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### Course: Statistics and probability>Unit 9

Lesson 6: Binomial mean and standard deviation formulas

# 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.

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• At why 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.
• Those rules (sample st. dev or population st. dev) do not apply to random variables, only to data sets.
• do the equations here for mean and SD, only apply if we've established that it is a binomial?
• Yes, only for binomial random variables.
• 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).
• 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.
• So the variance and standard deviation of the "defective parts" and the "non-defective parts" are identical values. Yes?
• 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 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!
• 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?
• All you need is to use substitution.

std = sqrt(np(1-p))
expected_value = np
(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)