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[parent] table of probabilities of standard normal distribution (Definition)

Below is a table of the values of the area (probabilities) $ \Phi(z)$ under the standard normal distribution function % latex2html id marker 639 $ N(1,0)=\operatorname{exp}(-x^2/2)$ given by

$\displaystyle \Phi(z) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^z N(1,0) \, dx\,,$
evaluated from $ -\infty$ to various $ z$-scores. The values are rounded to the nearest ten thousandths.
z-score 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998
3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998
3.6 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.8 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999
3.9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
4.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Graphically, this looks like

yunit=4cm,xunit=4
\begin{pspicture} % latex2html id marker 254 (-2,-0.2)(2,1) \psaxes{->}(0,0)(-2,... ...endpsclip \rput[t](0.4,-0.05){$z$} \rput[t](-0.1,0.4){$\Phi(z)$} \end{pspicture}

where the curve is the probability density function $ N(0,1)$ of the standard normal distribution (with mean 0 and standard deviation $ 1$), $ z$ on the $ x$-axis is the $ z$-score, and $ \Phi(z)$ (represented by the light gray region) is the area bounded by $ N(0,1)$, the $ x$-axis, and $ x\le z$.

Finding $ \Phi(z)$ from $ z$

Given a $ z$-score, one can easily find $ \Phi(z)$ as follows:

  1. round the $ z$-score $ z$ to the nearest hundredths decimal place; for example, if $ z=1.2345$, then rounding it to the hundredths gives you $ 1.23$.
  2. if $ 0\le z\le 4$, write $ z=a+b$, where $ a$ is the truncation of $ r$ at the tenths place, and $ b=r-a$; for example, if $ z=1.23$, then $ a=1.2$ and $ b=0.03$.
  3. locate $ a$ in the first column of the table, and then locate $ b$ in the first row of the table
  4. find the value in the cell corresponding to row $ a$ and column $ b$; this value is $ \Phi(z)$; for example, if $ a=1.2$ and $ b=0.03$, then the corresponding value is $ 0.8907$.

If $ z>4$, then $ \Phi(z)=1$ when rounded to the nearest ten thousandths. If $ z<0$, then we will not be able to use the table above. However, since $ N(0,1)$ is an even function, $ \Phi(z)$, the area bounded by $ N(0,1)$, the $ x$-axis, and $ x\le z$ is the same as the area bounded by $ N(0,1)$, the $ x$-axis, and $ x\ge -z$, which is equal to $ 1-\Phi(-z)$. These two facts can be summarized:

  1. If $ z>4$, then $ \Phi(z)=1$ when rounded to the nearest ten thousandths to the right of the decimal point.
  2. If $ z<0$, then use the formula $ \Phi(z)=1-\Phi(-z)$ before applying the table. For example, $ \Phi(-1.23)=1-\Phi(1.23)=1-0.8907=0.1093$.

Also, we may use linear interpolation to find (approximate) $ \Phi(z)$ for any arbitrary $ z$-score. For example, if we want to compute $ \Phi(1.234)$, then we first find $ \Phi(1.23)$ and $ \Phi(1.24)$. Then

$\displaystyle \Phi(1.234)\approx 0.6\cdot \Phi(1.23)+ 0.4\cdot \Phi(1.24)=0.6\cdot 0.8907+0.4\cdot 0.8925 \approx 0.8914.$

Finding $ z$ from $ \Phi(z)$

Given $ \Phi(z)$, we may use the table to find $ z$. The process works in reverse of the process presented in the previous section:

  1. round $ r=\Phi(z)$ to the nearest ten thousandths; for example if $ \Phi(z)=0.91236$, then $ r=0.9124$ after rounding
  2. if $ 0.5\le r\le 1$, then find the cell in the table with value as close to $ r$ as possible; for example, for $ r=0.9124$, the closest value that can be found in the table is $ 0.9131$
  3. if this cell is found, then find the corresponding value $ a$ in the first column and $ b$ in the first row, and $ z^*=a+b$ is the approximate $ z$-score that we are looking for; for example, $ 0.9131$ corresponds to $ a=1.3$ and $ b=0.06$ so that $ z^*=1.36$.
  4. if $ \Phi(z)<0.5$, then use $ r=1-\Phi(z)$ to find $ z^*$ using the first three steps above. Then $ z=-z^*$ is the $ z$-score that we are looking for.

Note that if $ \Phi(z)=1$, then any $ z\ge 3.9$ will work. Also, linear interpolation can again be applied to get better approximations of the $ z$-scores given $ \Phi(z)$. For example, $ \Phi(z)=0.91236$ is between $ 0.9115$ and $ 0.9131$, two consecutive values found in the table, and can be written

$\displaystyle 0.91236 \approx 0.4625 \cdot 0.9115 + 0.5375 \cdot 0.9131.$
So, the $ z$-score corresponding to $ 0.91236$ can be obtained similarly
$\displaystyle 0.4625 \cdot 1.35 + 0.5375 \cdot 1.36 \approx 1.3554 \approx z,$
where $ 1.35$ is the $ z$-score for $ 0.9115$ and $ 1.36$ is the $ z$-score for $ 0.9131$.



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See Also: area under Gaussian curve


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Cross-references: consecutive, approximations, section, linear interpolation, decimal point, right, even function, cell, row, column, place, tenths, truncation, rounding, decimal place, hundredths, bounded, region, standard deviation, mean, probability density function, curve, ten thousandths, rounded to, function, standard normal distribution, area
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This is version 6 of table of probabilities of standard normal distribution, born on 2007-08-06, modified 2007-12-18.
Object id is 9833, canonical name is TableOfProbabilitiesOfStandardNormalDistribution.
Accessed 1549 times total.

Classification:
AMS MSC: 60E05 (Probability theory and stochastic processes :: Distribution theory :: Distributions: general theory)
 62E15 (Statistics :: Distribution theory :: Exact distribution theory)
 62Q05 (Statistics :: Statistical tables)
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