Every question so far has run forward: here is a value, what fraction of the curve lies below it? Real work usually runs the other way. A growth chart names the 10th percentile and asks for the weight. A scholarship names the top 5% and asks for the score. A reference range names the middle 95% and asks for both ends. Backward, the area is what you are handed — and the value is what you have to find.
The 90th percentile
the area you were givensampled values inside itsampled values outsideμ (and z = 0)the cutoff you solved for
The whole table, built backward
Each z below is a property of the standard normal alone — it never changes, no matter which variable you are studying. Only the last number in each tile moves when you change μ and σ. Tap one to jump there.
Name an areaand read a value back out
The variable
Its two parameters — the only things that are ever given
Which slice of the curve were you handed?
90.0
Area to the left—
z that cuts it—standard normal
The value—
Check: area re‑integrated—
The backward recipe, in four moves.(1) Draw and shade the area you were handed, so you can see which side of the curve it is on. (2) Convert it to a left area — every inverse lookup ever built, on paper or in software, answers the question “how much is below?” and nothing else. (3) Invert to get z*. (4) Un-standardize with x = μ + zσ: multiplying by σ converts standard deviations back into real units, and adding μ puts the answer in the right place on the axis. Note that step 4 is just step 3 of standardizing solved for x — and because z carries its own sign, a below-average percentile takes care of itself. Never subtract by hand.
Where this goes wrong. The single most common error is answering the wrong side: “top 10%” is not an area of 0.10 to look up, it is an area of 0.90 to look up, because the inverse works from the left. Switch to Top p% and watch the equation spell the conversion out. Two more cautions. These cutoffs are only as good as the normal model — run this recipe on income, hospital length of stay, or anything else with a long right tail and the answer will be wrong, sometimes badly, no matter how clean the arithmetic. And a percentile is a location, not a verdict: the 10th percentile of birth weight is a description of where a baby sits among babies, not a diagnosis.
Does the cutoff actually cut there?
The arithmetic above never looked at data — it came from the model. So let us go get some data. Each draw generates real observations from this exact normal distribution, then simply counts how many landed inside the shaded region. If the backward work was right, that count should hover around the percentage you asked for.