CME 106 - Introduction to Probability and Statistics for Engineers

Distribution tables

By Afshine Amidi and Shervine Amidi

Standard Normal distribution

Notations

Let us note the random variable $Z$ that follows the standard normal distribution, i.e. which is such that:

\[Z\sim\mathcal{N}(0,1)\]

We note $z_\alpha$ as follows:

\[z_\alpha=\Phi_Z^{-1}(1-\alpha)\quad\textrm{i.e}\quad \alpha=\int_{z_{\alpha}}^{+\infty}f_Z(z)dz=1-\Phi_Z(z_\alpha)\]

with $\Phi_Z$ the cumulative distribution of $Z$


Distribution table

For one-tailed tests, the value of interest is $z_{\alpha}$, whereas for two-tailed tests, we have to look at $z_{\frac{\alpha}{2}}$

Confidence level 80% 85% 90% 95% 98% 99% 99.5% 99.9%
$\alpha$ 0.20 0.15 0.10 0.05 0.02 0.01 0.005 0.001
$z_{\alpha}$ 0.842 1.036 1.282 1.645 2.054 2.326 2.576 3.090
$z_{\frac{\alpha}{2}}$ 1.282 1.440 1.645 1.960 2.326 2.576 2.807 3.291


$t$ distribution

Notations

Let us note the random variable $T$ that follows a $t$ distribution of $n$ degrees of freedom, i.e. which is such that:

\[T\sim t_n\]

We note $t_\alpha$ as follows:

\[t_\alpha=\Phi_T^{-1}(1-\alpha)\quad\textrm{i.e}\quad \alpha=\int_{t_{\alpha}}^{+\infty}f_T(t)dt=1-\Phi_T(t_\alpha)\]

with $\Phi_T$ the cumulative distribution of $T$


Distribution table

$\alpha$ \ $n$ 1 2 3 4 5 6 7 8 9 10 11 12 13
0.20 1.38 1.06 0.98 0.94 0.92 0.91 0.90 0.89 0.88 0.88 0.88 0.87 0.87
0.10 3.08 1.89 1.64 1.53 1.48 1.44 1.41 1.40 1.38 1.37 1.36 1.36 1.35
0.05 6.31 2.92 2.35 2.13 2.02 1.94 1.89 1.86 1.83 1.81 1.80 1.78 1.77
0.025 12.7 4.30 3.18 2.78 2.57 2.45 2.36 2.31 2.26 2.23 2.20 2.18 2.16
0.01 31.8 6.96 4.54 3.75 3.36 3.14 3.00 2.90 2.82 2.76 2.72 2.68 2.65
0.005 63.7 9.92 5.84 4.60 4.03 3.71 3.50 3.36 3.25 3.17 3.11 3.05 3.01
0.001 318.3 22.3 10.2 7.17 5.89 5.21 4.79 4.50 4.30 4.14 4.03 3.93 3.85
$\alpha$ \ $n$ 15 18 20 22 24 26 28 30 40 50 100 200 $+\infty$
0.20 0.87 0.86 0.86 0.86 0.86 0.86 0.85 0.85 0.85 0.85 0.85 0.84 0.84
0.10 1.34 1.33 1.33 1.32 1.32 1.31 1.31 1.31 1.30 1.30 1.29 1.29 1.28
0.05 1.75 1.73 1.72 1.72 1.71 1.71 1.70 1.70 1.68 1.68 1.66 1.65 1.65
0.025 2.13 2.10 2.09 2.07 2.06 2.06 2.05 2.04 2.02 2.01 1.98 1.97 1.96
0.01 2.60 2.55 2.53 2.51 2.49 2.48 2.47 2.46 2.42 2.40 2.36 2.35 2.33
0.005 2.95 2.88 2.85 2.82 2.80 2.78 2.76 2.75 2.70 2.68 2.63 2.60 2.58
0.001 3.73 3.61 3.55 3.50 3.47 3.43 3.41 3.39 3.31 3.26 3.17 3.13 3.09


$\chi^2$ distribution

Notations

Let us note the random variable $K$ that follows a $\chi^2$ distribution of $n$ degrees of freedom, i.e. which is such that:

\[K\sim \chi_n^2\]
We note $q$ the quantile of the distribution.

Distribution table

$q$ \ $n$ 1 2 3 4 5 6 7 8 9 10 11 13
0.005 0.00 0.01 0.07 0.21 0.41 0.68 0.99 1.34 1.73 2.16 2.60 3.57
0.01 0.00 0.02 0.11 0.30 0.55 0.87 1.24 1.65 2.09 2.56 3.05 4.11
0.025 0.00 0.05 0.22 0.48 0.83 1.24 1.69 2.18 2.70 3.25 3.82 5.01
0.05 0.00 0.10 0.35 0.71 1.15 1.64 2.17 2.73 3.33 3.94 4.57 5.89
0.95 3.84 5.99 7.81 9.49 11.07 12.59 14.07 15.51 16.92 18.31 19.68 22.36
0.975 5.02 7.38 9.35 11.14 12.83 14.45 16.01 17.53 19.02 20.48 21.92 24.74
0.99 6.63 9.21 11.34 13.28 15.09 16.81 18.48 20.09 21.67 23.21 24.72 27.69
0.995 7.88 10.60 12.84 14.86 16.75 18.55 20.28 21.95 23.59 25.19 26.76 29.82
$q$ \ $n$ 15 18 20 22 24 26 28 30 40 50 100
0.005 4.60 6.26 7.43 8.64 9.89 11.16 12.46 13.79 20.71 27.99 67.33
0.01 5.23 7.01 8.26 9.54 10.86 12.20 13.56 14.95 22.16 29.71 70.06
0.025 6.26 8.23 9.59 10.98 12.40 13.84 15.31 16.79 24.43 32.36 74.22
0.05 7.26 9.39 10.85 12.34 13.85 15.38 16.93 18.49 26.51 34.76 77.93
0.95 25.00 28.87 31.41 33.92 36.42 38.89 41.34 43.77 55.76 67.50 124.34
0.975 27.49 31.53 34.17 36.78 39.36 41.92 44.46 46.98 59.34 71.42 129.56
0.99 30.58 34.81 37.57 40.29 42.98 45.64 48.28 50.89 63.69 76.15 135.81
0.995 32.80 37.16 40.00 42.80 45.56 48.29 50.99 53.67 66.77 79.49 140.17