In mathematics, “average” refers to a specific arithmetic calculation, while “mean” can be synonymous with “average” or refer to an entirely different type of calculation. A statistical mean of discrete random variables and an arithmetic mean are calculated in the same manner as average; for all intents and purposes, they are the same.

### Statistics

To understand the difference between mean and average, we need to understand how mean is calculated in statistics. In statistics, a distribution is the set of all possible values for terms that represent defined events. For example, all of the test results of junior high school history class would be a distribution. Distributions are made up of variables. Our example of test results illustrates a discrete random variable--random because the result is not known before hand and discrete because the value is precise and isolated (in other words, the test result must be one number between 0 and 100). Another type of random variable is the continuous random variable. A continuous random variable differs from a discrete random variable in that the value of a continuous random variable can fall anywhere within an unbroken and unbounded interval or span (a temperature, for example). Finding the mean of continuous random variables is significantly more difficult than finding the mean of discrete random variables.

### Mean of Discrete Random Variables

To arrive at the statistical mean of a distribution of discrete random variables, simply add up all the values and divide that total by the number of values in the distribution. This value is the mathematical average of all the terms in the distribution.

### Mean of Continuous Random Variables

The mean of a continuous random variable is the biggest difference between mean and average. The mean of a distribution of continuous random variables is obtained by integrating the product of the variable with its probability as defined by the distribution. If we wanted to find the mean of a distribution of temperature readings, we would need to integrate the probability of each temperature appearing in our measurements before we could calculate the mean of this distribution, a significant difference from finding the mean of a distribution of discrete random variable, which requires no probability factor. Statisticians call this mean the “expected value.”

### Arithmetical Mean and Average

In arithmetic, “mean” is a common abbreviation of “arithmetical mean,” a value obtained by taking a set of number, say, (7, 5, 2, 1, 1, 6, 3, 3 ). There are eight numbers in this example, but we can have as many as we want. Add all the elements and then divide by the number of elements to arrive at our “arithmetic mean” or “average”--(7+5+2+1+1+6+3+3)/8 = 28/8 = 3.5. In this case, “mean” and “average” are synonymous.

### Geometric Mean

However, another type of mathematical mean is the “geometric mean,” which is obtained by the following method: multiply all the elements of a set of numbers and then take the Xth root, where X equals the numbers of elements in the set. For example: (7_5_2_1_1_6_3*3)^(1/8) = 2.66179.

### Harmonic Mean

Yet another type of mathematical mean is the “harmonic mean,” which is obtained in much the same way as the “arithmetic mean,” with the principal difference being that the calculation is inverted: 8/(1/7+1/5+1/2+1/1+1/1+1/6+1/3+1/3) = 2.17621.