# The Critical Value Operator

The critiicalValue (critVal) operator is the inverse of the probability (prob) operator.  Like the distribution form of the probability operator, it usually takes a continuous distribution function as its left operand and a relational function as the right operand.    The left argument consists of the distribution parameters.  The right argument must be a probability, and the result is a particular value of the probability distribution.

The syntax of the criticalValue operator is:

[Parameters]  distributionFunction critVal relationalFunction Probability

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

What is the 95th percentile of the normal distribution?

```   normal critVal > .95 1.6449```

The above expression can be interpreted as the value of the standard normal distribution which is greater than 95% of all values.

The criticalValue operator can also be used in lieu of the t, chi-square and F tables:

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 tDist critVal < .01 3.3649```

The above expression can be interpreted as the value of the Student t distribution which is exactly less than 1% of all values.

For a 2-sided test with the same degrees of freedom and significance level:

```   5 tDist critVal between .01 ¯4.0321 4.0321```

The above expression can be interpreted as the two values of the Student t distribution which are between 1% of all values, that is greater than 0.5% and less than 0.5% of all values.

What are the lower and upper bounds of a chi-Square distribution with 14 degrees of freedom that correspond to a 5% significance level?

```   14 chiSquare critVal between .05 5.6287 26.1189```

What is the critical Value of the F-distribution with 3 degrees of freedom in the numerator and 9 degrees of freedom in the denominator?

```   3 9 fDist critVal < .05 3.8625```