The sample mean represents an average. For example, if a sample data set contains the numbers 10, 12, 14, 15 and 16, the mean would be calculated by adding all of the numbers together and dividing them by five. In this case, the sample mean would be approximately 13.4. One of the properties of a sample mean is that it is equal to or identical to the population mean. Therefore, if the sample mean is determined to be 13.4, then it is assumed that the population's mean is also 13.4.
If the population's distribution is normal, then it is assumed that the distribution of the sample mean is also normal. A normal distribution means that 50 percent of the data set is greater than the mean and 50 percent is less than the mean. Since all of the variables in a data set will not be equal to the mean, they must either fall above or below it. For a sample data set that contains 10 independent variables, five of those variables will be less than the mean, and five will be greater than the mean.
The sample variance is used to estimate the variance of the population. It is considered to be an unbiased estimate, since it is independent from the sample mean. The sample variance is calculated by first subtracting each variable in the data set from the sample mean. All of these figures are then added together and squared. Finally, the squared figure is divided by the amount of numbers in the data set minus one. A variance is a prediction of how much an independent variable is expected to deviate from the mean. For example, a sample data set that has a variance of three would be interpreted as the average population variable is expected to be either three figures above or below the mean.
The Central Limit Theorem states that as sample sizes become larger, the sample mean's distribution will become normal. Not every sample population is normally distributed. There are some that are positively or negatively skewed, with the majority of the independent variables falling above or below the mean. When sample sizes increase, more of those independent variables will be evenly distributed to either side of the sample mean. This is why many researchers and statisticians stress the importance of obtaining a large enough sample size so that the results are as accurate as possible.