/ ztables Public

Ever wonder how Standard Normal and Student-t Distribution z-tables are generated?

# jlinoff/ztables

Switch branches/tags
Nothing to show

A tag already exists with the provided branch name. Many Git commands accept both tag and branch names, so creating this branch may cause unexpected behavior. Are you sure you want to create this branch?

## Files

Failed to load latest commit information.
Type
Name
Commit time

# ztables

Ever wonder how Standard Normal and Student-t Distribution z-tables are generated?

## Introduction

This program shows how to generate z-tables for Standard Normal Distributions and Student-t Distributions. It also shows how to generate the z-values for specific probabilities from those distributions without using any special libraries.

I have tested the program using Python 2.7.10 and 3.5.

## Discussion

The goal is to show how to solve the probability density functions (PDFs) and use those solutions to approximate the area under a curve in order to obtain the z-value or, in the case of the z-value lookup, show how to do a binary search over the probabilities to find the desired z-value.

It is meant to be a learning tool for those that are interested in how the tables are generated. For production code use the scipy and numpy packages. They are optimized and much better debugged.

I explicitly chose to use the trapezoidal rule to approximate the definite integral representing the area under the curve because it is simple, reasonably fast and accurate enough. You could, of course, choose any other approximation method (like Simpson's Rule).

I also chose the Lanczos approximation for solving the gamma function for the t-distribution PDF. The algorithm for the approximation was obtained from page 214 of "Numerical Recipes in C, 2nd Edition".

Here is an example that shows you how to generate the z-table that contains the probabilities for the standard normal distribution using the default parameters. This is the table that you often see in text books.

``````\$ ztables.py -s

z-Table for Standard Normal Distribution (10,000)

z     .00     .01     .02     .03     .04     .05     .06     .07     .08     .09
=====  ======  ======  ======  ======  ======  ======  ======  ======  ======  ======
-3.4  0.0003  0.0003  0.0003  0.0003  0.0003  0.0003  0.0003  0.0003  0.0003  0.0002
-3.3  0.0005  0.0005  0.0005  0.0004  0.0004  0.0004  0.0004  0.0004  0.0004  0.0003
-3.2  0.0007  0.0007  0.0006  0.0006  0.0006  0.0006  0.0006  0.0005  0.0005  0.0005
-3.1  0.0010  0.0009  0.0009  0.0009  0.0008  0.0008  0.0008  0.0008  0.0007  0.0007
-3.0  0.0013  0.0013  0.0013  0.0012  0.0012  0.0011  0.0011  0.0011  0.0010  0.0010
.
.

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
3.4  0.9997  0.9997  0.9997  0.9997  0.9997  0.9997  0.9997  0.9997  0.9997  0.9998
``````

Here is an example that shows how to determine the z values for 95%, 98% and 99% probabilities using the Standard Normal Distribution and various Student-t distributions. It shows how the distributions converge on the SND values as the DOF increases.

``````\$ ./ztables.py -s -t 10 -t 20 -t 30 -t 100 -t 200 -p 0.95 -p 0.98 -p 0.99

Probabilities to z-values Table

Probability   SND    T-10     T-20     T-30     T-100    T-200
===========  =====  =======  =======  =======  =======  =======
95.00%     1.960    2.229    2.086    2.042    1.984    1.972
98.00%     2.326    2.765    2.528    2.457    2.364    2.345
99.00%     2.576    3.172    2.845    2.750    2.626    2.601
``````

In the sections below some of the code fragments are provided as well as the program options and some sample outputs.

## Program Options

These are the program options.

Option Long Description
-h --help Help.
-i INT --intervals INT The number intervals used for the area calculation.
-lÂ FLOAT --lower-boundÂ FLOAT The leftmost z-value to report.
-pÂ FLOAT --probability FLOAT Find the z-value for this probability. Must be in the range 0.001 to .9999.
-s --snd Generate the SND table unless -p specified. If -p is specified use the SND to find the z-value.
-t DOF --tdist DOF Generate the t-dist table with DOF degrees of freedom unless -p specified. If -p is specified use the t-dist to find the z-value.
-uÂ FLOAT --upper-boundÂ FLOAT The rightmost z-value to report.
-v --verbose Increase the level of verbosity. Only used for debugging -p.
-V --version Print the program version and exit.

## SND PDF

This is the code for the probability density function (PDF) for the standard normal distribution (SND).

```def pdf_snd(x):
'''
Calculate the probability density function (PDF) for a standard
normal distribution.

s = standard deviation (1 for a standard normal distribution)
u = mean (0 for a standard normal distribution)

This is the height of the curve at x.

It is exactly the same as pdf_nd(x, 1, 0) but is somewhat more
efficient.
'''
dx2 = float(x) ** 2
den = math.sqrt(2 * math.pi)
exp = math.e ** - (dx2 / 2)
y =  exp / den
return y```

## PDF for the normal distribution

This is the code for the probability density function (PDF) for the normal distribution where the mean and the standard deviation are explicitly specified.

```def pdf_nd(x, s=1.0, u=0.0):
'''
Calculate the probability density function (PDF) for a normal
distribution.

s = standard deviation (1 for a standard normal distribution)
u = mean (0 for a standard normal distribution)

This is the height of the curve at x.
'''
dx = float(x) - float(u)
dx2 = dx * dx
xden = 2 * (s ** 2)
den = s * math.sqrt(2 * math.pi)
exp = math.e ** ( -dx2 / xden )
y =  exp / den
return y```

## T-dist PDF

This is the code for the t-distribution PDF.

```def pdf_t(x, dof):
'''
Calculate the probability density function (PDF) at x for a
student-t distribution with dof degrees of freedom.

This is basically the height of the curve at x.
'''
assert dof > 2

x1 = gamma((float(dof) + 1.0) / 2.0)
x2 = math.sqrt(dof * math.pi) * gamma((float(dof) / 2.0))
x3 = 1.0 + (float((x ** 2)) / float(dof))
x4 = float((dof + 1)) / 2.0
x5 = x3 ** -x4

y = (x1 * x5) / x2
return y```

## Gamma Function

This is the code that estimates the gamma function for t-distribution calculations.

```def gamma(x):
'''
Gamma function.

Uses the Lanczos approximation and natural logarithms.

For integer values of x we can use the exact value of (x-1)!.

gamma(1/2) = 1.77245385091
gamma(3/2) = 0.886226925453
gamma(5/2) = 1.32934038818
gamma(7/2) = 3.32335097045
gamma(4)   = 6.0
'''
if (x - int(x)) == 0:
# Optimization for integer values: (x-1)!.
return reduce(lambda a, b: a * b, [float(i) for i in range(1, int(x))])

# Lanczos approximation, page 214 of Numerical Recipes in C.
c = [76.18009172947146,
-86.50532032941677,
24.01409824083091,
-1.231739572450155,
0.1208650973866179e-2,
-0.5395239384953e-5,
]
c0 = 1.000000000190015
c1 = 2.5066282746310005
x1 = float(x) + 5.5
x2 = (float(x) + 0.5) * math.log(x1)
x3 = x1 - x2
x4 = c0
x5 = float(x)
for i in range(6):
x5 += 1.0
x4 += c[i] / x5
x6 = math.log((c1 * x4) / float(x))
x7 = -x3 + x6  # ln(gamma(x))
g = math.exp(x7)
return g```

## Estimate the Area Under the Curve using the Trapezoidal Rule

This function shows how to estimate the area under the curve using the Trapezoidal Rule. There many other methods available calculating area under the curve for definite integrals but this one is reasonably fast and sufficiently accurate for this purpose.

```def area_under_curve(x1, x2, intervals, fct, *args, **kwargs):
'''
Calculate the approximate area under a curve using trapezoidal
approximation.

It breaks the interval between x1 and x2 into trapezoids whose
width is fixed (proportional to how the interval is sliced). The
height of each rectangle is the pdf function value for x at the
start of the interval. The accumulation of the areas provides an
estimate of the area under the curve.

The greater the number of intervals the better the estimate is at
the cost of performance.
'''
assert x2 > x1  # just a sanity check
assert intervals > 1  # another sanity check

total_area = 0.0
width = (float(x2) - float(x1)) / float(intervals)

x = float(x1)
py = float(fct(x, *args, **kwargs))
for i in range(intervals):
y = float(fct(x, *args, **kwargs))
rectangle_area = width * y  # area of rectangle at x with height y
triangle_area = ((y - py) * width) / 2.0  # adjustment based on height change
total_area += rectangle_area + triangle_area  # trapezoid area
x += width  # advance to the next edge
py = y  # remember the previous height

## Here is the SND Table.

Here is the Standard Normal Distribution Table generated by this program using the command: '`./ztables.py -s`'.

``````z-Table for Standard Normal Distribution (10,000)

z     .00     .01     .02     .03     .04     .05     .06     .07     .08     .09
=====  ======  ======  ======  ======  ======  ======  ======  ======  ======  ======
-3.4  0.0003  0.0003  0.0003  0.0003  0.0003  0.0003  0.0003  0.0003  0.0003  0.0002
-3.3  0.0005  0.0005  0.0005  0.0004  0.0004  0.0004  0.0004  0.0004  0.0004  0.0003
-3.2  0.0007  0.0007  0.0006  0.0006  0.0006  0.0006  0.0006  0.0005  0.0005  0.0005
-3.1  0.0010  0.0009  0.0009  0.0009  0.0008  0.0008  0.0008  0.0008  0.0007  0.0007
-3.0  0.0013  0.0013  0.0013  0.0012  0.0012  0.0011  0.0011  0.0011  0.0010  0.0010
-2.9  0.0019  0.0018  0.0018  0.0017  0.0016  0.0016  0.0015  0.0015  0.0014  0.0014
-2.8  0.0026  0.0025  0.0024  0.0023  0.0023  0.0022  0.0021  0.0021  0.0020  0.0019
-2.7  0.0035  0.0034  0.0033  0.0032  0.0031  0.0030  0.0029  0.0028  0.0027  0.0026
-2.6  0.0047  0.0045  0.0044  0.0043  0.0041  0.0040  0.0039  0.0038  0.0037  0.0036
-2.5  0.0062  0.0060  0.0059  0.0057  0.0055  0.0054  0.0052  0.0051  0.0049  0.0048
-2.4  0.0082  0.0080  0.0078  0.0075  0.0073  0.0071  0.0069  0.0068  0.0066  0.0064
-2.3  0.0107  0.0104  0.0102  0.0099  0.0096  0.0094  0.0091  0.0089  0.0087  0.0084
-2.2  0.0139  0.0136  0.0132  0.0129  0.0125  0.0122  0.0119  0.0116  0.0113  0.0110
-2.1  0.0179  0.0174  0.0170  0.0166  0.0162  0.0158  0.0154  0.0150  0.0146  0.0143
-2.0  0.0228  0.0222  0.0217  0.0212  0.0207  0.0202  0.0197  0.0192  0.0188  0.0183
-1.9  0.0287  0.0281  0.0274  0.0268  0.0262  0.0256  0.0250  0.0244  0.0239  0.0233
-1.8  0.0359  0.0351  0.0344  0.0336  0.0329  0.0322  0.0314  0.0307  0.0301  0.0294
-1.7  0.0446  0.0436  0.0427  0.0418  0.0409  0.0401  0.0392  0.0384  0.0375  0.0367
-1.6  0.0548  0.0537  0.0526  0.0516  0.0505  0.0495  0.0485  0.0475  0.0465  0.0455
-1.5  0.0668  0.0655  0.0643  0.0630  0.0618  0.0606  0.0594  0.0582  0.0571  0.0559
-1.4  0.0808  0.0793  0.0778  0.0764  0.0749  0.0735  0.0721  0.0708  0.0694  0.0681
-1.3  0.0968  0.0951  0.0934  0.0918  0.0901  0.0885  0.0869  0.0853  0.0838  0.0823
-1.2  0.1151  0.1131  0.1112  0.1093  0.1075  0.1056  0.1038  0.1020  0.1003  0.0985
-1.1  0.1357  0.1335  0.1314  0.1292  0.1271  0.1251  0.1230  0.1210  0.1190  0.1170
-1.0  0.1587  0.1562  0.1539  0.1515  0.1492  0.1469  0.1446  0.1423  0.1401  0.1379
-0.9  0.1841  0.1814  0.1788  0.1762  0.1736  0.1711  0.1685  0.1660  0.1635  0.1611
-0.8  0.2119  0.2090  0.2061  0.2033  0.2005  0.1977  0.1949  0.1922  0.1894  0.1867
-0.7  0.2420  0.2389  0.2358  0.2327  0.2296  0.2266  0.2236  0.2206  0.2177  0.2148
-0.6  0.2743  0.2709  0.2676  0.2643  0.2611  0.2578  0.2546  0.2514  0.2483  0.2451
-0.5  0.3085  0.3050  0.3015  0.2981  0.2946  0.2912  0.2877  0.2843  0.2810  0.2776
-0.4  0.3446  0.3409  0.3372  0.3336  0.3300  0.3264  0.3228  0.3192  0.3156  0.3121
-0.3  0.3821  0.3783  0.3745  0.3707  0.3669  0.3632  0.3594  0.3557  0.3520  0.3483
-0.2  0.4207  0.4168  0.4129  0.4090  0.4052  0.4013  0.3974  0.3936  0.3897  0.3859
-0.1  0.4602  0.4562  0.4522  0.4483  0.4443  0.4404  0.4364  0.4325  0.4286  0.4247
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
3.4  0.9997  0.9997  0.9997  0.9997  0.9997  0.9997  0.9997  0.9997  0.9997  0.9998
``````

## Here is a Student-t Table with 20 DOF.

Here is the Student-t Table with 20 degrees of freedom (DOF) generated by this program using the command '`ztables.py -t 20`'.

``````z-Table for Student-t Distribution (10,000, 20 DOF)

z     .00     .01     .02     .03     .04     .05     .06     .07     .08     .09
=====  ======  ======  ======  ======  ======  ======  ======  ======  ======  ======
-3.4  0.0014  0.0014  0.0014  0.0013  0.0013  0.0013  0.0012  0.0012  0.0012  0.0012
-3.3  0.0018  0.0017  0.0017  0.0017  0.0016  0.0016  0.0016  0.0015  0.0015  0.0015
-3.2  0.0022  0.0022  0.0021  0.0021  0.0021  0.0020  0.0020  0.0019  0.0019  0.0018
-3.1  0.0028  0.0028  0.0027  0.0026  0.0026  0.0025  0.0025  0.0024  0.0024  0.0023
-3.0  0.0035  0.0035  0.0034  0.0033  0.0032  0.0032  0.0031  0.0030  0.0030  0.0029
-2.9  0.0044  0.0043  0.0042  0.0041  0.0040  0.0040  0.0039  0.0038  0.0037  0.0036
-2.8  0.0055  0.0054  0.0053  0.0052  0.0051  0.0049  0.0048  0.0047  0.0046  0.0045
-2.7  0.0069  0.0067  0.0066  0.0065  0.0063  0.0062  0.0060  0.0059  0.0058  0.0057
-2.6  0.0086  0.0084  0.0082  0.0080  0.0079  0.0077  0.0075  0.0074  0.0072  0.0070
-2.5  0.0106  0.0104  0.0102  0.0100  0.0097  0.0095  0.0093  0.0091  0.0089  0.0088
-2.4  0.0131  0.0129  0.0126  0.0123  0.0121  0.0118  0.0116  0.0113  0.0111  0.0108
-2.3  0.0162  0.0158  0.0155  0.0152  0.0149  0.0146  0.0143  0.0140  0.0137  0.0134
-2.2  0.0199  0.0195  0.0191  0.0187  0.0183  0.0179  0.0176  0.0172  0.0169  0.0165
-2.1  0.0243  0.0238  0.0234  0.0229  0.0224  0.0220  0.0215  0.0211  0.0207  0.0203
-2.0  0.0296  0.0291  0.0285  0.0279  0.0274  0.0269  0.0263  0.0258  0.0253  0.0248
-1.9  0.0360  0.0353  0.0346  0.0340  0.0333  0.0327  0.0320  0.0314  0.0308  0.0302
-1.8  0.0435  0.0427  0.0419  0.0411  0.0403  0.0396  0.0388  0.0381  0.0374  0.0367
-1.7  0.0523  0.0514  0.0504  0.0495  0.0486  0.0477  0.0468  0.0460  0.0451  0.0443
-1.6  0.0626  0.0615  0.0604  0.0594  0.0583  0.0573  0.0563  0.0552  0.0543  0.0533
-1.5  0.0746  0.0733  0.0721  0.0708  0.0696  0.0684  0.0672  0.0661  0.0649  0.0638
-1.4  0.0884  0.0870  0.0855  0.0841  0.0827  0.0813  0.0799  0.0786  0.0772  0.0759
-1.3  0.1042  0.1025  0.1009  0.0992  0.0976  0.0960  0.0945  0.0929  0.0914  0.0899
-1.2  0.1221  0.1202  0.1183  0.1165  0.1147  0.1129  0.1111  0.1093  0.1076  0.1059
-1.1  0.1422  0.1401  0.1380  0.1359  0.1339  0.1319  0.1299  0.1279  0.1259  0.1240
-1.0  0.1646  0.1623  0.1600  0.1577  0.1554  0.1531  0.1509  0.1487  0.1465  0.1443
-0.9  0.1894  0.1868  0.1843  0.1817  0.1792  0.1767  0.1743  0.1718  0.1694  0.1670
-0.8  0.2166  0.2137  0.2109  0.2082  0.2054  0.2027  0.2000  0.1973  0.1947  0.1920
-0.7  0.2460  0.2429  0.2399  0.2369  0.2339  0.2310  0.2281  0.2251  0.2223  0.2194
-0.6  0.2776  0.2744  0.2711  0.2679  0.2647  0.2615  0.2584  0.2553  0.2521  0.2491
-0.5  0.3113  0.3078  0.3044  0.3010  0.2976  0.2942  0.2908  0.2875  0.2842  0.2809
-0.4  0.3467  0.3431  0.3395  0.3359  0.3323  0.3288  0.3252  0.3217  0.3182  0.3147
-0.3  0.3836  0.3799  0.3761  0.3724  0.3687  0.3650  0.3613  0.3576  0.3540  0.3503
-0.2  0.4217  0.4179  0.4141  0.4102  0.4064  0.4026  0.3988  0.3950  0.3912  0.3874
-0.1  0.4607  0.4568  0.4528  0.4489  0.4450  0.4411  0.4372  0.4334  0.4295  0.4256
0.0  0.5000  0.5039  0.5079  0.5118  0.5158  0.5197  0.5236  0.5276  0.5315  0.5354
0.1  0.5393  0.5432  0.5472  0.5511  0.5550  0.5589  0.5628  0.5666  0.5705  0.5744
0.2  0.5782  0.5821  0.5859  0.5898  0.5936  0.5974  0.6012  0.6050  0.6088  0.6126
0.3  0.6164  0.6201  0.6239  0.6276  0.6313  0.6350  0.6387  0.6424  0.6460  0.6497
0.4  0.6533  0.6569  0.6605  0.6641  0.6677  0.6712  0.6748  0.6783  0.6818  0.6853
0.5  0.6887  0.6922  0.6956  0.6990  0.7024  0.7058  0.7092  0.7125  0.7158  0.7191
0.6  0.7224  0.7256  0.7289  0.7321  0.7353  0.7385  0.7416  0.7447  0.7478  0.7509
0.7  0.7540  0.7570  0.7601  0.7631  0.7661  0.7690  0.7719  0.7748  0.7777  0.7806
0.8  0.7834  0.7863  0.7891  0.7918  0.7946  0.7973  0.8000  0.8027  0.8053  0.8080
0.9  0.8106  0.8132  0.8157  0.8183  0.8208  0.8233  0.8257  0.8282  0.8306  0.8330
1.0  0.8354  0.8377  0.8400  0.8423  0.8446  0.8469  0.8491  0.8513  0.8535  0.8557
1.1  0.8578  0.8599  0.8620  0.8641  0.8661  0.8681  0.8701  0.8721  0.8741  0.8760
1.2  0.8779  0.8798  0.8817  0.8835  0.8853  0.8871  0.8889  0.8907  0.8924  0.8941
1.3  0.8958  0.8975  0.8991  0.9008  0.9024  0.9040  0.9055  0.9071  0.9086  0.9101
1.4  0.9116  0.9130  0.9145  0.9159  0.9173  0.9187  0.9201  0.9214  0.9228  0.9241
1.5  0.9254  0.9267  0.9279  0.9292  0.9304  0.9316  0.9328  0.9339  0.9351  0.9362
1.6  0.9374  0.9385  0.9396  0.9406  0.9417  0.9427  0.9437  0.9448  0.9457  0.9467
1.7  0.9477  0.9486  0.9496  0.9505  0.9514  0.9523  0.9532  0.9540  0.9549  0.9557
1.8  0.9565  0.9573  0.9581  0.9589  0.9597  0.9604  0.9612  0.9619  0.9626  0.9633
1.9  0.9640  0.9647  0.9654  0.9660  0.9667  0.9673  0.9680  0.9686  0.9692  0.9698
2.0  0.9704  0.9709  0.9715  0.9721  0.9726  0.9731  0.9737  0.9742  0.9747  0.9752
2.1  0.9757  0.9762  0.9766  0.9771  0.9776  0.9780  0.9785  0.9789  0.9793  0.9797
2.2  0.9801  0.9805  0.9809  0.9813  0.9817  0.9821  0.9824  0.9828  0.9831  0.9835
2.3  0.9838  0.9842  0.9845  0.9848  0.9851  0.9854  0.9857  0.9860  0.9863  0.9866
2.4  0.9869  0.9871  0.9874  0.9877  0.9879  0.9882  0.9884  0.9887  0.9889  0.9892
2.5  0.9894  0.9896  0.9898  0.9900  0.9903  0.9905  0.9907  0.9909  0.9911  0.9912
2.6  0.9914  0.9916  0.9918  0.9920  0.9921  0.9923  0.9925  0.9926  0.9928  0.9930
2.7  0.9931  0.9933  0.9934  0.9935  0.9937  0.9938  0.9940  0.9941  0.9942  0.9943
2.8  0.9945  0.9946  0.9947  0.9948  0.9949  0.9951  0.9952  0.9953  0.9954  0.9955
2.9  0.9956  0.9957  0.9958  0.9959  0.9960  0.9960  0.9961  0.9962  0.9963  0.9964
3.0  0.9965  0.9965  0.9966  0.9967  0.9968  0.9968  0.9969  0.9970  0.9970  0.9971
3.1  0.9972  0.9972  0.9973  0.9974  0.9974  0.9975  0.9975  0.9976  0.9976  0.9977
3.2  0.9978  0.9978  0.9979  0.9979  0.9979  0.9980  0.9980  0.9981  0.9981  0.9982
3.3  0.9982  0.9983  0.9983  0.9983  0.9984  0.9984  0.9984  0.9985  0.9985  0.9985
3.4  0.9986  0.9986  0.9986  0.9987  0.9987  0.9987  0.9988  0.9988  0.9988  0.9988
``````

## Final

Enjoy!

Ever wonder how Standard Normal and Student-t Distribution z-tables are generated?