Z-table

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MamtaWardhani
Published Feb 17, 2025Updated Feb 20, 2025
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The Z-table is a statistical reference that shows cumulative probabilities for z-scores in a standard normal distribution (which has a mean of 0 and standard deviation of 1). The table helps find probabilities associated with z-scores, where:

  • Positive z-values represent values above the mean.
  • Negative z-values represent values below the mean.

Types of Z-Tables

Z-tables come in different formats depending on how probabilities are calculated:

  • Cumulative Z-table (Left-Tail): The most commonly used table, showing the probability of a value occurring below a given Z-score (area to the left).
  • Right-Tail Z-table: Displays the probability of a value occurring above a given Z-score (area to the right). It is used when focusing on upper-tail probabilities.
  • Two-Tailed Tests: Used in hypothesis testing, where deviations in both directions are considered. To find probabilities, sum the areas in both tails or subtract from 1 accordingly.

Z-Distribution Table for Positive Z-Values

Shows the cumulative probabilities of values occurring below positive Z-scores in a standard normal distribution:

Z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 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 0.87899 0.88098 0.88296
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 0.97614 0.97669
2.0 0.97725 0.97778 0.97831 0.97882 0.97932 0.97982 0.98030 0.98077 0.98124 0.98169
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 0.98499 0.98537 0.98574
2.2 0.98610 0.98645 0.98679 0.98713 0.98746 0.98778 0.98810 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 0.99597 0.99609 0.99620 0.99631 0.99642
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 0.99824 0.99830 0.99834 0.99839 0.99844 0.99849 0.99854 0.99858
3.0 0.99865 0.99869 0.99874 0.99878 0.99882 0.99886 0.99889 0.99893 0.99896 0.99899
3.1 0.99903 0.99906 0.99909 0.99912 0.99915 0.99918 0.99920 0.99923 0.99925 0.99927
3.2 0.99929 0.99931 0.99934 0.99936 0.99938 0.99940 0.99942 0.99944 0.99946 0.99947
3.3 0.99949 0.99951 0.99952 0.99954 0.99955 0.99957 0.99958 0.99959 0.99961 0.99962
3.4 0.99963 0.99964 0.99965 0.99966 0.99967 0.99968 0.99969 0.99970 0.99971 0.99972
3.5 0.99973 0.99974 0.99975 0.99975 0.99976 0.99977 0.99977 0.99978 0.99978 0.99979
3.6 0.99980 0.99980 0.99981 0.99981 0.99982 0.99982 0.99983 0.99983 0.99983 0.99984
3.7 0.99984 0.99985 0.99985 0.99985 0.99986 0.99986 0.99986 0.99986 0.99987 0.99987
3.8 0.99987 0.99987 0.99988 0.99988 0.99988 0.99988 0.99988 0.99989 0.99989 0.99989
3.9 0.99989 0.99989 0.99989 0.99990 0.99990 0.99990 0.99990 0.99990 0.99990 0.99991

Z-Distribution Table for Negative Z-Values

Displays the cumulative probabilities of values occurring below negative Z-scores, representing values to the left of the mean:

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 0.01049 0.01026 0.01004 0.00982 0.00961 0.00940 0.00919 0.00899 0.00879
-2.2 0.01390 0.01361 0.01321 0.01297 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 0.01969 0.01921 0.01873 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 0.03145 0.03076 0.03008 0.02941
-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 0.05156 0.05050 0.04947 0.04846 0.04746 0.04648 0.04552
-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 0.07778 0.07636 0.07493 0.07352 0.07215 0.07078 0.06944 0.06811
-1.3 0.09680 0.09510 0.09342 0.09176 0.09012 0.08851 0.08693 0.08536 0.08380 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 0.13360 0.13136 0.12924 0.12714 0.12507 0.12302 0.12099 0.11899 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 0.17621 0.17361 0.17106 0.16850 0.16600 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 0.23272 0.22962 0.22655 0.22352 0.22055 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 0.42464
0.0 0.50000 0.49601 0.49202 0.48803 0.48405 0.48006 0.47608 0.47210 0.46812 0.46414

How to Use the Z-Table

  1. Calculate the Z-Score: Determine the z-score based on the data point’s distance from the mean.
  2. Locate the Z-Score: Find the first decimal in the left column and the second decimal in the top row; their intersection gives the probability.
  3. Interpret the Probability:
    • Left-tail table: The value represents the probability to the left of Z.
    • Right-tail: Subtract from 1 to get the probability to the right.
    • Between two z-scores: Subtract the smaller probability from the larger one to find the probability of a value falling between two z-scores.

Example 1

To find the z-value corresponding to a cumulative probability, locate the probability in the table and find the associated z-score.

To find the z-score for a cumulative probability of 0.975:

  • Look for 0.975 in the table.
  • The closest value is 0.9750, which corresponds to z = 1.96 (row 1.9, column 0.06).

Thus, the z-score for a cumulative probability of 0.975 is 1.96.

Example 2

To find the cumulative probability (p-value) for a given z-score, locate the z-score in the table and read the corresponding probability.

To find the cumulative probability for z = -1.5:

  • Locate z = -1.5 in the table (row -1.5).
  • Find the probability under column 0.00, which is 0.0668.

Thus, the cumulative probability (p-value) for z = -1.5 is 0.0668.

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