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

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.

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Could you explain what you mean by "Ratio and group information"? –  whuber May 3 '12 at 16:24
Thanks @whuber for your notice. I have the values of Nicotine-G/Nicotine (Ratio) and a group variable which characterize weather the previous ratio belong to White or Black. –  MYaseen208 May 3 '12 at 16:32
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## 1 Answer

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)


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Thanks for your help and time. Would you mind to guide me how to make two qqnorm plots for different groups on the same plot. Thanks –  MYaseen208 May 3 '12 at 18:08
To get the most control over your plots, calculate the $p_i$ and $\Phi^{-1}(p_i)$ yourself, then use plot and points to overlay as many such sets as you like. –  whuber May 3 '12 at 20:05
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