Average

• The usual function for finding the average for a set of data is to sum the data points and divide by how many of them there are. The average, when calculated this way, is also called the arithmetic mean.
  
  

Weighted Mean

• A variation on the arithmetic mean is to weight the values of a set. This can be done as a shortcut in summing up all the elements of a set. For example, if a probability distribution is handy, it is easier to find the average age of a population by multiplying each age by the proportion of the population. This is a type of "weighted mean," using probabilities as weights.
  
  

Median

• One of the shortcomings of the arithmetic mean is that it can be skewed by extremes. For example, higher-end incomes may prevent the arithmetic mean of income from giving a representative figure of what normal workers make. A solution is to find the median, the value at which half of the data points are above and half are below. That way, a few employees who make enormous amounts do not figure in significantly.
  
  The median can be found by ordering the data points and finding the middle point. If a function, f(x), describes the income distribution, then the area under its curve up to the median equals one-half. Therefore, another function, F(x), called the cumulative distribution function, equals one-half at x = median. F(x) is found from f(x) using calculus, integrating from minus infinity up to x. Setting F(x) to one-half, you can then solve for x to get the median.
  
  

Mode

• What may be an even more desirable type of average in some circumstances is the mode, which is the value that has the highest frequency in a set. Such a number would be helpful for, say, determining the typical salary at a factory. Application of the mode would be particularly meaningful when data points clump together, e.g., salaries at a factory.
  
  If a continuous function describes the distribution of data points, then the mode can be found using calculus by setting the function's derivative to zero and solving for x. This is because a tangent line at the peak of the function has zero slope. So setting the integral of f(x) to 0.5 solves for the median, while setting its derivative to zero solves for its mode.
  
  

Root Mean Squared (RMS)

• Sometimes the average in the above sense is not useful. For example, in molecular motion, when the probability of a molecule's velocity being to the left or to the right is equal, then the three averages above are each zero. But the average magnitude of velocity is still of interest for calculating, for instance, to solve for temperature, pressure and kinetic energy. The solution is to find the average of the square of the velocity, so that all terms being averaged are positive. Information about velocity magnitude can be conveyed, instead of being wiped out by cancellation of positives and negatives during summation.
  
  

Probability Distribution Functions

• Aside from continuous probability distributions, some discrete distributions have simple formulas for the arithmetic mean. The mean for the binomial distribution is np, where n is the number of trials and p is the probability of success. The Poisson distribution with parameter lambda has a mean equal to lambda itself.