What’s the difference between the two modes?+−
"Probability between values" finds the chance a randomly drawn value falls within a range you specify. "Value at percentile" works backward — you give a target percentile (like 95th) and it returns the corresponding value.
Does this assume my data is normally distributed?+−
Yes — every calculation here assumes the underlying data genuinely follows (or closely approximates) a normal distribution. For data with strong skew or heavy outliers, treat results as an approximation, or check normality first.
How does this relate to the empirical rule?+−
The empirical rule (68-95-99.7) is this calculator's three special cases: μ±1σ ≈ 68%, μ±2σ ≈ 95%, μ±3σ ≈ 99.7%. This calculator computes the exact probability for ANY range, not just those three.
Why does the percentile mode use bisection instead of a formula?+−
There's no closed-form algebraic formula for the inverse of the normal CDF — it's a genuinely transcendental problem. Bisection (repeatedly narrowing a search range) is a standard, numerically robust way to solve it, and this implementation is checked against known critical z-values (e.g. the 95th percentile gives z=1.6449) before being trusted.