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How do distribution tables, such as t-table and z-scores, are calculated? For example, the formula of $t$ is as follows:

$t=\frac{m_a-m_b}{\sqrt{\frac{s^2}{n_a}+\frac{s^2}{n_b}}}$

How did they calculate a generic table without means or sample size?

Commissar Vasili Karlovic
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    With numerical integration. – Joaquin San Apr 02 '18 at 22:37
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    If the CDF of a distribution can be expressed in closed form then tables are not needed. For the distributions often tabled (std normal, t, chi-squared, F, etc.) the PDFs can be written in closed form, but not the CDF's. Then numerical integration is used (an impressive feat before modern computers) to make tables. The first example at -this page_ shows one method of numerical integration. // For distn's of some tests (e.g., Wilcoxon, GOF) simulation is used. – BruceET Apr 02 '18 at 23:15

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With the increasing availability of reliable statistical software (commercial software such as SAS, SPSS and Minitab, and freeware such as R and Python), it is becoming increasingly common to use software instead of tables. Often the required numerical integration is done afresh for each instance. Sometimes, key parts of the table are stored in memory and retrieved on command. Here are some examples of probabilities and percentage points from standard normal and t distributions from R statistical software:

pnorm(1.96)  # P(Z < 1.96)
## 0.9750021
qnorm(.975)  # c such that P(Z < c) = 0.975
## 1.959964
pt(0, 7)     # if X ~ T(df=7), find P(X < 0)
## 0.5
qt(.95, 20)  # if X ~ T(df=20), find c such that P(X < c) = .95
## 1.724718
BruceET
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