The terms mean, median, mode, and range describe properties of statistical distributions. In statistics, a distribution is the set of all possible values for terms that represent defined events. The value of a term, when expressed as a variable, is called a random variable.

There are two major types of statistical distributions. The first type has a discrete random variable. This means that every term has a precise, isolated numerical value. An example of a distribution with a discrete random variable is the set of results for a test taken by a class in school. The second major type of distribution has a continuous random variable. In this situation, a term can acquire any value within an unbroken interval or span. Such a distribution is called a probability density function. This is the sort of function that might, for example, be used by a computer in an attempt to forecast the path of a weather system.

**mean**

The most common expression for the mean of a statistical distribution with a discrete random variable is the mathematical average of all the terms. To calculate it, add up the values of all the terms and then divide by the number of terms. This expression is also called the arithmetic mean. There are other expressions for the mean of a finite set of terms but these forms are rarely used in statistics. The mean of a statistical distribution with a continuous random variable, also called the expected value, is obtained by integrating the product of the variable with its probability as defined by the distribution. The expected value is denoted by the lowercase Greek letter mu (µ).

**median**

The median of a distribution with a discrete random variable depends on whether the number of terms in the distribution is even or odd. If the number of terms is odd, then the median is the value of the term in the middle. This is the value such that the number of terms having values greater than or equal to it is the same as the number of terms having values less than or equal to it. If the number of terms is even, then the median is the average of the two terms in the middle, such that the number of terms having values greater than or equal to it is the same as the number of terms having values less than or equal to it. The median of a distribution with a continuous random variable is the value m such that the probability is at least 1/2 (50%) that a randomly chosen point on the function will be less than or equal to m, and the probability is at least 1/2 that a randomly chosen point on the function will be greater than or equal to m.

**mode**

The mode of a distribution with a discrete random variable is the value of the term that occurs the most often. It is not uncommon for a distribution with a discrete random variable to have more than one mode, especially if there are not many terms. This happens when two or more terms occur with equal frequency, and more often than any of the others. A distribution with two modes is called bimodal. A distribution with three modes is called trimodal. The mode of a distribution with a continuous random variable is the maximum value of the function. As with discrete distributions, there may be more than one mode.

**range**

The range of a distribution with a discrete random variable is the difference between the maximum value and the minimum value. For a distribution with a continuous random variable, the range is the difference between the two extreme points on the distribution curve, where the value of the function falls to zero. For any value outside the range of a distribution, the value of the function is equal to 0.

*Contributor(s): Andrew Giles-Peters*

*This was last updated in*September 2005

*Posted by:*Margaret Rouse

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