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I have encoutnered this formula used to calcualte cumulative probabilty based on the number of samples:

$$fi = \frac{i-3/8}{n+1/4}$$

some website uses the decimal form: $$fi = \frac{i-0.375}{n+0.25}$$

My question is how does one derive this formula?

One reference is here https://brownmath.com/ti83/nchk89.htm#Theory

user97662
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    Can you add some references, where you found this? – user2974951 Oct 21 '21 at 05:47
  • Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Oct 21 '21 at 05:55
  • I've seen those formulas in [plotting positions](https://en.wikipedia.org/wiki/Q%E2%80%93Q_plot#Heuristics) in a Q-Q-plot. – COOLSerdash Oct 21 '21 at 07:14
  • It is based empirically on finding *plotting points* for a normal QQ plot that tend to work well, even at the extremes, for genuinely Normal data. – whuber Oct 21 '21 at 16:19
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    A reference for the formula $\frac{i-\alpha}{n+1-2\alpha}$ (here with $\alpha=\frac38$) is Blom, G. (1958), Statistical estimates and transformed beta variables, New York: John Wiley and Sons – Glen_b Oct 22 '21 at 00:19
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    There's some excellent answers on a related question here: https://stats.stackexchange.com/questions/9001/approximate-order-statistics-for-normal-random-variables (and indeed on a number of other questions on site). The Blom formula (not always with the same $\alpha$) is also used for other distributions but that discussion of the normal covers the main considerations. – Glen_b Oct 22 '21 at 00:31

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