What does mean by “the cumulative frequency in the form of probits”?

Question

What does mean by the "the cumulative frequency in the form of probits" (see diagram)? How to use this methodology when one has only the Ratio and group information. I wonder how to compute these probits for the cumulative frequencies (preferably in R). Thanks in advance for your help and time.

Answer 1

This is a lognormal probability plot labeled with percentages rather than lognormal quantiles.

Specifically, let the ordered data be written $x_1 \le x_2 \le \cdots \le x_n$ and let $\Phi$ be the standard Normal cumulative distribution function. Form a parallel sequence of plotting points corresponding to percentages of the data; a convenient and simple rule is to associate $p_i = \frac{i-1/2}{n}$ with $x_i$. Make a scatterplot of these data via the ordered pairs

$$\left(\log(x_i), \Phi^{-1}(p_i)\right)$$

Label the x-axis with the values of the $x_i$ (not their logarithms) and label the y-axis with the values of the $p_i$ (not their Normal quantiles).

In R, qqnorm almost accomplishes all of this, except it labels the y-axis with the Normal quantiles. You can supply custom labels if you like:

x <- exp(rnorm(500))           # Sample data
qqnorm(x, datax=TRUE, log="x") # Probability plot with a logarithmic data axis
percents <- c(0.001, 0.025, 0.165, 0.500, 0.835, 0.975, 0.999)
mtext(as.character(percents), side=4, at=qnorm(percents), cex=0.8)