16.2 Credible Intervals for Student's t Distribution

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

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

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

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

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

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 $\bar{x}=5.2$ and sample deviation $s=0.3$ , 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 $s/\sqrt{N}$ 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.