How this calculator works
A percentile is the value with a specified cumulative area to its left. An upper-tail input reverses that area convention. Central mode splits the remaining area equally between both tails and returns two bounds. Probabilities of exactly zero or one would require infinite cutoffs and are excluded.
Formula
Left: x = μ + σΦ⁻¹(p). Right: x = μ - σΦ⁻¹(p). Central bounds: μ ± σΦ⁻¹((1+p)/2).
Worked example
The standard normal 97.5th percentile is 1.959964. A central 95% interval with mean 10 and SD 2 runs from 6.080072 to 13.919928.
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Frequently asked questions
Why does a 95% left cutoff differ from a central 95% interval?
The central interval allocates 2.5% to each tail. A 95% left cutoff allocates 5% to the right tail.
Can the cutoff be negative?
Yes. A normal variable can take any real value.
Which units does the answer use?
The same unit as the mean and standard deviation.