This page provides a sample size table for Pearson's Correlation Coefficient rho (ρ), as calculated using the Sample Size for Pearson's Correlation Coefficient Program Page . The table provides sample size for :

• Power (1-β) of 0.8, 0.9, and 0.95
• Probability of Type I Error (α) of 0.1, 0.05, 0.01, and 0.001
• One or two tail models
• Correlation Coefficient (ρ) at 0.02 intervals

Sample size calculation is based on the algorithm as described by Machin et.al., where the correlation coefficient ρ is firstly transformed to Fisher's Z, and the estimation is based on Z assuming Z is normally distributed.

Fisher's Z Z = log((1+ρ) / (1-ρ)) / 2
Sample Size n = ((z_α+z_β) / Z)² + 3 for one tail model
For 2 tail model the α value is haved. e.g. for p=0.05, z_α for 1 tail=1.64, and for 2 tail=1.96

Power is calculated by reversing the same formula, so that z_β = Z(sqrt(n-3))-z_α, then converting z_β to probability β. Again value of z_α depends on α and whether 1 or 2 tail model

Confidence interval is also calculated by reversing the formula, where

Z = (z_α+z_β) / sqrt(n-3)
SE = sqrt(n-3)
Confidence interval = Z ±z_α(SE)
Z and the confidence intervals are then converted to correlation coefficient where ρ=exp(2Z-1) / exp(2Z+1)
The intervals are not symmetrical, being wider towards 0 and narrower towards 1 and -1

On whether to use a one or two tail model

• If the user intends to establish 95% confidence interval for comparison with other correlation coefficients, or to use the results in a future meta-analysis, then a two tail model is appropriate.
• More commonly, however, the interest is whether a significant correlation exists in the data observed, whether one of the tails overlaps the null value of zero (0). In this case, the one tail model suffices, and requires a smaller sample size.
• In most publications, therefore, when the one or two tail model is not mentioned, the default is usually the one tail model.

Example

To establish an expected correlation coefficient (ρ) of 0.6, significant at the α<0.05 and a power (1-β) of 0.8, requires 16 pairs of x/y values, taking the default position of a one tail model.

	Power(1-β)=0.8								Power(1-β)=0.9								Power(1-β)=0.95
	α=0.1		α=0.05		α=0.01		α=0.001		α=0.1		α=0.05		α=0.01		α=0.001		α=0.1		α=0.05		α=0.01		α=0.001
ρ	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.02	11269	15455	15455	19619	25086	29192	38640	42677	16422	21406	21406	26264	32536	37190	47771	52248	21406	27051	27051	32481	39418	44526	56040	60881
0.04	2817	3862	3862	4903	6268	7294	9654	10663	4104	5349	5349	6563	8129	9292	11935	13053	5349	6759	6759	8116	9849	11124	14001	15210
0.06	1252	1716	1716	2177	2784	3239	4286	4734	1823	2376	2376	2914	3610	4126	5299	5795	2376	3001	3001	3603	4373	4939	6216	6752
0.08	704	964	964	1224	1564	1820	2408	2659	1025	1335	1335	1637	2028	2317	2976	3255	1335	1686	1686	2024	2456	2774	3491	3792
0.10	450	617	617	782	999	1163	1538	1699	655	853	853	1046	1296	1480	1901	2079	853	1078	1078	1293	1569	1772	2230	2422
0.12	313	428	428	542	693	806	1066	1177	454	592	592	725	898	1026	1317	1440	592	747	747	896	1087	1228	1545	1678
0.14	229	314	314	398	508	591	781	862	333	434	434	532	658	752	965	1055	434	547	547	657	797	899	1131	1229
0.16	176	240	240	304	388	451	596	658	255	331	331	406	502	574	736	805	331	418	418	501	608	686	863	938
0.18	139	189	189	240	306	355	469	518	201	261	261	320	396	452	580	634	261	329	329	395	479	540	680	738
0.20	112	153	153	193	247	287	379	418	162	211	211	258	319	365	468	511	211	266	266	319	386	436	548	595
0.22	93	126	126	159	203	236	312	344	134	174	174	213	263	300	385	420	174	219	219	262	318	359	451	489
0.24	78	106	106	134	170	197	261	288	112	145	145	178	220	251	322	351	145	183	183	219	266	300	377	409
0.26	66	90	90	113	144	167	221	244	95	123	123	151	186	213	272	298	123	155	155	186	225	254	319	346
0.28	57	77	77	97	124	144	189	209	82	106	106	130	160	182	233	255	106	133	133	160	193	218	273	297
0.30	50	67	67	84	107	124	164	181	71	92	92	112	138	158	202	221	92	116	116	138	167	188	237	257
0.32	44	59	59	74	94	109	143	158	62	80	80	98	121	138	176	193	80	101	101	121	146	165	206	224
0.34	39	52	52	65	83	96	126	139	55	71	71	86	106	121	155	169	71	89	89	106	128	145	181	197
0.36	34	46	46	58	73	85	111	123	49	63	63	77	94	107	137	150	63	79	79	94	114	128	160	174
0.38	31	41	41	52	65	76	99	109	44	56	56	68	84	96	122	133	56	70	70	84	101	114	143	155
0.40	28	37	37	46	59	68	89	98	39	50	50	61	75	85	109	119	50	63	63	75	90	102	127	138
0.42	25	33	33	42	53	61	80	88	35	45	45	55	68	77	98	107	45	57	57	67	81	91	114	124
0.44	23	30	30	38	48	55	72	79	32	41	41	50	61	69	88	96	41	51	51	61	73	82	103	112
0.46	21	28	28	34	43	50	65	72	29	37	37	45	55	63	80	87	37	46	46	55	66	75	93	101
0.48	19	25	25	31	39	45	59	65	27	34	34	41	50	57	72	79	34	42	42	50	60	68	85	92
0.50	18	23	23	29	36	41	54	59	24	31	31	37	46	52	66	72	31	39	39	46	55	62	77	83
	Power(1-β)=0.8								Power(1-β)=0.9								Power(1-β)=0.95
	α=0.1		α=0.05		α=0.01		α=0.001		α=0.1		α=0.05		α=0.01		α=0.001		α=0.1		α=0.05		α=0.01		α=0.001
ρ	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail	1 tail	2 tail
0.52	16	21	21	26	33	38	49	54	22	28	28	34	42	47	60	66	28	35	35	42	50	56	70	76
0.54	15	20	20	24	30	35	45	49	21	26	26	31	38	43	55	60	26	32	32	38	46	51	64	69
0.56	14	18	18	22	28	32	41	45	19	24	24	29	35	40	50	55	24	30	30	35	42	47	59	63
0.58	13	17	17	21	26	29	38	42	18	22	22	27	32	37	46	50	22	27	27	32	39	43	54	58
0.60	12	16	16	19	24	27	35	38	16	21	21	25	30	34	42	46	21	25	25	30	36	40	49	53
0.62	11	15	15	18	22	25	32	35	15	19	19	23	27	31	39	42	19	23	23	27	33	37	45	49
0.64	11	14	14	16	20	23	30	32	14	18	18	21	25	29	36	39	18	22	22	25	30	34	42	45
0.66	10	13	13	15	19	21	27	30	13	16	16	19	23	26	33	36	16	20	20	23	28	31	38	41
0.68	9	12	12	14	17	20	25	28	12	15	15	18	22	24	31	33	15	19	19	22	26	29	35	38
0.70	9	11	11	13	16	18	23	25	12	14	14	17	20	23	28	31	14	17	17	20	24	26	33	35
0.72	8	10	10	12	15	17	22	23	11	13	13	16	19	21	26	28	13	16	16	19	22	24	30	32
0.74	8	10	10	12	14	16	20	22	10	12	12	14	17	19	24	26	12	15	15	17	20	23	28	30
0.76	8	9	9	11	13	15	18	20	10	12	12	13	16	18	22	24	12	14	14	16	19	21	25	27
0.78	7	9	9	10	12	14	17	18	9	11	11	12	15	16	20	22	11	13	13	15	17	19	23	25
0.80	7	8	8	9	11	13	16	17	8	10	10	12	14	15	19	20	10	12	12	14	16	18	21	23
0.82	6	8	8	9	10	12	14	16	8	9	9	11	13	14	17	18	9	11	11	13	15	16	20	21
0.84	6	7	7	8	10	11	13	14	7	9	9	10	12	13	16	17	9	10	10	12	13	15	18	19
0.86	6	7	7	8	9	10	12	13	7	8	8	9	11	12	14	15	8	9	9	11	12	14	16	17
0.88	6	6	6	7	8	9	11	12	7	8	8	9	10	11	13	14	8	9	9	10	11	12	15	16
0.90	5	6	6	7	8	8	10	11	6	7	7	8	9	10	12	13	7	8	8	9	10	11	13	14
0.92	5	6	6	6	7	8	9	10	6	7	7	7	8	9	11	11	7	7	7	8	9	10	12	13
0.94	5	5	5	6	6	7	8	9	5	6	6	7	7	8	9	10	6	7	7	7	8	9	10	11
0.96	5	5	5	5	6	6	7	8	5	6	6	6	7	7	8	9	6	6	6	7	7	8	9	10
0.98	4	5	5	5	5	5	6	6	5	5	5	5	6	6	7	7	5	5	5	6	6	7	7	8

Machin D, Campbell M, Fayers, P, Pinol A (1997) Sample Size Tables for Clinical Studies. Second Ed. Blackwell Science IBSN 0-86542-870-0 p. 168-172

Altman DG, Machin D, Bryant TN and Gardner MJ. (2000) Statistics with Confidence Second Edition. BMJ Books ISBN 0 7279 1375 1. p. 89-92