 vote)

- How to explain that mean=10 and standard deviation =3.13 in this context? what does they represent? Thank you sir(1 vote)
- Mean: On average, 10 out of 500 chips will be defective

Standard Deviation: If you repeat the experience a large number of times (testing 500 chips), approximately 68% of the sample means will be between -3.13+10 and +3.13+10.(3 votes)

- Suppose that X is a binomial random variable with a mean of 10 and a standard deviation of 2. What is the probability of success in any trial?(2 votes)
- All you need is to use substitution.

std = sqrt(np(1-p))

expected_value = np(1 vote)

- How do you deal with multiple percentage variables. He gives 2% in this example, but what if 10 parts were 2%, 20 parts were 10%, 50 parts were 49%, 35 parts were 80%. Is it a weighted average? What's the formula in this context?

Thanks(1 vote) - The board of examiners that administers the real estate brokers examination in a certain state found that the mean score on the test was 426 and the standard deviation was 72. If the board wants to set the passing score so that only the best 80% of all applicants pass, what is the passing score ? Assume that the scores are normally distributed(1 vote)
- Why is the mean not calculated here using 1*p + 0*(1-p)? And similarly for variance?(1 vote)
- The formula that you are mentioning is used, if there is only one trial( which means it is a Bernoulli distribution)distribution). for Binomial distribution there are n trials, so either you enumerate all of them and multiply them with their corresponding probabilities (which is bit hard). so go for the readymade expectation and variance formula of binomial distribution that sal uses in this video. (also see the previous videos on this thread)(1 vote)

- how do you interpret the mean of a binomial distribution in context? is it the number of trials needed to get the first success or is it number of successes in the given number of trials?(1 vote)
- The mean of a binomial distribution is the expected value (long-run average) of the number of successes in the given number of trials.

By the way, the expected value of the number of trials needed to get the first success would be the mean of a geometric, not binomial, distribution.

Have a blessed, wonderful day!(1 vote)

## 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.