Find the standard deviation with normal probabilities using R

Question

Let's say that there is a variable A that has a Normal distribution N(μ,σ).

I have two probabilities when P(A>a) and P(A<b), where a<b, and the given probability is expressed in %.(as an example)

With this information can R find the standard deviation? I don't know which commands to use? qnorm, dnorm,.. so I can get the Standard Deviation.

What I tried to do was, knowing that a = 100, b = 200 , P(A>a) = 5% and P(A<b)=15%:

1. Use the standarized Normal distribution considering μ = 0, σ = 1 (But i dont know how to put it in R, so I can get what I want)

2. See the probability in the normal distribution table and calculate Z ..., but it didn't work.

Is there a way R can find the standard deviation with just these information??

Answer 1

Your problem as stated is impossible, check that your inequalities and values are correct.

You give the example that p(A > 100) = 5% which means that the p( A < 100 ) = 95% which means that p( A < 200 ) must be greater than 95% (all the probability between 100 and 200 adds to the 95%), but you also say that p( A < 200 ) = 15%. There is no set of numbers that can give you a probability that is both greater than 95% and equal to 15%.

Once you fix the problem definition to something that works there are a couple of options. Using Ryacas you may be able to solve directly (2 equations and 2 unkowns), but since this is based on the integral of the normal I don't know if it would work or not.

Another option would be to use optim or similar programs to find(approximate) a solution. Create an objective function that takes 2 parameters, the mean and sd of the normal, then computes the sum of the squared differences between the stated percentages and those computed based on current guesses. The objective function will be 0 at the "correct" mean and standard deviation and positive everywhere else. Then pass this function to optim to find the minimum.