Determine a normal distribution given its quantile information

Question

I was wondering how I could have R tell me the SD (as an argument in the qnorm() built in R) for a normal distribution whose 95% limit values are already known?

As an example, I know the two 95% limit values for my normal are 158, and 168, respectively. So, in the below R code SD is shown as "x". If "y" (the answer of this simple qnorm() function) needs to be (158, 168), then can R tell me what should be x?

y <- qnorm(c(.025,.975), 163, x)

Answer 1

A general procedure for Normal distribution

Suppose we have a Normal distribution X ~ N(mu, sigma), with unknown mean mu and unknown standard deviation sigma. And we aim to solve for mu and sigma, given two quantile equations:

Pr(X < q1) = alpha1
Pr(X < q2) = alpha2

We consider standardization: Z = (X - mu) / sigma, so that

Pr(Z < (q1 - mu) / sigma) = alpha1
Pr(Z < (q2 - mu) / sigma) = alpha2

In other words,

(q1 - mu) / sigma = qnorm(alpha1)
(q2 - mu) / sigma = qnorm(alpha2)

The RHS is explicitly known, and we define beta1 = qnorm(alpha1), beta2 = qnorm(alpha2). Now, the above simplifies to a system of 2 linear equations:

mu + beta1 * sigma = q1
mu + beta2 * sigma = q2

This system has coefficient matrix:

1  beta1
1  beta2

with determinant beta2 - beta1. The only situation for singularity is beta2 = beta1. As long as the system is non-singular, we can use solve to solve for mu and sigma.

Think about what the singularity situation means. qnorm is strictly monotone for Normal distribution. So beta1 = beta2 is as same as alpha1 = alpha2. But this can be easily avoided as it is under your specification, so in the following I will not check singularity.

Wrap up above into an estimation function:

est <- function(q, alpha) {
  beta <- qnorm(alpha)
  setNames(solve(cbind(1, beta), q), c("mu", "sigma"))
  }

Let's have a test:

x <- est(c(158, 168), c(0.025, 0.975))
#        mu      sigma 
#163.000000   2.551067 

## verification
qnorm(c(0.025, 0.975), x[1], x[2])
# [1] 158 168

---

We can also do something arbitrary:

x <- est(c(1, 5), c(0.1, 0.4))
#      mu    sigma 
#5.985590 3.890277 

## verification
qnorm(c(0.1, 0.4), x[1], x[2])
# [1] 1 5