Inverse normal interactive: set a tail probability, read the z-score from Φ⁻¹, then read the value x = μ + zσ off a live ruler

Normal problems run backwards

A forward normal question hands you a value and asks for a probability. An inverse question hands you the probability first and asks which value, mean or spread produced it, and the curve itself never changes: only the ruler under it does. Set the tail probability to 25% and decide before you look: is that z positive or negative, and how does it compare with the 75th percentile?

Inverse normal
50th percentile
Set the probability in the tail. Watch the shaded region, read z on the first ruler, then x on the second.
50.0% below → Z = 0.00 → X = 100.0
Which tail does the probability describe?
Probability in that tail50%
1% 25% 50% 75% 99%
Mean μ100
0 50 100 150 200
Standard deviation σ15
1 10 20 30
z = Φ⁻¹(area to the left)
0.00
x = μ + zσ
100.0
Which kind of question?

The nth percentile is the value with n% of the distribution below it, so a percentile question is an inverse normal question in disguise. Read the probability off the question first, feed it to Φ⁻¹ to get z, and only then bring in the units with x = μ + zσ. Setting the probability to 25% gives z ≈ −0.67 and 75% gives z ≈ +0.67: the same distance from the mean on opposite sides, because the curve is symmetric. Notice that moving μ or σ does not move the shaded region at all; the probability lives on the z ruler, and μ and σ only decide what the x ruler says underneath. On a CAS or calculator this is the invNorm command, which takes the area to the left by default.

Know This
Every inverse normal problem is probability → z through Φ⁻¹, then the single equation x = μ + zσ solved for whichever letter is missing.