This Negative Z Table can help you find the values that are left of the mean. Table entries for Z define the area under the standard normal curve to the left of the Z. The negative score represents the corresponding values that are less than the mean.
Z | 0 | 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 | 0.0001 | 0.0001 | 0.0001 | 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 | 0.0002 | 0.00019 | 0.00019 | 0.00018 | 0.00017 | 0.00017 |
-3.4 | 0.00034 | 0.00032 | 0.00031 | 0.0003 | 0.00029 | 0.00028 | 0.00027 | 0.00026 | 0.00025 | 0.00024 |
-3.3 | 0.00048 | 0.00047 | 0.00045 | 0.00043 | 0.00042 | 0.0004 | 0.00039 | 0.00038 | 0.00036 | 0.00035 |
-3.2 | 0.00069 | 0.00066 | 0.00064 | 0.00062 | 0.0006 | 0.00058 | 0.00056 | 0.00054 | 0.00052 | 0.0005 |
-3.1 | 0.00097 | 0.00094 | 0.0009 | 0.00087 | 0.00084 | 0.00082 | 0.00079 | 0.00076 | 0.00074 | 0.00071 |
-3 | 0.00135 | 0.00131 | 0.00126 | 0.00122 | 0.00118 | 0.00114 | 0.00111 | 0.00107 | 0.00104 | 0.001 |
-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 | 0.0024 | 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 | 0.0028 | 0.00272 | 0.00264 |
-2.6 | 0.00466 | 0.00453 | 0.0044 | 0.00427 | 0.00415 | 0.00402 | 0.00391 | 0.00379 | 0.00368 | 0.00357 |
-2.5 | 0.00621 | 0.00604 | 0.00587 | 0.0057 | 0.00554 | 0.00539 | 0.00523 | 0.00508 | 0.00494 | 0.0048 |
-2.4 | 0.0082 | 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 | 0.0099 | 0.00964 | 0.00939 | 0.00914 | 0.00889 | 0.00866 | 0.00842 |
-2.2 | 0.0139 | 0.01355 | 0.01321 | 0.01287 | 0.01255 | 0.01222 | 0.01191 | 0.0116 | 0.0113 | 0.01101 |
-2.1 | 0.01786 | 0.01743 | 0.017 | 0.01659 | 0.01618 | 0.01578 | 0.01539 | 0.015 | 0.01463 | 0.01426 |
-2 | 0.02275 | 0.02222 | 0.02169 | 0.02118 | 0.02068 | 0.02018 | 0.0197 | 0.01923 | 0.01876 | 0.01831 |
-1.9 | 0.02872 | 0.02807 | 0.02743 | 0.0268 | 0.02619 | 0.02559 | 0.025 | 0.02442 | 0.02385 | 0.0233 |
-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 | 0.0392 | 0.03836 | 0.03754 | 0.03673 |
-1.6 | 0.0548 | 0.0537 | 0.05262 | 0.05155 | 0.0505 | 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 | 0.0778 | 0.07636 | 0.07493 | 0.07353 | 0.07215 | 0.07078 | 0.06944 | 0.06811 |
-1.3 | 0.0968 | 0.0951 | 0.09342 | 0.09176 | 0.09012 | 0.08851 | 0.08691 | 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 | 0.1335 | 0.13136 | 0.12924 | 0.12714 | 0.12507 | 0.12302 | 0.121 | 0.119 | 0.11702 |
-1 | 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 | 0.2327 | 0.22965 | 0.22663 | 0.22363 | 0.22065 | 0.2177 | 0.21476 |
-0.6 | 0.27425 | 0.27093 | 0.26763 | 0.26435 | 0.26109 | 0.25785 | 0.25463 | 0.25143 | 0.24825 | 0.2451 |
-0.5 | 0.30854 | 0.30503 | 0.30153 | 0.29806 | 0.2946 | 0.29116 | 0.28774 | 0.28434 | 0.28096 | 0.2776 |
-0.4 | 0.34458 | 0.3409 | 0.33724 | 0.3336 | 0.32997 | 0.32636 | 0.32276 | 0.31918 | 0.31561 | 0.31207 |
-0.3 | 0.38209 | 0.37828 | 0.37448 | 0.3707 | 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 | 0.4562 | 0.45224 | 0.44828 | 0.44433 | 0.44038 | 0.43644 | 0.43251 | 0.42858 | 0.42465 |
0 | 0.5 | 0.49601 | 0.49202 | 0.48803 | 0.48405 | 0.48006 | 0.47608 | 0.4721 | 0.46812 | 0.46414 |
The positive Z-score table is used to find the values that are right from the mean. The table entries for Z define the area under the standard normal curve to the left of the Z. Positive Z-score represents the corresponding values greater than the mean.
Z | 0 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 |
---|---|---|---|---|---|---|---|---|---|---|
0 | 0.5 | 0.50399 | 0.50798 | 0.51197 | 0.51595 | 0.51994 | 0.52392 | 0.5279 | 0.53188 | 0.53586 |
0.1 | 0.53983 | 0.5438 | 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 | 0.6293 | 0.63307 | 0.63683 | 0.64058 | 0.64431 | 0.64803 | 0.65173 |
0.4 | 0.65542 | 0.6591 | 0.66276 | 0.6664 | 0.67003 | 0.67364 | 0.67724 | 0.68082 | 0.68439 | 0.68793 |
0.5 | 0.69146 | 0.69497 | 0.69847 | 0.70194 | 0.7054 | 0.70884 | 0.71226 | 0.71566 | 0.71904 | 0.7224 |
0.6 | 0.72575 | 0.72907 | 0.73237 | 0.73565 | 0.73891 | 0.74215 | 0.74537 | 0.74857 | 0.75175 | 0.7549 |
0.7 | 0.75804 | 0.76115 | 0.76424 | 0.7673 | 0.77035 | 0.77337 | 0.77637 | 0.77935 | 0.7823 | 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 | 0.84134 | 0.84375 | 0.84614 | 0.84849 | 0.85083 | 0.85314 | 0.85543 | 0.85769 | 0.85993 | 0.86214 |
1.1 | 0.86433 | 0.8665 | 0.86864 | 0.87076 | 0.87286 | 0.87493 | 0.87698 | 0.879 | 0.881 | 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 | 0.9032 | 0.9049 | 0.90658 | 0.90824 | 0.90988 | 0.91149 | 0.91309 | 0.91466 | 0.91621 | 0.91774 |
1.4 | 0.91924 | 0.92073 | 0.9222 | 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 | 0.9452 | 0.9463 | 0.94738 | 0.94845 | 0.9495 | 0.95053 | 0.95154 | 0.95254 | 0.95352 | 0.95449 |
1.7 | 0.95543 | 0.95637 | 0.95728 | 0.95818 | 0.95907 | 0.95994 | 0.9608 | 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 | 0.9732 | 0.97381 | 0.97441 | 0.975 | 0.97558 | 0.97615 | 0.9767 |
2 | 0.97725 | 0.97778 | 0.97831 | 0.97882 | 0.97932 | 0.97982 | 0.9803 | 0.98077 | 0.98124 | 0.98169 |
2.1 | 0.98214 | 0.98257 | 0.983 | 0.98341 | 0.98382 | 0.98422 | 0.98461 | 0.985 | 0.98537 | 0.98574 |
2.2 | 0.9861 | 0.98645 | 0.98679 | 0.98713 | 0.98745 | 0.98778 | 0.98809 | 0.9884 | 0.9887 | 0.98899 |
2.3 | 0.98928 | 0.98956 | 0.98983 | 0.9901 | 0.99036 | 0.99061 | 0.99086 | 0.99111 | 0.99134 | 0.99158 |
2.4 | 0.9918 | 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 | 0.9943 | 0.99446 | 0.99461 | 0.99477 | 0.99492 | 0.99506 | 0.9952 |
2.6 | 0.99534 | 0.99547 | 0.9956 | 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 | 0.9972 | 0.99728 | 0.99736 |
2.8 | 0.99744 | 0.99752 | 0.9976 | 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 | 0.99865 | 0.99869 | 0.99874 | 0.99878 | 0.99882 | 0.99886 | 0.99889 | 0.99893 | 0.99896 | 0.999 |
3.1 | 0.99903 | 0.99906 | 0.9991 | 0.99913 | 0.99916 | 0.99918 | 0.99921 | 0.99924 | 0.99926 | 0.99929 |
3.2 | 0.99931 | 0.99934 | 0.99936 | 0.99938 | 0.9994 | 0.99942 | 0.99944 | 0.99946 | 0.99948 | 0.9995 |
3.3 | 0.99952 | 0.99953 | 0.99955 | 0.99957 | 0.99958 | 0.9996 | 0.99961 | 0.99962 | 0.99964 | 0.99965 |
3.4 | 0.99966 | 0.99968 | 0.99969 | 0.9997 | 0.99971 | 0.99972 | 0.99973 | 0.99974 | 0.99975 | 0.99976 |
3.5 | 0.99977 | 0.99978 | 0.99978 | 0.99979 | 0.9998 | 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 | 0.9999 | 0.9999 | 0.9999 | 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 |
Let's take an example to understand how the z-score is applied in real-life situations and how it can be determined.
Suppose there are 100 students in class. John is among them and scored 80 out of 100 in mathematics. The average score of the full class is 63 and the standard deviation is 30. Find out the z-score.
The Z-score is calculated using the following formula:
Z-score = |
|
Where,
x = Observed Value,
μ = Mean,
σ = Standard Deviation.
In John's case, the values are:
x = 80,
µ = 63,
σ = 30.
Now place the above values in the formula:
Z-score = |
|
= 0.57 |
You can also use the Z-score Calculator to find it with ease.
To check how well John performed, it needs to find the percentage of the highest and lowest scores of other students. For that, the Z-Table (or Standard Normal Distribution Table) is used. So, keep it handy.
There are two z-tables with positive and negative values. If the z-score is positive, use the positive one and, for the negative z-score, use the negative one.
In John's case, the z-score is 0.57 and it's positive. So, we will look into the positive Z-score table.
Follow the instructions below to find the corresponding probability from the table.
In our case, it's '0.5'.
In this case, it's '0.07'.
As a result, the intersecting value is 0.71566.
Multiply the resultant value by 100 to get a percentage score. Or you can use the Online Percentage Calculator to do so.
It means almost 72% of students got lower marks than John and 28% got higher.
72% out of 100 = 72.
So, it defines John performed better than the other 72 students.