AP Notes, Outlines, Study Guides, Vocabulary, Practice Exams and more!

Sample Mean and Variance

Sample Mean and Variance

Consider a set of independent and identically distributed random variables

X1, X2, ..., Xn, where E(Xi) = m and Var(Xi) = s2, i = 1, 2, ..., n. These random variables can be considered as elements of a random sample from an infinite population having a probability distribution with mean m and variance s2. For this random sample, the sample mean and variance are given by the following equations:


The sampling distribution of the mean is the probability distribution of the mean of a random sample. Its mean and variance can be easily calculated as follows:


The sampling distribution of the mean has the same mean as the original population, but its variance is smaller than that of the original population by a factor of 1/n. The square root of the variance of the sample mean is also known as the standard error of the mean.

The sampling distribution of the sample variance is the probability distribution of the sample variance of a random sample. Its mean is calculated as follows:




The above computations show why the factor in the equation for the sample variance is (1/(n-1)) instead of (1/n)--this is true so that the expected value of the sample variance will equal the variance of the population.

The variance of the sample variance has a mathematical form that depends on the probability distribution of the parent population. Thus, its value will differ from one probability distribution to another, even if the distributions have the same mean.


Subject X2: 

Need Help?

We hope your visit has been a productive one. If you're having any problems, or would like to give some feedback, we'd love to hear from you.

For general help, questions, and suggestions, try our dedicated support forums.

If you need to contact the Course-Notes.Org web experience team, please use our contact form.

Need Notes?

While we strive to provide the most comprehensive notes for as many high school textbooks as possible, there are certainly going to be some that we miss. Drop us a note and let us know which textbooks you need. Be sure to include which edition of the textbook you are using! If we see enough demand, we'll do whatever we can to get those notes up on the site for you!