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

1,2

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.

Links

Table of n, a(n) for n=1..91.

Eric Weisstein's World of Mathematics, Quantile Function.

Wikipedia, Probit.

Example

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

Mathematica

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

Prog

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

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).

Sequence in context: A198840 A211268 A021612 * A201587 A110756 A200698

Adjacent sequences: A329278 A329279 A329280 * A329282 A329283 A329284

Keyword

nonn,cons

Author

Jianing Song, Nov 12 2019

Status

approved