Approximate Calculation Method for Noncentral t-Distribution Quantile

Table 3

Corollary results of coefficient k (C = 0.7).

	0.7
n	Exact solution	Method 1	Relative error (%)	Method 2	Relative error (%)	Method 3	Relative error (%)
3	1.056856	0.823382	−22.09	—	—	1.041203	−1.48
4	0.92804	0.770563	−16.97	0.982941	5.92	0.919138	−0.96
5	0.861097	0.742317	−13.79	0.850421	−1.24	0.855142	−0.69
6	0.818649	0.723444	−11.63	0.799625	−2.32	0.814274	−0.53
7	0.788632	0.709445	−10.04	0.769229	−2.46	0.785322	−0.42
8	0.766156	0.698382	−8.85	0.747888	−2.38	0.763447	−0.35
9	0.748413	0.689293	−7.9	0.731641	−2.24	0.74618	−0.3
10	0.73401	0.681582	−7.14	0.718652	−2.09	0.732107	−0.26
20	0.663438	0.639206	−3.65	0.655701	−1.17	0.662707	−0.11
30	0.635261	0.619653	−2.46	0.6302	−0.8	0.634827	−0.07
40	0.619144	0.607677	−1.85	0.615421	−0.6	0.61886	−0.05
Error range			−22.09∼−1.85		−2.46∼5.92		−1.48∼−0.05
	0.8
n	Exact solution	Method 1	Relative error (%)	Method 2	Relative error (%)	Method 3	Relative error (%)
3	1.528369	1.412346	−7.59	—	—	1.507871	−1.34
4	1.343264	1.267934	−5.61	1.458218	8.56	1.333216	−0.75
5	1.251379	1.1953	−4.48	1.248493	−0.23	1.245424	−0.48
6	1.19489	1.150102	−3.75	1.173717	−1.77	1.190894	−0.33
7	1.15592	1.118601	−3.23	1.131398	−2.12	1.153027	−0.25
8	1.1271	1.09512	−2.84	1.102841	−2.15	1.12485	−0.2
9	1.104595	1.076739	−2.52	1.081717	−2.07	1.102872	−0.16
10	1.0865	1.061783	−2.27	1.06519	−1.96	1.085136	−0.13
20	1.00039	0.988779	−1.16	0.988889	−1.15	0.999985	−0.04
30	0.967035	0.959519	−0.78	0.959368	−0.79	0.966858	−0.02
40	0.948261	0.942748	−0.58	0.942525	−0.6	0.948161	−0.01
Error range			−7.59∼−0.58		−2.15∼8.56		−1.34∼−0.01
	0.9
n	Exact solution	Method 1	Relative error (%)	Method 2	Relative error (%)	Method 3	Relative error (%)
3	2.219999	2.162173	−2.6	—	—	2.190339	−1.34
4	1.943907	1.912762	−1.6	2.14235	10.21	1.931962	−0.61
5	1.81224	1.790519	−1.2	1.819984	0.43	1.806181	−0.33
6	1.733163	1.716398	−0.97	1.708671	−1.41	1.729754	−0.2
7	1.679532	1.665911	−0.81	1.647691	−1.9	1.67754	−0.12
8	1.640374	1.628914	−0.7	1.60763	−2	1.63918	−0.07
9	1.610364	1.600302	−0.62	1.578625	−1.97	1.609567	−0.05
10	1.586351	1.577401	−0.56	1.556317	−1.89	1.585873	−0.03
20	1.474504	1.470441	−0.28	1.457608	−1.15	1.474811	0.02
30	1.432615	1.429924	−0.19	1.421102	−0.8	1.432884	0.02
40	1.409306	1.407292	−0.14	1.400619	−0.62	1.409556	0.02
Error range			−2.6∼−0.14		−2∼10.21		−1.34∼0.02
	0.95
n	Exact solution	Method 1	Relative error (%)	Method 2	Relative error (%)	Method 3	Relative error (%)
3	2.8089821	2.769737	−1.4	—	—	2.769095	−1.42
4	2.4530226	2.438936	−0.57	2.717347	10.8	2.437778	−0.62
5	2.2858326	2.278508	−0.32	2.300963	0.66	2.278716	−0.31
6	2.186787	2.181884	−0.22	2.158486	−1.29	2.183009	−0.17
7	2.120093	2.116571	−0.17	2.081324	−1.83	2.118108	−0.09
8	2.071657	2.069042	−0.13	2.031161	−1.95	2.070709	−0.05
9	2.034752	2.032585	−0.11	1.995165	−1.95	2.034297	−0.02
10	2.0052348	2.003434	−0.09	1.967689	−1.87	2.00528	0
20	1.8702006	1.869459	−0.04	1.848538	−1.16	1.870855	0.03
30	1.820288	1.819858	−0.02	1.805492	−0.81	1.820876	0.03
40	1.7927441	1.7925	−0.01	1.781563	−0.62	1.793272	0.03
Error range			−1.4∼−0.01		−1.95∼10.8		−1.42∼0.03