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Hometable of probabilities of standard normal distribution

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# table of probabilities of standard normal distribution

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

$\Phi(z)=\frac{1}{\sqrt{2\pi}}\int_{{-\infty}}^{z}N(1,0)\,dx\,,$ |

evaluated from $-\infty$ to various $z$-scores. The values are rounded to the nearest ten thousandths.

z-score | \red0.00 | \red0.01 | \red0.02 | \red0.03 | \red0.04 | \red0.05 | \red0.06 | \red0.07 | \red0.08 | \red0.09 |
---|---|---|---|---|---|---|---|---|---|---|

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 |

\blue0.2 | \blue0.5793 | \blue0.5832 | \blue0.5871 | \blue0.5910 | \blue0.5948 | \blue0.5987 | \blue0.6026 | \blue0.6064 | \blue0.6103 | \blue0.6141 |

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 |

\blue0.5 | \blue0.6915 | \blue0.6950 | \blue0.6985 | \blue0.7019 | \blue0.7054 | \blue0.7088 | \blue0.7123 | \blue0.7157 | \blue0.7190 | \blue0.7224 |

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 |

\blue0.8 | \blue0.7881 | \blue0.7910 | \blue0.7939 | \blue0.7967 | \blue0.7995 | \blue0.8023 | \blue0.8051 | \blue0.8078 | \blue0.8106 | \blue0.8133 |

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 |

\blue1.1 | \blue0.8643 | \blue0.8665 | \blue0.8686 | \blue0.8708 | \blue0.8729 | \blue0.8749 | \blue0.8770 | \blue0.8790 | \blue0.8810 | \blue0.8830 |

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 |

\blue1.4 | \blue0.9192 | \blue0.9207 | \blue0.9222 | \blue0.9236 | \blue0.9251 | \blue0.9265 | \blue0.9279 | \blue0.9292 | \blue0.9306 | \blue0.9319 |

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 |

\blue1.7 | \blue0.9554 | \blue0.9564 | \blue0.9573 | \blue0.9582 | \blue0.9591 | \blue0.9599 | \blue0.9608 | \blue0.9616 | \blue0.9625 | \blue0.9633 |

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 |

\blue2.0 | \blue0.9772 | \blue0.9778 | \blue0.9783 | \blue0.9788 | \blue0.9793 | \blue0.9798 | \blue0.9803 | \blue0.9808 | \blue0.9812 | \blue0.9817 |

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 |

\blue2.3 | \blue0.9893 | \blue0.9896 | \blue0.9898 | \blue0.9901 | \blue0.9904 | \blue0.9906 | \blue0.9909 | \blue0.9911 | \blue0.9913 | \blue0.9916 |

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 |

\blue2.6 | \blue0.9953 | \blue0.9955 | \blue0.9956 | \blue0.9957 | \blue0.9959 | \blue0.9960 | \blue0.9961 | \blue0.9962 | \blue0.9963 | \blue0.9964 |

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 |

\blue2.9 | \blue0.9981 | \blue0.9982 | \blue0.9982 | \blue0.9983 | \blue0.9984 | \blue0.9984 | \blue0.9985 | \blue0.9985 | \blue0.9986 | \blue0.9986 |

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 |

\blue3.2 | \blue0.9993 | \blue0.9993 | \blue0.9994 | \blue0.9994 | \blue0.9994 | \blue0.9994 | \blue0.9994 | \blue0.9995 | \blue0.9995 | \blue0.9995 |

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 |

\blue3.4 | \blue0.9997 | \blue0.9997 | \blue0.9997 | \blue0.9997 | \blue0.9997 | \blue0.9997 | \blue0.9997 | \blue0.9997 | \blue0.9997 | \blue0.9998 |

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 |

\blue3.7 | \blue0.9999 | \blue0.9999 | \blue0.9999 | \blue0.9999 | \blue0.9999 | \blue0.9999 | \blue0.9999 | \blue0.9999 | \blue0.9999 | \blue0.9999 |

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 |

\blue4.0 | \blue1.0000 | \blue1.0000 | \blue1.0000 | \blue1.0000 | \blue1.0000 | \blue1.0000 | \blue1.0000 | \blue1.0000 | \blue1.0000 | \blue1.0000 |

Graphically, this looks like

yunit=4cm,xunit=4 \pspicture(-2,-0.2)(2,1) \psaxes-¿(0,0)(-2,0)(2,1) \uput[-90](2,0)x\uput[0](0,0.9)y \psplot-1.81.82.718282 x 2 mul 2 exp neg 2 div exp 0.75 mul \psclip \pscustom[linestyle=none]\moveto(-1.8,0)\psplot-1.81.82.718282 x 2 mul 2 exp neg 2 div exp 0.75 mul \lineto(0.4,-1) \pscustom[linestyle=none]\moveto(-1.8,0)\psplot01.80\lineto(0.4,1) \psframe*[linecolor=lightgray](-1.8,0)(0.4,2) \endpsclip\rput[t](0.4,-0.05)$z$ \rput[t](-0.1,0.4)$\Phi(z)$

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 $x\leq z$.

# 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 $0\leq z\leq 4$, 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. 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 $x\leq z$ is the same as the area bounded by $N(0,1)$, the $x$-axis, and $x\geq-z$, 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

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

# 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 $0.5\leq r\leq 1$, 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 $z\geq 3.9$ 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

$0.91236\approx 0.4625\cdot 0.9115+0.5375\cdot 0.9131.$ |

So, the $z$-score corresponding to $0.91236$ can be obtained similarly

$0.4625\cdot 1.35+0.5375\cdot 1.36\approx 1.3554\approx z,$ |

where $1.35$ is the $z$-score for $0.9115$ and $1.36$ is the $z$-score for $0.9131$.

## Mathematics Subject Classification

62E15*no label found*62Q05

*no label found*60E05

*no label found*

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