Standard normal table

Template:Short description

In statistics, a standard normal table, also called the unit normal table or Z table,^[1] is a mathematical table for the values of Template:Math, the cumulative distribution function of the normal distribution. It is used to find the probability that a statistic is observed below, above, or between values on the standard normal distribution, and by extension, any normal distribution. Since probability tables cannot be printed for every normal distribution, as there are an infinite variety of normal distributions, it is common practice to convert a normal to a standard normal (known as a z-score) and then use the standard normal table to find probabilities.^[2]

Normal and standard normal distribution

Normal distributions are symmetrical, bell-shaped distributions that are useful in describing real-world data. The standard normal distribution, represented by Template:Mvar, is the normal distribution having a mean of 0 and a standard deviation of 1.

Conversion

Template:Main If Template:Mvar is a random variable from a normal distribution with mean Template:Mvar and standard deviation Template:Mvar, its Z-score may be calculated from Template:Mvar by subtracting Template:Mvar and dividing by the standard deviation:

Z = \frac{X - μ}{σ}

If $\overline{X}$ is the mean of a sample of size Template:Mvar from some population in which the mean is Template:Mvar and the standard deviation is Template:Mvar, the standard error is Template:Tmath

Z = \frac{\overline{X} - μ}{σ / \sqrt{n}}

If $\sum X$ is the total of a sample of size Template:Mvar from some population in which the mean is Template:Mvar and the standard deviation is Template:Mvar, the expected total is Template:Mvar and the standard error is Template:Tmath

Z = \frac{\sum X - n μ}{σ \sqrt{n}}

Reading a Z table

Formatting / layout

Template:Mvar tables are typically composed as follows:

The label for rows contains the integer part and the first decimal place of Template:Mvar.
The label for columns contains the second decimal place of Template:Mvar.
The values within the table are the probabilities corresponding to the table type. These probabilities are calculations of the area under the normal curve from the starting point (0 for cumulative from mean, negative infinity for cumulative and positive infinity for complementary cumulative) to Template:Mvar.

Example: To find 0.69, one would look down the rows to find 0.6 and then across the columns to 0.09 which would yield a probability of 0.25490 for a cumulative from mean table or 0.75490 from a cumulative table.

To find a negative value such as –0.83, one could use a cumulative table for negative z-values^[3] which yield a probability of 0.20327.

But since the normal distribution curve is symmetrical, probabilities for only positive values of Template:Mvar are typically given. The user might have to use a complementary operation on the absolute value of Template:Mvar, as in the example below.

Types of tables

Template:Mvar tables use at least three different conventions:

Cumulative from mean: gives a probability that a statistic is between 0 (mean) and Template:Mvar. Example: Template:Math.
Cumulative: gives a probability that a statistic is less than Template:Mvar. This equates to the area of the distribution below Template:Mvar. Example: Template:Math.
Complementary cumulative: gives a probability that a statistic is greater than Template:Mvar. This equates to the area of the distribution above Template:Mvar.; Example: Find Template:Math. Since this is the portion of the area above Template:Mvar, the proportion that is greater than Template:Mvar is found by subtracting Template:Mvar from 1. That is Template:Math or Template:Math.

Table examples

Cumulative from minus infinity to Z

This table gives a probability that a statistic is between minus infinity and Template:Mvar.

f (z) = Φ (z)

The values are calculated using the cumulative distribution function of a standard normal distribution with mean of zero and standard deviation of one, usually denoted with the capital Greek letter $Φ$ (phi), is the integral

Φ (z) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{z} e^{- t^{2} / 2} d t

$Φ$ (z) is related to the error function, or Template:Math.

Φ (z) = \frac{1}{2} [1 + \erf (\frac{z}{\sqrt{2}})]

Note that for Template:Math, one obtains (after multiplying by 2 to account for the Template:Math interval) the results Template:Math, characteristic of the 68–95–99.7 rule.

Cumulative (less than Z)

This table gives a probability that a statistic is less than Template:Mvar (i.e. between negative infinity and Template:Mvar).

z	−0.00	−0.01	−0.02	−0.03	−0.04	−0.05	−0.06	−0.07	−0.08	−0.09
−1.9	0.02872	0.02807	0.02743	0.02680	0.02619	0.02559	0.02500	0.02442	0.02385	0.02330
−1.8	0.03593	0.03515	0.03438	0.03362	0.03288	0.03216	0.03144	0.03074	0.03005	0.02938
−1.7	0.04457	0.04363	0.04272	0.04182	0.04093	0.04006	0.03920	0.03836	0.03754	0.03673
−1.6	0.05480	0.05370	0.05262	0.05155	0.05050	0.04947	0.04846	0.04746	0.04648	0.04551
−1.5	0.06681	0.06552	0.06426	0.06301	0.06178	0.06057	0.05938	0.05821	0.05705	0.05592

−1.4	0.08076	0.07927	0.07780	0.07636	0.07493	0.07353	0.07215	0.07078	0.06944	0.06811
−1.3	0.09680	0.09510	0.09342	0.09176	0.09012	0.08851	0.08692	0.08534	0.08379	0.08226
−1.2	0.11507	0.11314	0.11123	0.10935	0.10749	0.10565	0.10383	0.10204	0.10027	0.09853
−1.1	0.13567	0.13350	0.13136	0.12924	0.12714	0.12507	0.12302	0.12100	0.11900	0.11702
−1.0	0.15866	0.15625	0.15386	0.15151	0.14917	0.14686	0.14457	0.14231	0.14007	0.13786

−0.9	0.18406	0.18141	0.17879	0.17619	0.17361	0.17106	0.16853	0.16602	0.16354	0.16109
−0.8	0.21186	0.20897	0.20611	0.20327	0.20045	0.19766	0.19489	0.19215	0.18943	0.18673
−0.7	0.24196	0.23885	0.23576	0.23270	0.22965	0.22663	0.22363	0.22065	0.21770	0.21476
−0.6	0.27425	0.27093	0.26763	0.26435	0.26109	0.25785	0.25463	0.25143	0.24825	0.24510
−0.5	0.30854	0.30503	0.30153	0.29806	0.29460	0.29116	0.28774	0.28434	0.28096	0.27760

−0.4	0.34458	0.34090	0.33724	0.33360	0.32997	0.32636	0.32276	0.31918	0.31561	0.31207
−0.3	0.38209	0.37828	0.37448	0.37070	0.36693	0.36317	0.35942	0.35569	0.35197	0.34827
−0.2	0.42074	0.41683	0.41294	0.40905	0.40517	0.40129	0.39743	0.39358	0.38974	0.38591
−0.1	0.46017	0.45620	0.45224	0.44828	0.44433	0.44038	0.43644	0.43251	0.42858	0.42465
−0.0	0.50000	0.49601	0.49202	0.48803	0.48405	0.48006	0.47608	0.47210	0.46812	0.46414
z	−0.00	−0.01	−0.02	−0.03	−0.04	−0.05	−0.06	−0.07	−0.08	−0.09

z	+ 0.00	+ 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	0.56360	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	0.91308	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

z	+0.00	+0.01	+0.02	+0.03	+0.04	+0.05	+0.06	+0.07	+0.08	+0.09

^[4]

Complementary cumulative

This table gives a probability that a statistic is greater than Template:Mvar. : $f (z) = 1 - Φ (z)$

z	+0.00	+0.01	+0.02	+0.03	+0.04	+0.05	+0.06	+0.07	+0.08	+0.09
0.0	0.50000	0.49601	0.49202	0.48803	0.48405	0.48006	0.47608	0.47210	0.46812	0.46414
0.1	0.46017	0.45620	0.45224	0.44828	0.44433	0.44038	0.43640	0.43251	0.42858	0.42465
0.2	0.42074	0.41683	0.41294	0.40905	0.40517	0.40129	0.39743	0.39358	0.38974	0.38591
0.3	0.38209	0.37828	0.37448	0.37070	0.36693	0.36317	0.35942	0.35569	0.35197	0.34827
0.4	0.34458	0.34090	0.33724	0.33360	0.32997	0.32636	0.32276	0.31918	0.31561	0.31207

0.5	0.30854	0.30503	0.30153	0.29806	0.29460	0.29116	0.28774	0.28434	0.28096	0.27760
0.6	0.27425	0.27093	0.26763	0.26435	0.26109	0.25785	0.25463	0.25143	0.24825	0.24510
0.7	0.24196	0.23885	0.23576	0.23270	0.22965	0.22663	0.22363	0.22065	0.21770	0.21476
0.8	0.21186	0.20897	0.20611	0.20327	0.20045	0.19766	0.19489	0.19215	0.18943	0.18673
0.9	0.18406	0.18141	0.17879	0.17619	0.17361	0.17106	0.16853	0.16602	0.16354	0.16109

1.0	0.15866	0.15625	0.15386	0.15151	0.14917	0.14686	0.14457	0.14231	0.14007	0.13786
1.1	0.13567	0.13350	0.13136	0.12924	0.12714	0.12507	0.12302	0.12100	0.11900	0.11702
1.2	0.11507	0.11314	0.11123	0.10935	0.10749	0.10565	0.10383	0.10204	0.10027	0.09853
1.3	0.09680	0.09510	0.09342	0.09176	0.09012	0.08851	0.08692	0.08534	0.08379	0.08226
1.4	0.08076	0.07927	0.07780	0.07636	0.07493	0.07353	0.07215	0.07078	0.06944	0.06811

1.5	0.06681	0.06552	0.06426	0.06301	0.06178	0.06057	0.05938	0.05821	0.05705	0.05592
1.6	0.05480	0.05370	0.05262	0.05155	0.05050	0.04947	0.04846	0.04746	0.04648	0.04551
1.7	0.04457	0.04363	0.04272	0.04182	0.04093	0.04006	0.03920	0.03836	0.03754	0.03673
1.8	0.03593	0.03515	0.03438	0.03362	0.03288	0.03216	0.03144	0.03074	0.03005	0.02938
1.9	0.02872	0.02807	0.02743	0.02680	0.02619	0.02559	0.02500	0.02442	0.02385	0.02330

2.0	0.02275	0.02222	0.02169	0.02118	0.02068	0.02018	0.01970	0.01923	0.01876	0.01831
2.1	0.01786	0.01743	0.01700	0.01659	0.01618	0.01578	0.01539	0.01500	0.01463	0.01426
2.2	0.01390	0.01355	0.01321	0.01287	0.01255	0.01222	0.01191	0.01160	0.01130	0.01101
2.3	0.01072	0.01044	0.01017	0.00990	0.00964	0.00939	0.00914	0.00889	0.00866	0.00842
2.4	0.00820	0.00798	0.00776	0.00755	0.00734	0.00714	0.00695	0.00676	0.00657	0.00639

2.5	0.00621	0.00604	0.00587	0.00570	0.00554	0.00539	0.00523	0.00508	0.00494	0.00480
2.6	0.00466	0.00453	0.00440	0.00427	0.00415	0.00402	0.00391	0.00379	0.00368	0.00357
2.7	0.00347	0.00336	0.00326	0.00317	0.00307	0.00298	0.00289	0.00280	0.00272	0.00264
2.8	0.00256	0.00248	0.00240	0.00233	0.00226	0.00219	0.00212	0.00205	0.00199	0.00193
2.9	0.00187	0.00181	0.00175	0.00169	0.00164	0.00159	0.00154	0.00149	0.00144	0.00139

3.0	0.00135	0.00131	0.00126	0.00122	0.00118	0.00114	0.00111	0.00107	0.00104	0.00100
3.1	0.00097	0.00094	0.00090	0.00087	0.00084	0.00082	0.00079	0.00076	0.00074	0.00071
3.2	0.00069	0.00066	0.00064	0.00062	0.00060	0.00058	0.00056	0.00054	0.00052	0.00050
3.3	0.00048	0.00047	0.00045	0.00043	0.00042	0.00040	0.00039	0.00038	0.00036	0.00035
3.4	0.00034	0.00032	0.00031	0.00030	0.00029	0.00028	0.00027	0.00026	0.00025	0.00024

3.5	0.00023	0.00022	0.00022	0.00021	0.00020	0.00019	0.00019	0.00018	0.00017	0.00017
3.6	0.00016	0.00015	0.00015	0.00014	0.00014	0.00013	0.00013	0.00012	0.00012	0.00011
3.7	0.00011	0.00010	0.00010	0.00010	0.00009	0.00009	0.00008	0.00008	0.00008	0.00008
3.8	0.00007	0.00007	0.00007	0.00006	0.00006	0.00006	0.00006	0.00005	0.00005	0.00005
3.9	0.00005	0.00005	0.00004	0.00004	0.00004	0.00004	0.00004	0.00004	0.00003	0.00003

4.0	0.00003	0.00003	0.00003	0.00003	0.00003	0.00003	0.00002	0.00002	0.00002	0.00002

^[5] This table gives a probability that a statistic is greater than Template:Mvar, for large integer Template:Mvar values.

z	+0	+1	+2	+3	+4	+5	+6	+7	+8	+9
0	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val
10	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val
20	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val
30	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val
40	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val
50	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val
60	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val
70	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val	Template:Val

Examples of use

A professor's exam scores are approximately distributed normally with mean 80 and standard deviation 5. Only a cumulative from mean table is available.

What is the probability that a student scores an 82 or less? $\begin{matrix} P (X \leq 82) & = P (Z \leq \frac{82 - 80}{5}) \\ = P (Z \leq 0.40) \\ = 0.15542 + 0.5 \\ = 0.65542 \end{matrix}$
What is the probability that a student scores a 90 or more? $\begin{matrix} P (X \geq 90) & = P (Z \geq \frac{90 - 80}{5}) \\ = P (Z \geq 2.00) \\ = 1 - P (Z \leq 2.00) \\ = 1 - (0.47725 + 0.5) \\ = 0.02275 \end{matrix}$
What is the probability that a student scores a 74 or less? $\begin{matrix} P (X \leq 74) & = P (Z \leq \frac{74 - 80}{5}) \\ = P (Z \leq - 1.20) \end{matrix}$ Since this table does not include negatives, the process involves the following additional step: $\begin{matrix} = & P (Z \geq 1.20) \\ = & 1 - (0.38493 + 0.5) \\ = & 0.11507 \end{matrix}$
What is the probability that a student scores between 74 and 82? $\begin{matrix} P (74 \leq X \leq 82) & = P (X \leq 82) - P (X \leq 74) \\ = 0.65542 - 0.11507 \\ = 0.54035 \end{matrix}$
What is the probability that an average of three scores is 82 or less? $\begin{matrix} P (X \leq 82) & = P (Z \leq \frac{82 - 80}{5 / \sqrt{3}}) \\ = P (Z \leq 0.69) \\ = 0.2549 + 0.5 \\ = 0.7549 \end{matrix}$

References

Template:Reflist

↑ Template:Cite web
↑ Template:Cite book
↑ Template:Cite web
↑ 0.5 + each value in Cumulative from mean table
↑ 0.5 − each value in Cumulative from mean (0 to Z) table

[1] Template:Cite web

[2] Template:Cite book

[3] Template:Cite web

[4] 0.5 + each value in Cumulative from mean table

[5] 0.5 − each value in Cumulative from mean (0 to Z) table

[1]

[2]

[3]

[4]

[5]

Standard normal table

Contents

Normal and standard normal distribution

Conversion

Reading a Z table

Formatting / layout

Types of tables

Table examples

Cumulative from minus infinity to Z

Cumulative (less than Z)

Complementary cumulative

Examples of use

See also

References

Navigation menu

Standard normal table

Normal and standard normal distribution

Conversion

Reading a Z table

Formatting / layout

Types of tables

Table examples

Cumulative from minus infinity to Z

Cumulative (less than Z)

Complementary cumulative

Examples of use

See also

References

Navigation menu

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