Student-t, chi-square and F distributions: tcdf, invT, χ²cdf and Fcdf in Scientific calculator plus 991 for Android

The Student-t, chi-square and F distributions are the ones behind t-tests, chi-square tests and the comparison of two variances. Where the normal distribution takes a mean and a standard deviation, these take degrees of freedom. On the calculator each has a cumulative function for a probability and a density function for the height of its curve, and the t distribution has an inverse as well.

Before you start

All seven are on the distribution menu, SHIFT 3 DISTR; type part of a name in its search box to narrow the list, as on the normal distribution. Each entry shows its arguments in order. The chi-square names begin with χ², so choose them from the menu.

• df is the degrees of freedom. For Student-t it must be greater than 0; for chi-square it must be a whole number greater than 0.
• The F functions take two degrees of freedom, a numerator df and a denominator df, because the F distribution compares two variances. Each must be a whole number greater than 0.
• A chi-square or F value is never negative, so a probability below a value starts its interval at 0.

Student-t

invT(area, df) runs tcdf backwards: given the area to the left of a point, it returns the t-value that point sits at. This is the number a printed t-table would give.

The t-value with 95% of the area to its left, on 24 degrees of freedom:

That is the value a t-table lists for 5% in one tail. For a two-sided 95% interval, put in 0.975 instead, leaving 2.5% in each tail.

tcdf(lowerbound, upperbound, df) gives the probability that a t value falls between the two bounds. Between −2 and 3 on 18 degrees of freedom:

tpdf(x, df) gives the density: the height of the curve at x, not a probability. At x = 2.2 on 2 degrees of freedom:

Chi-square

χ²cdf(lowerbound, upperbound, df) gives the probability that a chi-square value falls between the two bounds. The probability below 19.023 on 9 degrees of freedom:

So 19.023 is the 97.5% point of that distribution. The interval starts at 0 because a chi-square value is never negative.

χ²pdf(x, df) gives the height of the chi-square curve at x. At x = 2 on 9 degrees of freedom:

F

Fcdf(lowerbound, upperbound, numerator df, denominator df) gives the probability that an F value falls between the two bounds. The probability below 2.4523, with 24 and 19 degrees of freedom:

So 2.4523 is the 97.5% point of that distribution.

Fpdf(x, numerator df, denominator df) gives the height of the F curve at x. At x = 2, with 24 and 19 degrees of freedom:

At x = 5, with 4 and 3 degrees of freedom:

pdf, cdf and inv

• A cdf function gives a probability: the area under the curve between two bounds.
• Here, a pdf function (tpdf, χ²pdf, Fpdf) gives the height of the curve at one value. It is not a probability.
• invT goes the other way from tcdf, from an area to a value, as invNorm does for the normal distribution.

At a glance