A329281 Decimal expansion of the quantile z_0.95 of the standard normal distribution.

1, 6, 4, 4, 8, 5, 3, 6, 2, 6, 9, 5, 1, 4, 7, 2, 7, 1, 4, 8, 6, 3, 8, 4, 8, 9, 0, 7, 9, 9, 1, 6, 3, 2, 1, 3, 6, 0, 8, 3, 1, 9, 5, 7, 4, 4, 2, 7, 5, 3, 2, 2, 0, 7, 1, 7, 6, 9, 6, 7, 2, 0, 9, 4, 4, 0, 4, 1, 0, 6, 3, 5, 1, 9, 9, 4, 4, 6, 7, 4, 1, 7, 6, 6, 4, 8, 7, 8, 4, 8, 5

Offset: 1

Author

Jianing Song, Nov 12 2019

Keywords

• nonn
• cons

Comments

z_p is the number z such that Phi(z) = p, where Phi(x) = Integral_{t=-oo..x} (1/sqrt(2*Pi))*exp(-t^2/2)*dt is the cumulative distribution function of the standard normal distribution. This sequence gives z_0.95 (also called the 95th percentile).

This number can also be denoted as probit(0.95), where probit(p) is the inverse function of Phi(x). See the Wikipedia link below.

Examples

			If X ~ N(0,1), then P(X<=1.6448536269...) = 0.95, P(X<=-1.6448536269...) = 0.05.
		

Links

• Eric Weisstein's World of Mathematics, Quantile Function.
• Wikipedia, Probit.

Crossrefs

Quantiles of the standard normal distribution: A092678 (z_0.75), A329280 (z_0.9), this sequence (z_0.95), A329282 (z_0.99), A329283 (z_0.995), A329284 (z_0.999), A329285 (z_0.9995), A329286 (z_0.9999), A329287 (z_0.99999), A329363 (z_0.999999).

Programs

Mathematica

RealDigits[(x /. FindRoot[10*Erfc[x] == 1, {x, 1, 2}, WorkingPrecision -> 120]) * Sqrt[2]][[1]] (* Amiram Eldar, Aug 23 2024 *)

PARI

default(realprecision, 100); solve(x=0, 5, erfc(x)-2*0.05)*sqrt(2)