{{Short description|Table of probabilities related to the normal distribution}}
In statistics, a '''standard normal table''', also called the '''unit normal table''' or '''Z table''',<ref>{{cite web|title=Z Table. History of Z Table. Z Score|url=https://www.ztable.net/|accessdate=21 December 2018}}</ref> is a mathematical table for the values of {{math|Φ}}, 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.<ref>{{cite book |title=Elementary Statistics: Picturing the World|first1=Ron |last1=Larson|first2=Elizabeth|last2=Farber|publisher=清华大学出版社|year=2004|isbn=7-302-09723-2|page=214}}</ref>
== 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 {{mvar|Z}}, is the normal distribution having a mean of 0 and a standard deviation of 1.
=== Conversion === {{main|Standard normal deviate}} If {{mvar|X}} is a random variable from a normal distribution with mean {{mvar|μ}} and standard deviation {{mvar|σ}}, its Z-score may be calculated from {{mvar|X}} by subtracting {{mvar|μ}} and dividing by the standard deviation:
: <math>Z = \frac{X - \mu}{\sigma } </math> If <math>\overline{X}</math> is the mean of a sample of size {{mvar|n}} from some population in which the mean is {{mvar|μ}} and the standard deviation is {{mvar|σ}}, the standard error is {{tmath|\tfrac{\sigma}{\sqrt n}:}}
: <math>Z = \frac{\overline{X} - \mu}{\sigma / \sqrt n}</math>
If <math display="inline">\sum X</math> is the total of a sample of size {{mvar|n}} from some population in which the mean is {{mvar|μ}} and the standard deviation is {{mvar|σ}}, the expected total is {{mvar|nμ}} and the standard error is {{tmath|\sigma \sqrt n:}}
: <math>Z = \frac{\sum{X} - n\mu}{\sigma \sqrt{n}} </math>
== Reading a Z table == === Formatting / layout === {{mvar|Z}} tables are typically composed as follows: * The label for rows contains the integer part and the first decimal place of {{mvar|Z}}. * The label for columns contains the second decimal place of {{mvar|Z}}. * 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 {{mvar|Z}}.
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<ref>{{Cite web|url=https://ztable.io/ |title=How to use a Z Table |publisher=ztable.io |access-date=9 January 2023}}</ref> which yield a probability of '''0.20327'''.
But since the normal distribution curve is symmetrical, probabilities for only positive values of {{mvar|Z}} are typically given. The user might have to use a complementary operation on the absolute value of {{mvar|Z}}, as in the example below.
=== Types of tables === {{mvar|Z}} tables use at least three different conventions:
; Cumulative from mean: gives a probability that a statistic is between 0 (mean) and {{mvar|Z}}. Example: {{math|Prob(0 ≤ Z ≤ 0.69) {{=}} 0.2549}}. ; Cumulative: gives a probability that a statistic is less than {{mvar|Z}}. This equates to the area of the distribution below {{mvar|Z}}. Example: {{math|Prob(Z ≤ 0.69) {{=}} 0.7549}}. ; Complementary cumulative: gives a probability that a statistic is greater than {{mvar|Z}}. This equates to the area of the distribution above {{mvar|Z}}. : Example: Find {{math|Prob(''Z'' ≥ 0.69)}}. Since this is the portion of the area above {{mvar|Z}}, the proportion that is greater than {{mvar|Z}} is found by subtracting {{mvar|Z}} from 1. That is {{math|Prob(''Z'' ≥ 0.69) {{=}} 1 − Prob(Z ≤ 0.69)}} or {{math|Prob(''Z'' ≥ 0.69) {{=}} 1 − 0.7549 {{=}} 0.2451}}.
== Table examples == === Cumulative from minus infinity to Z === thumb|right|The values correspond to the shaded area for given {{mvar|Z}} This table gives a probability that a statistic is between minus infinity and {{mvar|Z}}.
: <math> f(z) = \Phi(z)</math>
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 <math>\Phi</math> (phi), is the integral
:<math>\Phi(z) = \frac 1 {\sqrt{2\pi}} \int_{-\infty}^z e^{-t^2/2} \, dt</math>
<math>\Phi</math>(z) is related to the error function, or {{math|erf(''z'')}}.
: <math> \Phi(z) = \frac12\left[1 + \operatorname{erf}\left( \frac z {\sqrt 2} \right) \right]</math>
Note that for {{math|''z'' {{=}} 1, 2, 3}}, one obtains (after multiplying by 2 to account for the {{math|[−''z'',''z'']}} interval) the results {{math|''f'' (''z'') {{=}} 0.6827, 0.9545, 0.9974}}, characteristic of the 68–95–99.7 rule.
=== Cumulative (less than Z) === This table gives a probability that a statistic is less than {{mvar|Z}} (i.e. between negative infinity and {{mvar|Z}}).
{| class="wikitable" |- ! ''z'' !! −0.00 !! −0.01 !! −0.02 !! −0.03 !! −0.04 !! −0.05 !! −0.06 !! −0.07 !! −0.08 !! −0.09 |- ! -3.9 | 0.00005 || 0.00005 || 0.00004 || 0.00004 || 0.00004 || 0.00004 || 0.00004 || 0.00004 || 0.00003 || 0.00003 |- ! -3.8 | 0.00007 || 0.00007 || 0.00007 || 0.00006 || 0.00006 || 0.00006 || 0.00006 || 0.00005 || 0.00005 || 0.00005 |- ! -3.7 | 0.00011 || 0.00010 || 0.00010 || 0.00010 || 0.00009 || 0.00009 || 0.00008 || 0.00008 || 0.00008 || 0.00008 |- ! -3.6 | 0.00016 || 0.00015 || 0.00015 || 0.00014 || 0.00014 || 0.00013 || 0.00013 || 0.00012 || 0.00012 || 0.00011 |- ! -3.5 | 0.00023 || 0.00022 || 0.00022 || 0.00021 || 0.00020 || 0.00019 || 0.00019 || 0.00018 || 0.00017 || 0.00017 |- ! -3.4 | 0.00034 || 0.00032 || 0.00031 || 0.00030 || 0.00029 || 0.00028 || 0.00027 || 0.00026 || 0.00025 || 0.00024 |- ! -3.3 | 0.00048 || 0.00047 || 0.00045 || 0.00043 || 0.00042 || 0.00040 || 0.00039 || 0.00038 || 0.00036 || 0.00035 |- ! -3.2 | 0.00069 || 0.00066 || 0.00064 || 0.00062 || 0.00060 || 0.00058 || 0.00056 || 0.00054 || 0.00052 || 0.00050 |- ! -3.1 | 0.00097 || 0.00094 || 0.00090 || 0.00087 || 0.00084 || 0.00082 || 0.00079 || 0.00076 || 0.00074 || 0.00071 |- ! -3.0 | 0.00135 || 0.00131 || 0.00126 || 0.00122 || 0.00118 || 0.00114 || 0.00111 || 0.00107 || 0.00104 || 0.00100 |- ! -2.9 | 0.00187 || 0.00181 || 0.00175 || 0.00169 || 0.00164 || 0.00159 || 0.00154 || 0.00149 || 0.00144 || 0.00139 |- ! -2.8 | 0.00256 || 0.00248 || 0.00240 || 0.00233 || 0.00226 || 0.00219 || 0.00212 || 0.00205 || 0.00199 || 0.00193 |- ! -2.7 | 0.00347 || 0.00336 || 0.00326 || 0.00317 || 0.00307 || 0.00298 || 0.00289 || 0.00280 || 0.00272 || 0.00264 |- ! -2.6 | 0.00466 || 0.00453 || 0.00440 || 0.00427 || 0.00415 || 0.00402 || 0.00391 || 0.00379 || 0.00368 || 0.00357 |- ! -2.5 | 0.00621 || 0.00604 || 0.00587 || 0.00570 || 0.00554 || 0.00539 || 0.00523 || 0.00508 || 0.00494 || 0.00480 |- ! -2.4 | 0.00820 || 0.00798 || 0.00776 || 0.00755 || 0.00734 || 0.00714 || 0.00695 || 0.00676 || 0.00657 || 0.00639 |- ! -2.3 | 0.01072 || 0.01044 || 0.01017 || 0.00990 || 0.00964 || 0.00939 || 0.00914 || 0.00889 || 0.00866 || 0.00842 |- ! -2.2 | 0.01390 || 0.01355 || 0.01321 || 0.01287 || 0.01255 || 0.01222 || 0.01191 || 0.01160 || 0.01130 || 0.01101 |- ! -2.1 | 0.01786 || 0.01743 || 0.01700 || 0.01659 || 0.01618 || 0.01578 || 0.01539 || 0.01500 || 0.01463 || 0.01426 |- ! -2.0 | 0.02275 || 0.02222 || 0.02169 || 0.02118 || 0.02068 || 0.02018 || 0.01970 || 0.01923 || 0.01876 || 0.01831 |- ! −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 |- | colspan="11" style="padding: 0;" | |- ! −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 |- | colspan="11" style="padding: 0;" | |- ! −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 |- | colspan="11" style="padding: 0;" | |- ! −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 |}
{| class="wikitable" |- ! ''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 |- | colspan="1" style="padding: 0;" | |- ! 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 |- | colspan="1" style="padding: 0;" | |- ! 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 |- | colspan="1" style="padding: 0;" | |- ! 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 |- | colspan="1" style="padding: 0;" | |- ! 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 |- | colspan="1" style="padding: 0;" | |- ! 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 |- | colspan="1" style="padding: 0;" | |- ! ''z'' !! +0.00 !! +0.01 !! +0.02 !! +0.03 !! +0.04 !! +0.05 !! +0.06 !! +0.07 !! +0.08 !! +0.09 |} <ref>0.5 + each value in ''Cumulative from mean'' table</ref>
=== Complementary cumulative === This table gives a probability that a statistic is greater than {{mvar|Z}}. :<math>f(z) = 1 - \Phi(z)</math>
{| class="wikitable" |- ! ''z'' !! +0.00 !! +0.01 !! +0.02 !! +0.03 !! +0.04 !! rowspan="50" style="padding:0;display:none;"| !! +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 |- | colspan="11" style="padding: 0;" | |- ! 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 |- | colspan="11" style="padding: 0;" | |- ! 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 |- | colspan="11" style="padding: 0;" | |- ! 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 |- | colspan="11" style="padding: 0;" | |- ! 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 |- | colspan="11" style="padding: 0;" | |- ! 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 |- | colspan="11" style="padding: 0;" | |- ! 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 |- | colspan="11" style="padding: 0;" | |- ! 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 |- | colspan="11" style="padding: 0;" | |- ! 4.0 | 0.00003 || 0.00003 || 0.00003 || 0.00003 || 0.00003 || 0.00003 || 0.00002 || 0.00002 || 0.00002 || 0.00002 |} <ref>0.5 − each value in ''Cumulative from mean'' (0 ''to'' Z) table</ref> This table gives a probability that a statistic is greater than {{mvar|Z}}, for large integer {{mvar|Z}} values.
{| class="wikitable" |- ! ''z'' !! +0 !! +1 !! +2 !! +3 !! +4 !! rowspan="38" style="padding:0;display:none;"| !! +5 !! +6 !! +7 !! +8 !! +9 |- ! 0 | {{val|5.00000|e=-1}} || {{val|1.58655|e=-1}} || {{val|2.27501|e=-2}} || {{val|1.34990|e=-3}} || {{val|3.16712|e=-5}} || {{val|2.86652|e=-7}} || {{val|9.86588|e=-10}} || {{val|1.27981|e=-12}} || {{val |6.22096|e=-16}} || {{val|1.12859|e=-19}} |- ! 10 | {{val|7.61985|e=-24}} || {{val|1.91066|e=-28}} || {{val|1.77648|e=-33}} || {{val|6.11716|e=-39}} || {{val|7.79354|e=-45}} || {{val|3.67097|e=-51}} || {{val|6.38875|e=-58}} || {{val|4.10600|e=-65}} || {{val|9.74095|e=-73}} || {{val|8.52722|e=-81}} |- ! 20 | {{val|2.75362|e=-89}} || {{val|3.27928|e=-98}} || {{val|1.43989|e=-107}} || {{val|2.33064|e=-117}} || {{val|1.39039|e=-127}} || {{val|3.05670|e=-138}} || {{val|2.47606|e=-149}} || {{val|7.38948|e=-161}} || {{val|8.12387|e=-173}} || {{val|3.28979|e=-185}} |- ! 30 | {{val|4.90671|e=-198}} || {{val|2.69525|e=-211}} || {{val|5.45208|e=-225}} || {{val|4.06119|e=-239}} || {{val|1.11390|e=-253}} || {{val|1.12491|e=-268}} || {{val|4.18262|e=-284}} || {{val|5.72557|e=-300}} || {{val|2.88543|e=-316}} || {{val|5.35312|e=-333}} |- ! 40 | {{val|3.65589|e=-350}} || {{val|9.19086|e=-368}} || {{val|8.50515|e=-386}} || {{val|2.89707|e=-404}} || {{val|3.63224|e=-423}} || {{val|1.67618|e=-442}} || {{val|2.84699|e=-462}} || {{val|1.77976|e=-482}} || {{val|4.09484|e=-503}} || {{val|3.46743|e=-524}} |- ! 50 | {{val|1.08060|e=-545}} || {{val|1.23937|e=-567}} || {{val|5.23127|e=-590}} || {{val|8.12606|e=-613}} || {{val|4.64529|e=-636}} || {{val|9.77237|e=-660}} || {{val|7.56547|e=-684}} || {{val|2.15534|e=-708}} || {{val|2.25962|e=-733}} || {{val|8.71741|e=-759}} |- ! 60 | {{val|1.23757|e=-784}} || {{val|6.46517|e=-811}} || {{val|1.24283|e=-837}} || {{val|8.79146|e=-865}} || {{val|2.28836|e=-892}} || {{val|2.19180|e=-920}} || {{val|7.72476|e=-949}} || {{val|1.00178|e=-977}} || {{val|4.78041|e=-1007}} || {{val|8.39374|e=-1037}} |- ! 70 | {{val|5.42304|e=-1067}} || {{val|1.28921|e=-1097}} || {{val|1.12771|e=-1128}} || {{val|3.62960|e=-1160}} || {{val|4.29841|e=-1192}} || {{val|1.87302|e=-1224}} || {{val|3.00302|e=-1257}} || {{val|1.77155|e=-1290}} || {{val|3.84530|e=-1324}} || {{val|3.07102|e=-1358}} |}
*
== See also == {{Portal|Mathematics}} {{Colbegin}} * 68–95–99.7 rule * ''t''-distribution table {{Colend}}
== References == {{reflist}}
Category:Normal distribution Category:Mathematical tables
== External links == * [https://math.arizona.edu/~rsims/ma464/standardnormaltable.pdf printable table] (from University of Arizona, USA)