Standard normal distribution table

Note: The standard normal distribution table is a supplement to the articles Normal Distribution and Central Limit Theorem . The table shows the 0-1 normal distribution.

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Graph of the half-sided curve of Φ _{0; 1} ( z )

Since the integral of the normal distribution

${\displaystyle F(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{x}e^{-{\frac {1}{2}}\left({\frac {t-\mu }{\sigma }}\right)^{2}}\mathrm {d} t}$

cannot be traced back to an elementary antiderivative , tables are usually used for the calculation. However, these do not apply to any - and values, but only to the standardized form of the Gaussian distribution , in which and is in each case (one also speaks of a 0-1 normal distribution , standard normal distribution or normalized normal distribution ). Nevertheless, the table is also useful for any - normal distributions, as these can be converted into a 0-1 distribution in a very simple way. The following table of the standard normal distribution is calculated accordingly ${\displaystyle \mu }$ ${\displaystyle \sigma }$ ${\displaystyle \mu =0}$ ${\displaystyle \sigma =1}$ ${\displaystyle \mu }$ ${\displaystyle \sigma }$

${\displaystyle \Phi _{0;1}(z)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{z}e^{-{\frac {1}{2}}t^{2}}\mathrm {d} t}$

(because and ) for . ${\displaystyle \mu =0}$ ${\displaystyle \sigma =1}$ ${\displaystyle z\in \mathbb {R} }$

Area under the graph of the standard normal distribution

${\displaystyle \Phi _{0;1}(z)\,}$ →

z	0	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
0.0	0.50000	0.50399	0.50798	0.51197	0.51595	0.51994	0.52392	0.52790	0.53188	0.53586
0.1	0.53983	0.54380	0.54776	0.55172	0.55567	0.55962	0.56356	0.56749	0.57142	0.57535
0.2	0.57926	0.58317	0.58706	0.59095	0.59483	0.59871	0.60257	0.60642	0.61026	0.61409
0.3	0.61791	0.62172	0.62552	0.62930	0.63307	0.63683	0.64058	0.64431	0.64803	0.65173
0.4	0.65542	0.65910	0.66276	0.66640	0.67003	0.67364	0.67724	0.68082	0.68439	0.68793
0.5	0.69146	0.69497	0.69847	0.70194	0.70540	0.70884	0.71226	0.71566	0.71904	0.72240
0.6	0.72575	0.72907	0.73237	0.73565	0.73891	0.74215	0.74537	0.74857	0.75175	0.75490
0.7	0.75804	0.76115	0.76424	0.76730	0.77035	0.77337	0.77637	0.77935	0.78230	0.78524
0.8	0.78814	0.79103	0.79389	0.79673	0.79955	0.80234	0.80511	0.80785	0.81057	0.81327
0.9	0.81594	0.81859	0.82121	0.82381	0.82639	0.82894	0.83147	0.83398	0.83646	0.83891
1.0	0.84134	0.84375	0.84614	0.84849	0.85083	0.85314	0.85543	0.85769	0.85993	0.86214
1.1	0.86433	0.86650	0.86864	0.87076	0.87286	0.87493	0.87698	0.87900	0.88100	0.88298
1.2	0.88493	0.88686	0.88877	0.89065	0.89251	0.89435	0.89617	0.89796	0.89973	0.90147
1.3	0.90320	0.90490	0.90658	0.90824	0.90988	0.91149	0.91309	0.91466	0.91621	0.91774
1.4	0.91924	0.92073	0.92220	0.92364	0.92507	0.92647	0.92785	0.92922	0.93056	0.93189
1.5	0.93319	0.93448	0.93574	0.93699	0.93822	0.93943	0.94062	0.94179	0.94295	0.94408
1.6	0.94520	0.94630	0.94738	0.94845	0.94950	0.95053	0.95154	0.95254	0.95352	0.95449
1.7	0.95543	0.95637	0.95728	0.95818	0.95907	0.95994	0.96080	0.96164	0.96246	0.96327
1.8	0.96407	0.96485	0.96562	0.96638	0.96712	0.96784	0.96856	0.96926	0.96995	0.97062
1.9	0.97128	0.97193	0.97257	0.97320	0.97381	0.97441	0.97500	0.97558	0.97615	0.97670
2.0	0.97725	0.97778	0.97831	0.97882	0.97932	0.97982	0.98030	0.98077	0.98124	0.98169
2.1	0.98214	0.98257	0.98300	0.98341	0.98382	0.98422	0.98461	0.98500	0.98537	0.98574
2.2	0.98610	0.98645	0.98679	0.98713	0.98745	0.98778	0.98809	0.98840	0.98870	0.98899
2.3	0.98928	0.98956	0.98983	0.99010	0.99036	0.99061	0.99086	0.99111	0.99134	0.99158
2.4	0.99180	0.99202	0.99224	0.99245	0.99266	0.99286	0.99305	0.99324	0.99343	0.99361
2.5	0.99379	0.99396	0.99413	0.99430	0.99446	0.99461	0.99477	0.99492	0.99506	0.99520
2.6	0.99534	0.99547	0.99560	0.99573	0.99585	0.99598	0.99609	0.99621	0.99632	0.99643
2.7	0.99653	0.99664	0.99674	0.99683	0.99693	0.99702	0.99711	0.99720	0.99728	0.99736
2.8	0.99744	0.99752	0.99760	0.99767	0.99774	0.99781	0.99788	0.99795	0.99801	0.99807
2.9	0.99813	0.99819	0.99825	0.99831	0.99836	0.99841	0.99846	0.99851	0.99856	0.99861
3.0	0.99865	0.99869	0.99874	0.99878	0.99882	0.99886	0.99889	0.99893	0.99896	0.99900
3.1	0.99903	0.99906	0.99910	0.99913	0.99916	0.99918	0.99921	0.99924	0.99926	0.99929
3.2	0.99931	0.99934	0.99936	0.99938	0.99940	0.99942	0.99944	0.99946	0.99948	0.99950
3.3	0.99952	0.99953	0.99955	0.99957	0.99958	0.99960	0.99961	0.99962	0.99964	0.99965
3.4	0.99966	0.99968	0.99969	0.99970	0.99971	0.99972	0.99973	0.99974	0.99975	0.99976
3.5	0.99977	0.99978	0.99978	0.99979	0.99980	0.99981	0.99981	0.99982	0.99983	0.99983
3.6	0.99984	0.99985	0.99985	0.99986	0.99986	0.99987	0.99987	0.99988	0.99988	0.99989
3.7	0.99989	0.99990	0.99990	0.99990	0.99991	0.99991	0.99992	0.99992	0.99992	0.99992
3.8	0.99993	0.99993	0.99993	0.99994	0.99994	0.99994	0.99994	0.99995	0.99995	0.99995
3.9	0.99995	0.99995	0.99996	0.99996	0.99996	0.99996	0.99996	0.99996	0.99997	0.99997
4.0	0.99997	0.99997	0.99997	0.99997	0.99997	0.99997	0.99998	0.99998	0.99998	0.99998

Note: Negative values ​​are not given for reasons of symmetry because is. ${\displaystyle \Phi (-z)=1-\Phi (z)}$

Working with the table

The probability for the standard normal distribution can be determined from the table . Because of the relationship (and thus also because of the symmetry of the Gaussian bell curve) only the positive values ​​of can be found here. ${\displaystyle \Phi (z)}$ ${\displaystyle \Phi (-z)=1-\Phi (z)}$ ${\displaystyle z}$

If the probability for values in the interval from 0 to 4.09 is sought, then up to the tenth is in the left margin of the table and the hundredth is in the header. The probability is located where the corresponding row and column intersect . ${\displaystyle \Phi (z)}$ ${\displaystyle z}$ ${\displaystyle z}$ ${\displaystyle \Phi (z)}$

If it exceeds the limit of 4.09, then applies ${\displaystyle z}$

${\displaystyle \Phi (z)\approx 1}$, For ${\displaystyle z>4{,}09.}$

Caution is advised with the reversal, in which a probability is given and the associated one is sought. The value that is closer to the given probability can be viewed here. Then you put the row and column of this value together. So is z. If, for example, the probability 0.90670 is given, the value 0.90658 (corresponds to one of 1.32) is selected in the table because this is much closer than the next possible value of 0.90824 (which is one of 1 , 33 would result). The more precise result for of 1.321 is obtained by the usual (linear) interpolation, which results here (0.90670 - 0.90658) / (0.90824 - 0.90658) = 12/166, which is around 0.1. By this 0.1 of the difference between 1.32 and 1.33, i.e. by 0.001, the lower value must be increased from 1.32 to 1.321. ${\displaystyle z}$ ${\displaystyle \Phi (z)}$ ${\displaystyle z}$ ${\displaystyle z}$ ${\displaystyle z}$ ${\displaystyle z}$

Note: If any - normal distribution has been transformed into the standard normal distribution, the probability read off in the table no longer has to be transformed back, since a transformation of the same area is present! (If, on the other hand , it was determined from the table, the limit must still be calculated by.) ${\displaystyle \mu }$ ${\displaystyle \sigma }$ ${\displaystyle z}$ ${\displaystyle x}$ ${\displaystyle x=z\sigma +\mu }$

Sample calculation

A normal distribution is given with the expected value of 5 and the standard deviation of 2. We are looking for the probability that the random variable lies between the values and . ${\displaystyle \mu }$ ${\displaystyle \sigma }$ ${\displaystyle X}$ ${\displaystyle x_{1}=3}$ ${\displaystyle x_{2}=7}$

If you look at the Gaussian bell curve, then this is the area under the graph of the probability density

${\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\,e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}$, with and , ${\displaystyle \mu =5}$ ${\displaystyle \sigma =2}$

which is limited by and . ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$

In order to be able to calculate the probability, the distribution function belonging to this probability density must be

${\displaystyle F(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{x}e^{-{\frac {1}{2}}\left({\frac {t-\mu }{\sigma }}\right)^{2}}\mathrm {d} t}$

can be transformed (which is formally described in the chapter Transformation of the normal distribution in the article Normal Distribution). The transformation shifts and compresses (or extends) the curve with the expected value of the standard deviation so that it corresponds to a 0-1 normal distribution. In doing so, however, the limits are shifted and the random variable is also transformed. ${\displaystyle \mu }$ ${\displaystyle \sigma }$ ${\displaystyle x_{1}}$ ${\displaystyle x_{2}}$ ${\displaystyle X}$

This is done through

${\displaystyle z={\frac {x-\mu }{\sigma }}}$ or. ${\displaystyle Z={\frac {X-\mu }{\sigma }}.}$

(This means that the transformation steps of the distribution function do not have to be calculated during the actual calculation, they only serve to understand how the z-formula comes about.)

Shown in the example:

${\displaystyle {\begin{aligned}P(3\leq X\leq 7)&=\\&=P\left({\frac {x_{1}-\mu }{\sigma }}\leq Z={\frac {X-\mu }{\sigma }}\leq {\frac {x_{2}-\mu }{\sigma }}\right)\\&=P(-1\leq Z\leq 1)\\&=P(Z\leq 1)-P(Z\leq -1)\\&=\Phi (1)-\Phi (-1).\end{aligned}}}$

While you can now determine the value for simply from the table, you have to consider that the area (or probability) you are looking for extends from up to the limit −1. Due to the symmetry of the bell curve, however, this is the same value as from +1 to . From the total area under the curve, which is 1 (= probability of a certain event), it is deducted, that is ${\displaystyle \Phi (1)}$ ${\displaystyle \Phi (-1)}$ ${\displaystyle -\infty }$ ${\displaystyle \infty }$ ${\displaystyle \Phi (+1)}$

${\displaystyle \Phi (-1)=1-\Phi (1).}$

Applied to the example results

${\displaystyle {\begin{aligned}\Phi (1)-\Phi (-1)&=\Phi (1)-(1-\Phi (1))\\&=2\Phi (1)-1\qquad \Phi (1){\text{ in der Tabelle nachschlagen}}\\&=2\cdot 0{,}84134-1\\&=0{,}68268,\end{aligned}}}$

that is, the probability we are looking for is almost 70 percent.

Quantiles

In statistical applications, e.g. For example, in the context of hypothesis tests to find critical values, the question often arises: What is the value of the quantile , i.e. when does it apply ? ${\displaystyle q}$ ${\displaystyle z_{q}}$ ${\displaystyle \Phi (z_{q})=q}$

Are you looking for B. the 97.5% quantile , i.e. H. , then results from the table opposite (rounded to six or two decimal places). ${\displaystyle z_{0{,}975}}$ ${\displaystyle \Phi (z_{0{,}975})=0{,}975}$ ${\displaystyle z_{0{,}975}\approx 1{,}959960\approx 1{,}96}$

${\displaystyle q}$	0.750	0.800	0.900	0.950	0.975	0.990	0.995
${\displaystyle z_{q}}$	0.674490	0.841621	1.281550	1.644850	1.959960	2,326350	2.575830

literature

• Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .
• Christian Hesse : Applied probability theory . 1st edition. Vieweg, Wiesbaden 2003, ISBN 3-528-03183-2 , doi : 10.1007 / 978-3-663-01244-3 .

Web links

Wikibooks: Table of Standard  Normal Distribution - Learning and Teaching Materials