In another article I mention that statistics are simply mathematical characteristics of samples. The similar characteristics of the populations from which the samples are drawn are called parameters. The fundamental role of statistical analysis is to determine how close the statistics are to the parameters.
Probably the two most important mathematical characteristics with which statistical analysis is concerned are averages and measures of variability. Averages are also known as measures of central tendency, which pretty well sums up their function. If you have a set of data – the heights of a group of people, for example – ordered by size, the average attempts to tell you where the middle of that distribution is.
Two types of average are most important in statistical analysis. The first is the mean, or arithmetic average. The mean height of a set of heights is simply the result of adding up everyone's heights and dividng by the number of people. The median, on the other hand, is the height which has equal numbers of people with other heights above and below it. This is often a theoretical value obtained by dividing the two middle scores or by interpolating a value between them in some other way.
If the median is different from the mean, the distribution is said to be skewed. Of course, the mean and the median of a sample will rarely be exactly identical, so statisticians have devised formulas for determing the probability of different sizes of difference between mean and median in a sample if there is no difference in the population.
If the distribution is skewed, that means not only that the median is the better measure of central tendency but that if you want to do statistical testing some common testing techniques may (I repeat, may) be inadvisable.
Once you know the average for a sample, you can determine how close your sample average is to the average of the population. That requires knowledge of the other mathematical characteristic which statistical analysis deals with, variability. We'll look at that in another article.