16.2 Credible Intervals for Student's t Distribution

	Degrees of Freedom
Credible Interval	1	2	3	4	5	6	7	8
50.0%	$1.000 σ$	$0.816 σ$	$0.765 σ$	$0.741 σ$	$0.727 σ$	$0.718 σ$	$0.711 σ$	$0.706 σ$
68.0%	$1.819 σ$	$1.312 σ$	$1.189 σ$	$1.134 σ$	$1.104 σ$	$1.084 σ$	$1.070 σ$	$1.060 σ$
90.0%	$6.314 σ$	$2.920 σ$	$2.353 σ$	$2.132 σ$	$2.015 σ$	$1.943 σ$	$1.895 σ$	$1.860 σ$
95.0%	$12.706 σ$	$4.303 σ$	$3.182 σ$	$2.776 σ$	$2.571 σ$	$2.447 σ$	$2.365 σ$	$2.306 σ$
99.0%	$63.657 σ$	$9.925 σ$	$5.841 σ$	$4.604 σ$	$4.032 σ$	$3.707 σ$	$3.499 σ$	$3.355 σ$
99.8%	$318.309 σ$	$22.327 σ$	$10.215 σ$	$7.173 σ$	$5.893 σ$	$5.208 σ$	$4.785 σ$	$4.501 σ$
99.995%	$12732.395 σ$	$141.416 σ$	$35.298 σ$	$18.522 σ$	$12.893 σ$	$10.261 σ$	$8.783 σ$	$7.851 σ$

	Degrees of Freedom
Credible Interval	9	10	11	12	13	14	15	16
50.0%	$0.703 σ$	$0.700 σ$	$0.697 σ$	$0.695 σ$	$0.694 σ$	$0.692 σ$	$0.691 σ$	$0.690 σ$
68.0%	$1.053 σ$	$1.046 σ$	$1.041 σ$	$1.037 σ$	$1.034 σ$	$1.031 σ$	$1.029 σ$	$1.026 σ$
90.0%	$1.833 σ$	$1.812 σ$	$1.796 σ$	$1.782 σ$	$1.771 σ$	$1.761 σ$	$1.753 σ$	$1.746 σ$
95.0%	$2.262 σ$	$2.228 σ$	$2.201 σ$	$2.179 σ$	$2.160 σ$	$2.145 σ$	$2.131 σ$	$2.120 σ$
99.0%	$3.250 σ$	$3.169 σ$	$3.106 σ$	$3.055 σ$	$3.012 σ$	$2.977 σ$	$2.947 σ$	$2.921 σ$
99.8%	$4.297 σ$	$4.144 σ$	$4.025 σ$	$3.930 σ$	$3.852 σ$	$3.787 σ$	$3.733 σ$	$3.686 σ$
99.995%	$7.215 σ$	$6.757 σ$	$6.412 σ$	$6.143 σ$	$5.928 σ$	$5.753 σ$	$5.607 σ$	$5.484 σ$

	Degrees of Freedom
Credible Interval	17	18	19	20	21	22	23	24
50.0%	$0.689 σ$	$0.688 σ$	$0.688 σ$	$0.687 σ$	$0.686 σ$	$0.686 σ$	$0.685 σ$	$0.685 σ$
68.0%	$1.024 σ$	$1.023 σ$	$1.021 σ$	$1.020 σ$	$1.019 σ$	$1.017 σ$	$1.016 σ$	$1.015 σ$
90.0%	$1.740 σ$	$1.734 σ$	$1.729 σ$	$1.725 σ$	$1.721 σ$	$1.717 σ$	$1.714 σ$	$1.711 σ$
95.0%	$2.110 σ$	$2.101 σ$	$2.093 σ$	$2.086 σ$	$2.080 σ$	$2.074 σ$	$2.069 σ$	$2.064 σ$
99.0%	$2.898 σ$	$2.878 σ$	$2.861 σ$	$2.845 σ$	$2.831 σ$	$2.819 σ$	$2.807 σ$	$2.797 σ$
99.8%	$3.646 σ$	$3.610 σ$	$3.579 σ$	$3.552 σ$	$3.527 σ$	$3.505 σ$	$3.485 σ$	$3.467 σ$
99.995%	$5.379 σ$	$5.288 σ$	$5.209 σ$	$5.139 σ$	$5.077 σ$	$5.022 σ$	$4.972 σ$	$4.927 σ$

	Degrees of Freedom
Credible Interval	25	26	27	28	29	30	31	32
50.0%	$0.684 σ$	$0.684 σ$	$0.684 σ$	$0.683 σ$	$0.683 σ$	$0.683 σ$	$0.682 σ$	$0.682 σ$
68.0%	$1.015 σ$	$1.014 σ$	$1.013 σ$	$1.012 σ$	$1.012 σ$	$1.011 σ$	$1.011 σ$	$1.010 σ$
90.0%	$1.708 σ$	$1.706 σ$	$1.703 σ$	$1.701 σ$	$1.699 σ$	$1.697 σ$	$1.696 σ$	$1.694 σ$
95.0%	$2.060 σ$	$2.056 σ$	$2.052 σ$	$2.048 σ$	$2.045 σ$	$2.042 σ$	$2.040 σ$	$2.037 σ$
99.0%	$2.787 σ$	$2.779 σ$	$2.771 σ$	$2.763 σ$	$2.756 σ$	$2.750 σ$	$2.744 σ$	$2.738 σ$
99.8%	$3.450 σ$	$3.435 σ$	$3.421 σ$	$3.408 σ$	$3.396 σ$	$3.385 σ$	$3.375 σ$	$3.365 σ$
99.995%	$4.887 σ$	$4.849 σ$	$4.816 σ$	$4.784 σ$	$4.756 σ$	$4.729 σ$	$4.705 σ$	$4.682 σ$

	Degrees of Freedom
Credible Interval	33	34	35	36	37	38	39	40
50.0%	$0.682 σ$	$0.682 σ$	$0.682 σ$	$0.681 σ$	$0.681 σ$	$0.681 σ$	$0.681 σ$	$0.681 σ$
68.0%	$1.010 σ$	$1.009 σ$	$1.009 σ$	$1.008 σ$	$1.008 σ$	$1.008 σ$	$1.007 σ$	$1.007 σ$
90.0%	$1.692 σ$	$1.691 σ$	$1.690 σ$	$1.688 σ$	$1.687 σ$	$1.686 σ$	$1.685 σ$	$1.684 σ$
95.0%	$2.035 σ$	$2.032 σ$	$2.030 σ$	$2.028 σ$	$2.026 σ$	$2.024 σ$	$2.023 σ$	$2.021 σ$
99.0%	$2.733 σ$	$2.728 σ$	$2.724 σ$	$2.719 σ$	$2.715 σ$	$2.712 σ$	$2.708 σ$	$2.704 σ$
99.8%	$3.356 σ$	$3.348 σ$	$3.340 σ$	$3.333 σ$	$3.326 σ$	$3.319 σ$	$3.313 σ$	$3.307 σ$
99.995%	$4.660 σ$	$4.640 σ$	$4.622 σ$	$4.604 σ$	$4.588 σ$	$4.572 σ$	$4.558 σ$	$4.544 σ$

Example 1 Usage of the Credible Interval Table for the Student's $t$ Distribution

Given a set of 10 samples (9 degrees of freedom) with sample mean $x bar equals 5.2$ and sample deviation $s equals 0.3$ , 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 $s / the square root of N$ or $0.3 / \sqrt{10} = 0.095$ . Some of the credible intervals for this estimate then are the following

68% - $[5.2 - 1.053 \cdot 0.095, 5.2 + 1.053 \cdot 0.095] = [5.09, 5.3]$
95% - $[5.2 - 2.262 \cdot 0.095, 5.2 + 2.262 \cdot 0.095] = [4.99, 5.41]$
99.8% - $[5.2 - 4.297 \cdot 0.095, 5.2 + 4.297 \cdot 0.095] = [4.79, 5.61]$

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.