The Probability That a Measurement Falls within a Range of 𝑛 Standard Deviations from an Estimate of the Mean

Table 1

Comparisons of numerical integration of (1/√𝜋)∫^∞_−∞erf(𝑧+(𝑛/√2))𝑒^−𝑧2𝑑𝑧 and erf(𝑛/2).

𝑛	(1/√𝜋)∫^∞−∞erf(𝑧+𝑛/√2)𝑒^−𝑧2𝑑𝑧	erf(𝑛/2)
0.5	0.2763	0.2763
1.5	0.7112	0.7112
2.5	0.9229	0.9229
3.5	0.9867	0.9867
4.5	0.9985	0.9985