Statistics

The Statistics module is used to compute the mean or expected value or weighted average and standard deviation of a sample or population.

A sample screen that includes both the data and solution appears next.

Value. The first column contains the numerical values (x_i)

Frequency or Probability. The frequencies for each category are entered here. These must be nonnegative but do not need to be integer, nor do they need to sum to anything special (such as 1 or 100), since the program will total this column and scale the results. If they are less than one and sum top one then the program will assume that these are probabilities.

In the preceding screen we display a 5-category problem and its solution. The solution includes the following:

Total. This is the total of the frequency column and as mentioned before, is used for scaling. Since the total is 1 we assume that a probability distribution is being used.

Probability. This column represents the scaled frequency for each category given by the frequency divided by the total frequency. In this example it is identical to the original probabilities.

Cumulative. The cumulative probability is presented.The cumulative probability is simply the running sum of probabilities. For example, the cumulative probability for value 3 3 is .1 +.2 +.3 = .6.

Value*probability. This column is used to compute the weighted average or expected value of the given frequency distribution. In this example, the column total is 2.1 which is the weighted average of the two columns or the expected value or mean of the distribution.

x-xbar In order to compute the standard deviation we need to compute the values x_i-the average of the x_is

x-xbar^2 The previous value is squared.

(x-xbar^2)*p(x) The squared values are weighted buy the probabilities and summed. The variance for this data is 1.29.

Weighted averages

In the example below we have entered data that is different from before. First, the values range from 14 though 20 rather than from 1 to 5 and second the frequencies are each greater than 1. Thus, these are observations rather than a probability distribution.

The total number of observations is 200 so the probability column is computes as the frequency divided by 200. The average is 16.8 and the variance is 2.66.