In many situations the population statistics such as the mean and variance are unknown and a sample is obtained to estimate the population mean from the sample mean; however, in many planning stage of a sampling study, before any data is collected, we can speak of the sample mean only in terms of probability, i.e. it is an unknown random variable.

If we were to take a sample of the population and estimate the sample mean and repeat this experiment several times the calculated mean will have many values, some close to each other and some not so close, the distribution of the sample mean is a the distribution of a random variable.

1. Know how to determine a distribution of the sample mean.

By Example: If several experiments are conducted to find or determine the sample mean and one observed the relative frequencies for each value of the mean, then a frequency distribution or probability distribution can be constructed for the sample mean as follows:
 

Values of the sample mean, x (1)	Frequency of x (2)	Probability, P(x) (2) divided by N
230	50	0.17
260	100	0.33
290	100	0.33
320	50	0.17
	N =300

Each value of the statistics or mean with its relative or original frequencies is called a distribution of values.

2. Know how to calculate the expected value of the sample mean.

The expected value of the sample mean is the same as the regular value of the sample mean.

3. Know how to determine the standard error of the sample mean.

The standard deviation of a statistics (mean, p, etc.) is called the standard error.

The standard deviation or the standard error of the sample mean is.

The standard error of the mean,  is what is used as the standard deviation when making estimates about the sample or population mean.

Example: Suppose the sample standard deviation is  and the sample size and n = 14 then

4. Know how to calculate the standard error of the binomial proportional constant, p.

Standard error of p (p is a random variable with a probability distribution)

The standard error of the proportion,  is what is used as the standard deviation when making estimates about the sample or population proportion.

Example: Suppose p = 0.4 and n = 14 then (1-p)= 0.6 and