Estimating an unknown σ (or μ) from a given percentile

The reverse of a standard z-score lookup: given that a certain PERCENTAGE lies below/above a value, work BACKWARDS to find the missing standard deviation or mean.

The method

Use inverse normal on the GIVEN percentage to find the z-score for that cutoff — a percentage below 50% gives a NEGATIVE z, above 50% gives a POSITIVE z.
Substitute the z-score, the given x-value, and whichever of μ/σ IS known into z = (x − μ)/σ.
Rearrange to solve for the UNKNOWN parameter (σ or μ) algebraically — this is a linear equation once z is known.
Sanity-check the answer's sign and size against the context (a standard deviation must be positive; a mean should sit within the plausible range of the data).

Worked example Finding the standard deviation from a percentile A dataset has mean 114 minutes. 12% of values exceed 150 minutes. Find the standard deviation. CONVERT THE PERCENTAGE: 12% ABOVE 150 minutes means 88% lie below it — inverse normal on 0.88 gives z ≈ 1.175 (a positive z, since 150 is above the mean). SUBSTITUTE into z = (x − μ)/σ: 1.175 = (150 − 114)/σ = 36/σ. SOLVE for σ: σ = 36/1.175 ≈ 30.6 minutes. SANITY CHECK: a positive, plausible standard deviation relative to the 114-minute mean — consistent with the context. ✓ The critical first step is converting "12% ABOVE" into "88% BELOW" before reading the z-score — inverse-normal tables and calculators expect a below-the-cutoff (cumulative) probability. ✕

Using the given percentage directly as the cumulative probability without checking above/below

Feeding 0.12 straight into inverse normal, when the question actually describes the percentage ABOVE a value, gives entirely the wrong z-score (and the wrong sign). Always convert an "above" percentage to its complementary "below" percentage (100% − given%) before finding z.

Given	Convert to (before inverse normal)
X% below a value	Use X% directly
X% above a value	(100 − X)% below
X% between two values	Combine with a known tail to isolate one cumulative probability

Practice for this standard covers the whole topic — flashcards and quiz.