cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

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

Original entry on oeis.org

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

Views

Author

Jianing Song, Nov 12 2019

Keywords

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

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)

A329280 Decimal expansion of the quantile z_0.9 of the standard normal distribution.

Original entry on oeis.org

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

Views

Author

Jianing Song, Nov 12 2019

Keywords

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_0.9 (also called the 9th decile or the 90th percentile).
This number can also be denoted as probit(0.9), where probit(p) is the inverse function of Phi(x). See the Wikipedia link below.

Links

Crossrefs

Quantiles of the standard normal distribution: A092678 (z_0.75), this sequence (z_0.9), A329281 (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[Sqrt[2] InverseErfc[9/10], 10, 100][[1]] (* Jean-François Alcover, Sep 26 2020 *)
  • PARI
    default(realprecision, 100); solve(x=0, 5, erfc(x)-2*0.1)*sqrt(2)

Formula

If X ~ N(0,1), then P(X<=1.2815515655...) = 0.9, P(X<=-1.2815515655...) = 0.1.

A329282 Decimal expansion of the quantile z_0.99 of the standard normal distribution.

Original entry on oeis.org

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

Views

Author

Jianing Song, Nov 12 2019

Keywords

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_0.99 (also called the 99th percentile).
This number can also be denoted as probit(0.99), where probit(p) is the inverse function of Phi(x). See the Wikipedia link below.

Examples

			If X ~ N(0,1), then P(X<=2.3263478740...) = 0.99, P(X<=-2.3263478740...) = 0.01.

Links

Crossrefs

Quantiles of the standard normal distribution: A092678 (z_0.75), A329280 (z_0.9), A329281 (z_0.95), this sequence (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

  • PARI
    default(realprecision, 100); solve(x=0, 5, erfc(x)-2*0.01)*sqrt(2)

A329283 Decimal expansion of the quantile z_0.995 of the standard normal distribution.

Original entry on oeis.org

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

Views

Author

Jianing Song, Nov 12 2019

Keywords

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_0.995.
This number can also be denoted as probit(0.995), where probit(p) is the inverse function of Phi(x). See the Wikipedia link below.

Examples

			If X ~ N(0,1), then P(X<=2.5758293035...) = 0.995, P(X<=-2.5758293035...) = 0.005.

Links

Crossrefs

Quantiles of the standard normal distribution: A092678 (z_0.75), A329280 (z_0.9), A329281 (z_0.95), A329282 (z_0.99), this sequence (z_0.995), A329284 (z_0.999), A329285 (z_0.9995), A329286 (z_0.9999), A329287 (z_0.99999), A329363 (z_0.999999).

Programs

  • PARI
    default(realprecision, 100); solve(x=0, 5, erfc(x)-2*0.005)*sqrt(2)

A329284 Decimal expansion of the quantile z_0.999 of the standard normal distribution.

Original entry on oeis.org

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

Views

Author

Jianing Song, Nov 12 2019

Keywords

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_0.999.
This number can also be denoted as probit(0.999), where probit(p) is the inverse function of Phi(x). See the Wikipedia link below.

Examples

			If X ~ N(0,1), then P(X<=3.0902323061...) = 0.999, P(X<=-3.0902323061...) = 0.001.

Links

Crossrefs

Quantiles of the standard normal distribution: A092678 (z_0.75), A329280 (z_0.9), A329281 (z_0.95), A329282 (z_0.99), A329283 (z_0.995), this sequence (z_0.999), A329285 (z_0.9995), A329286 (z_0.9999), A329287 (z_0.99999), A329363 (z_0.999999).

Programs

  • PARI
    default(realprecision, 100); solve(x=0, 5, erfc(x)-2*0.001)*sqrt(2)

A329285 Decimal expansion of the quantile z_0.9995 of the standard normal distribution.

Original entry on oeis.org

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

Views

Author

Jianing Song, Nov 12 2019

Keywords

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_0.9995.
This number can also be denoted as probit(0.9995), where probit(p) is the inverse function of Phi(x). See the Wikipedia link below.

Examples

			If X ~ N(0,1), then P(X<=3.2905267314...) = 0.9995, P(X<=-3.2905267314...) = 0.0005.

Links

Crossrefs

Quantiles of the standard normal distribution: A092678 (z_0.75), A329280 (z_0.9), A329281 (z_0.95), A329282 (z_0.99), A329283 (z_0.995), A329284 (z_0.999), this sequence (z_0.9995), A329286 (z_0.9999), A329287 (z_0.99999), A329363 (z_0.999999).

Programs

  • PARI
    default(realprecision, 100); solve(x=0, 5, erfc(x)-2*0.0005)*sqrt(2)

A329286 Decimal expansion of the quantile z_0.9999 of the standard normal distribution.

Original entry on oeis.org

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

Views

Author

Jianing Song, Nov 12 2019

Keywords

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_0.9999.
This number can also be denoted as probit(0.9999), where probit(p) is the inverse function of Phi(x). See the Wikipedia link below.

Examples

			If X ~ N(0,1), then P(X<=3.7190164854...) = 0.9999, P(X<=-3.7190164854...) = 0.0001.

Links

Crossrefs

Quantiles of the standard normal distribution: A092678 (z_0.75), A329280 (z_0.9), A329281 (z_0.95), A329282 (z_0.99), A329283 (z_0.995), A329284 (z_0.999), A329285 (z_0.9995), this sequence (z_0.9999), A329287 (z_0.99999), A329363 (z_0.999999).

Programs

  • PARI
    default(realprecision, 100); solve(x=0, 5, erfc(x)-2*0.0001)*sqrt(2)

A329287 Decimal expansion of the quantile z_0.99999 of the standard normal distribution.

Original entry on oeis.org

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

Views

Author

Jianing Song, Nov 12 2019

Keywords

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_0.99999.
This number can also be denoted as probit(0.99999), where probit(p) is the inverse function of Phi(x). See the Wikipedia link below.

Examples

			If X ~ N(0,1), then P(X<=4.2648907939...) = 0.99999, P(X<=-4.2648907939...) = 0.00001.

Links

Crossrefs

Quantiles of the standard normal distribution: A092678 (z_0.75), A329280 (z_0.9), A329281 (z_0.95), A329282 (z_0.99), A329283 (z_0.995), A329284 (z_0.999), A329285 (z_0.9995), A329286 (z_0.9999), this sequence (z_0.99999), A329363 (z_0.999999).

Programs

  • PARI
    default(realprecision, 100); solve(x=0, 5, erfc(x)-2*0.00001)*sqrt(2)

A329363 Decimal expansion of the quantile z_0.999999 of the standard normal distribution.

Original entry on oeis.org

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

Views

Author

Jianing Song, Nov 12 2019

Keywords

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_0.999999.
This number can also be denoted as probit(0.999999), where probit(p) is the inverse function of Phi(x). See the Wikipedia link below.

Examples

			If X ~ N(0,1), then P(X<=4.7534243088...) = 0.999999, P(X<=-4.7534243088...) = 0.000001.

Links

Crossrefs

Quantiles of the standard normal distribution: A092678 (z_0.75), A329280 (z_0.9), A329281 (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), this sequence (z_0.999999).

Programs

  • PARI
    default(realprecision, 100); solve(x=0, 5, erfc(x)-2*0.000001)*sqrt(2)

A076668 Decimal expansion of sqrt(2/Pi).

Original entry on oeis.org

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

Views

Author

Zak Seidov, Oct 25 2002

Keywords

Comments

This is the limit of (n+1)!!/n!!/n^(1/2) at n_even->inf.
Expected value of |x - mu|/sigma for normal distribution with mean mu and standard deviation sigma (i.e., the normalized mean absolute deviation). - Stanislav Sykora, Jun 30 2017

Examples

			0.79788456080286535587989211986876373695171726232986931533...

Links

Crossrefs

Cf. A004730, A004731, A019727, A060294 (Buffon's constant 2/Pi), A092678 (probable error).

Programs

  • Magma
    pi:=Sqrt(2/Pi(RealField(110))); Reverse(Intseq(Floor(10^110*pi))); // Vincenzo Librandi, Jul 01 2017
  • Mathematica
    RealDigits[Sqrt[2/Pi],10,120][[1]] (* Harvey P. Dale, Feb 05 2012 *)
  • PARI
    sqrt(2/Pi) \\ G. C. Greubel, Sep 23 2017

Formula

Equals A087197*A002193. - R. J. Mathar Feb 05 2009
Equals integral_{-infinity..infinity} (1-erf(x)^2)/2 dx. - Jean-François Alcover, Feb 25 2015

Extensions

More terms and better description from Benoit Cloitre and Michael Somos, Oct 29 2002
Leading zero removed, offset changed by R. J. Mathar, Feb 05 2009
Showing 1-10 of 10 results.

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