Running the normal curve backwards
The z table and the forward calculators answer: given a value, how much area sits below it? The inverse question is at least as common: given an area, which value cuts it off? The top 5% of an exam. The 97.5th percentile that anchors a confidence interval. The value a measurement must beat to sit in the middle 95%. All of them ask for Φ−1, the inverse normal CDF, and the three modes above cover the three shapes the question takes: area below (left), area above (right), or area in the middle with the tails split evenly.
The landmarks are worth memorising. Left area 0.975 gives z = 1.9598, the 1.96 of every 95% interval. Left area 0.90 gives 1.2815: the 90th percentile. A right-tail area of 0.05 gives the same 1.6449 as a left area of 0.95, by symmetry. On a raw scale with mean 70 and SD 10, the 97.5th percentile is 70 + 1.9598 × 10 = 89.60: scores above that sit in the top 2.5%.
Percentiles, quantiles, critical values: one object
The same function wears several name tags across fields, which confuses students meeting it in a second subject. A percentile is the inverse CDF read as a rank (the 90th percentile = the value with left area 0.90). A quantile is the same thing with the area in decimals. A critical value is the inverse CDF evaluated at 1 − α/2 or 1 − α, depending on tails. If you can run this calculator, you already own all three concepts; only the vocabulary changes.
Reading z = 1.96 off a table for a one-tailed 95% question. Two-tailed 95% work uses the 0.975 quantile (1.96); one-tailed 95% uses the 0.95 quantile (1.645). Matching the quantile to the tail structure is the whole job.
Frequently asked questions
What z value leaves 95% of the area to the left?
1.6449. For 97.5% it is the famous 1.9598, and for 99.5% it is 2.5758.
How is this different from the z to p calculator?
Direction. That page takes a z and returns an area; this one takes an area and returns the z (or the raw score, when you supply a mean and SD).
What does 'between' mode assume about the tails?
That the leftover area splits evenly, half in each tail. That is the setup for standard confidence intervals and middle-percentage questions.