Math & Trigonometry
Square Root of n*Pi and Gaussian Normal Density Formulas (SQRTPI)
Calculate sqrt(n * pi) normalizing constants, standard normal peak height 1 / (sigma * sqrt(2*pi)), and Gaussian bell-curve ordinates using SQRT and PI.
Where sqrt(n * pi) appears in statistics and physics
Excel's SQRTPI(number) is shorthand for SQRT(number * PI()). At number = 2, SQRT(2 * PI()) = 2.506628 is the Gaussian normalizing denominator ensuring the normal probability density integrates to 1; at number = 1, SQRT(PI()) = 1.772454 equals Euler's Gamma(1/2).
Writing SQRT(n * PI()) directly works across every spreadsheet engine without Analysis ToolPak dependencies and plugs straight into the full Gaussian PDF: EXP(-0.5 * ((x - mu) / sigma)^2) / (sigma * SQRT(2 * PI())).
| Multiplier (n) | Exact Meaning | SQRT(n * PI()) | Application |
|---|---|---|---|
| 1 | sqrt(pi) = Gamma(0.5) | 1.772454 | Chi-square df=1 & error-function scaling |
| 2 | sqrt(2 * pi) | 2.506628 | Standard normal PDF denominator |
| 0 | Origin check | 0.000000 | Zero radius boundary |
| -1 | Negative domain | #NUM! | Invalid negative radicand guard |
SQRTPI Formula With Negative-Domain Guard
Evaluates SQRT(B2 * PI()) rounded to 6 decimals and returns "#NUM!" if the multiplier B2 is negative.
=IF(B2 < 0, "#NUM!", ROUND(SQRT(B2 * PI()), 6))Gaussian Probability Density Ordinate f(x; mu, sigma)
Evaluates the bell-curve density height at x (B5) for mean mu (B3) and standard deviation sigma (B4).
=ROUND(EXP(-0.5 * POWER((B5 - B3) / B4, 2)) / (B4 * SQRT(2 * PI())), 6)Try the example
Use the original values or change them to test your understanding.