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	$\pm\mathbf{z}$	Approximately
50.0%	$0.6745 \sigma$
68.0%	$0.9945 \sigma$	$1\sigma$
90.0%	$1.6449 \sigma$
95.0%	$1.9600 \sigma$	$2\sigma$
99.0%	$2.5758 \sigma$
99.8%	$3.0902 \sigma$	$3\sigma$
99.995%	$4.0556 \sigma$	$4\sigma$

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

Given a set of 10 samples with sample mean $\bar{x}=5.2$ and known deviation $\sigma=0.3$ , the best estimate for the mean parameter $\mu$ , representing the true value of the data, is the sample mean, $\hat{\mu}=5.2$ with uncertainty $\sigma/\sqrt{N}$ 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.