16.1 Credible Intervals for Standard Normal Distribution

Standard Normal curve with the central region between z=-1 and z=+1 shaded and annotated '68% CI at μ±1σ' — the picture heading the credible-interval table.

Credible Interval	$\pm 𝐳$	Approximately
50.0%	$0.6745 σ$
68.0%	$0.9945 σ$	$1 σ$
90.0%	$1.6449 σ$
95.0%	$1.9600 σ$	$2 σ$
99.0%	$2.5758 σ$
99.8%	$3.0902 σ$	$3 σ$
99.995%	$4.0556 σ$	$4 σ$

Example 1 Usage of the Credible Interval Table for the Normal Distribution

Given a set of 10 samples with sample mean $x bar equals 5.2$ and known deviation $σ equals 0.3$ , the best estimate for the mean parameter $μ$ , representing the true value of the data, is the sample mean, $μ hat equals 5.2$ with uncertainty $σ / the square root of N$ or $0.3 / \sqrt{10} = 0.095$ . Some of the credible intervals for this estimate then are the following

68% - $[5.2 - 0.9945 \cdot 0.095, 5.2 + 0.9945 \cdot 0.095] = [5.11, 5.29]$
95% - $[5.2 - 1.9600 \cdot 0.095, 5.2 + 1.9600 \cdot 0.095] = [5.01, 5.39]$
99.8% - $[5.2 - 3.0902 \cdot 0.095, 5.2 + 3.0902 \cdot 0.095] = [4.91, 5.49]$

or approximately

68% - $[5.2 - 1 \cdot 0.095, 5.2 + 1 \cdot 0.095] = [5.11, 5.29]$
95% - $[5.2 - 2 \cdot 0.095, 5.2 + 2 \cdot 0.095] = [5.01, 5.39]$
99.8% - $[5.2 - 3 \cdot 0.095, 5.2 + 3 \cdot 0.095] = [4.91, 5.49]$

Adapted from Statistical Inference for Everyone, by Brian Blais (Bryant University), licensed under CC BY-SA 4.0 (dual-licensed under the GNU FDL 1.2 or later; this adaptation uses the CC BY-SA grant). Changes were made; this adaptation is distributed under the same license. License: CC-BY-SA-4.0.