Quantiles of the standard normal distribution

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Numbers

$p$

$z_p$

1/10000:

-3.719016485455680564393660624508478304617319708272154684739481724786930643296726178907270327748896567

1/2000:

-3.290526731491894793221627035374649179216226925677390076993878286917996599649757864211744708056358681

1/1000:

-3.090232306167813541540399830107379205491008491865808855697171108543569142895145553122667241020237179

1/200:

-2.575829303548900760978576748603814117306017634276317376460486218862551207876418110849814656993675249

1/100:

-2.326347874040841100885606163346911723351817141532013069065640247890876626456034487356822930790951317

1/50:

-2.053748910631823052937351657740453446416247391900876563491144609618890019565675826300589798922539054

1/40:

-1.959963984540054235524594430520551527955550077869548398476952646361635274144882667798254709492814206

1/20:

-1.644853626951472714863848907991632136083195744275322071769672094404106351994467417664878485485753765

1/10:

-1.281551565544600466965103329448742818619907824352582659702648230565703324812245430155438161325028510

3/20:

-1.036433389493789579713244074673503366134740595985917627904454866389774789427345864699016720253281748

1/5:

-0.8416212335729142051787061213632481006262975340088792004064335393338886179605538554865122508088476230

1/4:

-0.6744897501960817432022270145413071853869044150498618956620937885948486792824430910954450447401677846

3/10:

-0.5244005127080407840382893250251225543253780354499781689059166758604298849812053729440109732777709288

2/5:

-0.2533471031357997987981961814242439387872107062853953615943364328979606026454771900450726684996133043

1/2:

comment: The median of the standard normal distribution is exactly zero.

3/5:

0.2533471031357997987981961814242439387872107062853953615943364328979606026454771900450726684996133043

7/10:

0.5244005127080407840382893250251225543253780354499781689059166758604298849812053729440109732777709288

3/4:

0.6744897501960817432022270145413071853869044150498618956620937885948486792824430910954450447401677846

4/5:

0.8416212335729142051787061213632481006262975340088792004064335393338886179605538554865122508088476230

17/20:

1.036433389493789579713244074673503366134740595985917627904454866389774789427345864699016720253281748

9/10:

1.281551565544600466965103329448742818619907824352582659702648230565703324812245430155438161325028510

19/20:

1.644853626951472714863848907991632136083195744275322071769672094404106351994467417664878485485753765

comment: This is the one-sided 5% upper critical value.

39/40:

1.959963984540054235524594430520551527955550077869548398476952646361635274144882667798254709492814206

comment: This is the positive cutoff for a two-sided 5% normal test.

49/50:

2.053748910631823052937351657740453446416247391900876563491144609618890019565675826300589798922539054

99/100:

2.326347874040841100885606163346911723351817141532013069065640247890876626456034487356822930790951317

comment: This is the one-sided 1% upper critical value.

199/200:

2.575829303548900760978576748603814117306017634276317376460486218862551207876418110849814656993675249

comment: This is the positive cutoff for a two-sided 1% normal test.

999/1000:

3.090232306167813541540399830107379205491008491865808855697171108543569142895145553122667241020237179

comment: This is the one-sided 0.1% upper critical value.

1999/2000:

3.290526731491894793221627035374649179216226925677390076993878286917996599649757864211744708056358681

9999/10000:

3.719016485455680564393660624508478304617319708272154684739481724786930643296726178907270327748896567

Definition

For the standard normal distribution with cumulative distribution function $\Phi$ [1], this table gives the lower-tail quantile $z_p=\inf\{x:\Phi(x)\ge p\}$.

Parameters

$p$

— lower-tail probability ($0<p<1$)

Formulas

(1)

$\Phi(x)=\frac{1}{2}\left(1+\operatorname{erf}(x/\sqrt{2})\right)$, so $z_p=\sqrt{2}\,\operatorname{erf}^{-1}(2p-1)$.

(2)

Equivalently, using the complementary error function, $z_p=\sqrt{2}\,\operatorname{erfc}^{-1}(2(1-p))$.

(3)

The standard normal distribution is symmetric, so $z_{1-p}=-z_p$.

(4)

If $z^{\mathrm{upper}}_{\alpha}$ denotes the upper-tail critical value, then $z^{\mathrm{upper}}_{\alpha}=z_{1-\alpha}$.

(5)

A two-sided normal test with significance level $\alpha$ uses the cutoffs $z_{\alpha/2}$ and $z_{1-\alpha/2}$.

(6)

If $\chi^2_{1,p}$ is the $p$-quantile of the chi-squared distribution with one degree of freedom, then $\chi^2_{1,p}=z_{(1+p)/2}^2$.

Comments

(7)

The table stores lower-tail probabilities. If a printed table is indexed by an upper-tail probability $\alpha$, its value is $z_{1-\alpha}$ here.

(8)

The parameter is written as an exact rational in lowest terms, so the familiar decimal probability $0.975$ is the row $p=39/40$. The twenty-nine rows are selected conventional confidence and significance levels and their complements, rather than a uniform base-ten grid: $p=999/1000$ is the individual $0.1\%$ tail, not one member of a thousand rows. The values $0.01\%$, $0.05\%$ and $0.25\%$, together with their complements, extend the usual printed levels to extreme tails; compare [2].

(9)

The row $p=1/2$ is exact: the standard normal distribution has median zero, so $z_{1/2}=0$ is stored as an integer.

(10)

Negative quantiles are stored as entries, not only recovered from the symmetry in (3), because a reader may arrive with the negative critical value itself.

Programs

(P1)

qnorm(0.975)

(P2)

Python

import mpmath as mp
mp.mp.dps = 110
p = mp.mpf(39) / 40
mp.sqrt(2) * mp.erfinv(2*p - 1)

Links

[1]

Wikipedia: Normal distribution

[2]

AQA: statistical tables, percentage points of the normal distribution

Similar tables

Values of the error function $\operatorname{erf}(x)$ — $\Phi(x)$ is an affine rescaling of $\operatorname{erf}(x/\sqrt{2})$, and the rows here store the inverse values selected by rational probabilities

Values of the complementary error function $\operatorname{erfc}(x)$ — upper-tail normal probabilities are written with $\operatorname{erfc}(x/\sqrt{2})/2$

Differential entropies of continuous probability distributions — includes the entropy of the same standard normal distribution

Kullback-Leibler divergences between probability distributions — includes divergences between normal distributions

Data properties

Entries are of type: real number

How well the digits are known: proven

How they were obtained:

Values are computed by bisection in Sage real ball arithmetic, evaluating $\Phi(x)$ through RealBall.erf at exact rational endpoints until the bracket determines 100 significant digits. The returned value is the final bracketing ball, not a rounded midpoint.

The stored entries were checked against the error-function relation in (1), the symmetry in (3), strict monotonicity in $p$, the exact median row, and an independent mpmath computation of $z_{39/40}$ using mp.sqrt(2) * mp.erfinv(2*p - 1) at 110 decimal digits.

Table is complete: no (it holds $z_p$ at the conventional confidence and significance levels and their complements: $p$ and $1-p$ for $p\in\{1/2,3/5,7/10,3/4,4/5,17/20,9/10,19/20,39/40,49/50,99/100,199/200,999/1000,1999/2000,9999/10000\}$, twenty-nine in all. These are the probabilities the subject actually uses -- $z_{0.975}=1.959963\ldots$ above all -- rather than a grid of a thousand of which twenty are ever wanted)

parameter selection: The twenty-nine probabilities are the conventional confidence and significance levels and their complements, listed exhaustively in complete-note; this is not a uniform grid. (Unknown key)