The **theoretical** operator can be used to calculate the actual result of a summary function applied to population with a particular distribution or the expected value of a summary function applied to a sample from that distribution. The probability distribution is the left operand and can be described by a probability function with a set of parameters as the left argument, or with two-column matrix representing a relative frequency distribution.

The syntax of the criticalValue operator is:

[*Parameters*] distributionFunction|relativeFrequency **theoretical** *summaryFunction ***0***|Value*

The parameters are optional for the normal and rectangular distributions, defaulting to 0 1. Some examples follow:

What is the expected number of heads if a fair coin is tossed ten times?

** 10 0.5 binomial theoretical mean 0
5**

What is the standard deviation?

** 10 0.5 binomial theoretical sdev 0
1.5811**

What is the expected number of spades in a bridge hand? .

** 13 13 52 hyperGeometric theoretical mean 0
3.25**

What is the interquartile range?

** 13 13 52 hyperGeometric theoretical iqr 0
2**

What is the skewness of a chiSquare distribution with 5 degrees of freedom?** **

.** 5 chiSquare theoretical skewness 0
1.2649**

Find the third quartile of a distribution whose minimum value is 5, maximum value is 10 and most frequent value is 9: What is the critical value of the test statistic for a one-sided upper-tail t-test with 5 degrees of freedom and a significance level of .01?

** 5 9 10 triangular theoretical quartile 3
8.87298**

From past experience, an insurance company knows there is a 50% chance there will be no claims in the next week, a 25% chance there will be exactly 1 claim, a 15% chance there will be 2 claims and a 10% chance there will be 3 claims. What is the expected number of claims?

** **** ****RF←0 1 2 3,[1.5]0.5 0.25 0.15 0.1
RF theoretical mean 0**

What is the variance?

**RF theoretical var 0
1.0275**