# Tail difference of quantiles of (symmetric) distribution functions

Assume, for example, $$z_\alpha$$ are $$\Phi^{-1}(\alpha)$$ quantiles from standard normal distribution, $$\alpha > 0$$.

If we are interested in the sum$$z_\alpha + z_{1 - \alpha}$$ for standard normal distribution it is equal to 0, if I'm not mistaken; and $$2\mu$$ for $$\mathcal{N}(\mu,1)$$.

Do we have this same or similar result hold for other distributions? I'm assuming $$z_\alpha + z_{1-\alpha}=0$$ should hold for all symmetric around 0 distributions, i.e., normal, student, cauchy etc. with appropriate parameters.

However, for other distributions, such as exponential, chi-squared and other - does the sum add up to anything meaningful/known, or is there nothing interesting about the sum?

First, the subscript-notation $$z_c$$ (for 'percentage points') refers to the number $$z_c$$ such that $$P(Z > z_c) = c,$$ where $$Z$$ is standard normal and $$0 < c < 1.$$ This notation has been used in some printed tables of the normal distribution. Similar subscript notation has been used for other commonly tabled distributions such as t and chi-squared.

Most modern software packages implement functions for the CDF and their inverses, called 'quantile functions'. For example, if $$Phi$$ is the CDF of standard normal then $$Phi(1.96) = 0.975$$ and $$\Phi^{-1}(.975) = 1.96.$$

In R statistical software $$\Phi$$ is denoted by pnorm and $$\Phi^{-1}$$ by qnorm. For example, with a little more accuracy than one sees in printed tables, one has:

pnorm(1.96);  qnorm(.975)
[1] 0.9750021
[1] 1.959964


So if we let $$\alpha = .05,$$ then for standard normal we have $$z_.05 = \Phi^{-1}(.95) \approx 1.645$$. in R:

 qnorm(.95)
[1] 1.644854


For $$\alpha = .01, .02, .05, .10,$$ we have $$z_\alpha = -z_{1-\alpha}$$ and $$z_\alpha + z_{1-\alpha} = 0,$$ as you say. By symmetry, in R we have a 5-place table:

 al = c(.01, .02, .05, .10)
L = qnorm(al);  U = qnorm(1-al); S = L + U
round(cbind(al, L, U, S), 5)

al        L       U S
[1,] 0.01 -2.32635 2.32635 0
[2,] 0.02 -2.05375 2.05375 0
[3,] 0.05 -1.64485 1.64485 0
[4,] 0.10 -1.28155 1.28155 0


Similarly, for distribution $$\mathsf{T}(\nu = 15),$$ also symmetrical about $$0,$$ we have the five-place table below, which you can compare to row $$\nu=15$$ of a printed t table.

al = c(.01, .02, .05, .10)
L = qt(al, 15);  U = qt(1-al, 15); S = L + U
round(cbind(al, L, U, S), 5)

al        L       U S
[1,] 0.01 -2.60248 2.60248 0
[2,] 0.02 -2.24854 2.24854 0
[3,] 0.05 -1.75305 1.75305 0
[4,] 0.10 -1.34061 1.34061 0


However, for an asymmetrical distribution such as $$\mathsf{Chisq}(\nu = 15),$$ the upper and lower cut-off points do not sum to $$0,$$ even if they are 'probability-symmetric', cutting the same probability from each tail of the distribution.

al = c(.01, .02, .05, .10)
L = qchisq(al, 15);  U = qchisq(1-al, 15); S = L + U
round(cbind(al, L, U, S), 5)

al       L        U        S
[1,] 0.01 5.22935 30.57791 35.80726
[2,] 0.02 5.98492 28.25950 34.24441
[3,] 0.05 7.26094 24.99579 32.25673
[4,] 0.10 8.54676 22.30713 30.85389


Maybe at this time of day I lack some perspective or imagination, but I do not immediately see any interesting pattern in the 'sum' column. [Perhaps it is worthwhile noting that the mean of this distribution is $$\mu = 15$$ and that the values in the 'sum' column are very roughly $$2\mu.]$$

Finally, you asked about the distribution $$\mathsf{Norm}(\mu = 2. \sigma=1),$$ which is symmetrical about $$\mu \ne 0,$$ and for which the 'sum' column is $$2\mu = 4.$$

al = c(.01, .02, .05, .10)
L = qnorm(al, 2, 1);  U = qnorm(1-al, 2, 1); S = L + U
round(cbind(al, L, U, S), 5)

al        L       U S
[1,] 0.01 -0.32635 4.32635 4
[2,] 0.02 -0.05375 4.05375 4
[3,] 0.05  0.35515 3.64485 4
[4,] 0.10  0.71845 3.28155 4

• Yes, my bad on the notation used, thanks for the clarification! – Nutle Apr 19 at 8:51
• Nothing wrong with your notation which is still (for a while anyhow) in common use. I changed to quantiles because they more easily match the functions in R. – BruceET Apr 19 at 9:01