Confidence and Tolerance Intervals

Exact Poisson Confidence Intervals for Counts and Rates

Exact confidence limits for the expected number of events. Divide by the exposure time to get a confidence interval for a rate.

x = 0 to 100 events
90%, 95% and 99%
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Exact Poisson CI

Rows: observed number of events x · Columns: lower and upper confidence limits for the expected count

Exact Poisson Confidence Interval Table
	90% CI		95% CI		99% CI
Observed count (x)	Lower	Upper	Lower	Upper	Lower	Upper
0	0.000	2.996	0.000	3.689	0.000	5.298
1	0.051	4.744	0.025	5.572	0.005	7.430
2	0.355	6.296	0.242	7.225	0.103	9.274
3	0.818	7.754	0.619	8.767	0.338	10.977
4	1.366	9.154	1.090	10.242	0.672	12.594
5	1.970	10.513	1.623	11.668	1.078	14.150
6	2.613	11.842	2.202	13.059	1.537	15.660
7	3.285	13.148	2.814	14.423	2.037	17.134
8	3.981	14.435	3.454	15.763	2.571	18.578
9	4.695	15.705	4.115	17.085	3.132	19.998
10	5.425	16.962	4.795	18.390	3.717	21.398
11	6.169	18.208	5.491	19.682	4.321	22.779
12	6.924	19.443	6.201	20.962	4.943	24.145
13	7.690	20.669	6.922	22.230	5.580	25.497
14	8.464	21.886	7.654	23.490	6.231	26.836
15	9.246	23.097	8.395	24.740	6.893	28.164
16	10.036	24.301	9.145	25.983	7.567	29.482
17	10.832	25.499	9.903	27.219	8.251	30.791
18	11.634	26.692	10.668	28.448	8.943	32.091
19	12.442	27.879	11.439	29.671	9.644	33.383
20	13.255	29.062	12.217	30.888	10.353	34.668
21	14.072	30.240	12.999	32.101	11.069	35.946
22	14.894	31.415	13.787	33.308	11.792	37.218
23	15.719	32.585	14.580	34.511	12.521	38.484
24	16.549	33.752	15.377	35.710	13.255	39.745
25	17.382	34.916	16.179	36.905	13.995	41.000
26	18.219	36.077	16.984	38.096	14.741	42.251
27	19.058	37.234	17.793	39.284	15.491	43.497
28	19.901	38.389	18.606	40.468	16.245	44.738
29	20.746	39.541	19.422	41.649	17.004	45.976
30	21.594	40.691	20.241	42.827	17.767	47.209
31	22.445	41.838	21.063	44.002	18.534	48.439
32	23.297	42.982	21.888	45.174	19.305	49.665
33	24.153	44.125	22.716	46.344	20.079	50.888
34	25.010	45.266	23.546	47.512	20.857	52.107
35	25.870	46.404	24.379	48.677	21.638	53.324
36	26.731	47.541	25.214	49.839	22.422	54.537
37	27.595	48.675	26.051	51.000	23.208	55.748
38	28.460	49.808	26.891	52.158	23.998	56.955
39	29.327	50.940	27.733	53.314	24.791	58.161
40	30.196	52.069	28.577	54.469	25.586	59.363
41	31.066	53.197	29.422	55.621	26.384	60.563
42	31.938	54.324	30.270	56.772	27.184	61.761
43	32.812	55.449	31.119	57.921	27.986	62.956
44	33.687	56.573	31.970	59.068	28.791	64.149
45	34.563	57.695	32.823	60.214	29.598	65.341
46	35.441	58.816	33.678	61.358	30.407	66.530
47	36.320	59.935	34.534	62.500	31.218	67.717
48	37.200	61.054	35.391	63.641	32.032	68.902
49	38.082	62.171	36.250	64.781	32.847	70.085
50	38.965	63.287	37.111	65.919	33.664	71.266
51	39.849	64.402	37.973	67.056	34.483	72.446
52	40.734	65.516	38.836	68.191	35.303	73.624
53	41.620	66.628	39.701	69.325	36.125	74.800
54	42.507	67.740	40.566	70.458	36.949	75.974
55	43.396	68.851	41.434	71.590	37.775	77.147
56	44.285	69.960	42.302	72.721	38.602	78.319
57	45.176	71.069	43.171	73.850	39.431	79.489
58	46.067	72.177	44.042	74.978	40.261	80.657
59	46.959	73.284	44.914	76.106	41.093	81.824
60	47.852	74.390	45.786	77.232	41.926	82.990
61	48.746	75.495	46.660	78.357	42.760	84.154
62	49.641	76.599	47.535	79.481	43.596	85.317
63	50.537	77.702	48.411	80.604	44.433	86.479
64	51.434	78.805	49.288	81.727	45.272	87.639
65	52.331	79.907	50.166	82.848	46.111	88.798
66	53.229	81.008	51.044	83.968	46.952	89.956
67	54.128	82.108	51.924	85.088	47.794	91.113
68	55.028	83.208	52.805	86.206	48.637	92.269
69	55.928	84.306	53.686	87.324	49.482	93.423
70	56.830	85.405	54.568	88.441	50.327	94.577
71	57.732	86.502	55.452	89.557	51.174	95.729
72	58.634	87.599	56.336	90.672	52.022	96.881
73	59.537	88.695	57.220	91.787	52.871	98.031
74	60.441	89.790	58.106	92.900	53.720	99.180
75	61.346	90.885	58.992	94.013	54.571	100.328
76	62.251	91.979	59.879	95.125	55.423	101.476
77	63.157	93.073	60.767	96.237	56.276	102.622
78	64.063	94.166	61.656	97.348	57.129	103.767
79	64.970	95.258	62.545	98.458	57.984	104.912
80	65.878	96.350	63.435	99.567	58.840	106.056
81	66.786	97.441	64.326	100.676	59.696	107.198
82	67.695	98.532	65.217	101.784	60.553	108.340
83	68.604	99.622	66.109	102.891	61.412	109.481
84	69.514	100.712	67.002	103.998	62.271	110.621
85	70.425	101.801	67.895	105.104	63.131	111.761
86	71.336	102.889	68.789	106.209	63.991	112.899
87	72.247	103.977	69.683	107.314	64.853	114.037
88	73.159	105.065	70.579	108.418	65.715	115.174
89	74.071	106.152	71.474	109.522	66.578	116.310
90	74.984	107.239	72.371	110.625	67.442	117.445
91	75.898	108.325	73.268	111.728	68.307	118.580
92	76.812	109.410	74.165	112.830	69.172	119.714
93	77.726	110.495	75.063	113.931	70.038	120.847
94	78.641	111.580	75.962	115.032	70.905	121.980
95	79.556	112.664	76.861	116.133	71.773	123.112
96	80.472	113.748	77.760	117.232	72.641	124.243
97	81.388	114.832	78.660	118.332	73.510	125.373
98	82.305	115.915	79.561	119.431	74.379	126.503
99	83.222	116.997	80.462	120.529	75.250	127.632
100	84.139	118.079	81.364	121.627	76.120	128.761

Tip: click any value to highlight its row and column.

How to read this table

Find the row for the observed number of events x.
Read the lower and upper limits for your confidence level.
For a rate, divide both limits by the total exposure (for example person-years), then multiply by your reporting unit (for example 1,000).

Worked example

A ward records 10 infections over 2,000 patient-days. Row 10 gives a 95% interval for the expected count of 4.795 to 18.390. Dividing by 2,000 and multiplying by 1,000 gives an infection rate of 5.0 per 1,000 patient-days, 95% CI 2.40 to 9.20.

With no events, the upper 95% limit is 3.689, so zero events in 2,000 patient-days is still consistent with a rate of up to 1.84 per 1,000 patient-days.

Click the underlined links to highlight the value in the table.

Calculator

For values between the rows and columns of the table. Results update as you type.

Formula and how the values were computed

Lower = ½ χ²(α/2; 2x) · Upper = ½ χ²(1 − α/2; 2x + 2)

Limits are computed exactly from the chi-square distribution (Garwood's method) and rounded to three decimals.

Frequently asked questions

When should I use an exact Poisson interval?

Whenever event counts are small, for example rare adverse events or infections. For large counts, x ± 1.96√x is a reasonable approximation.

Does this account for overdispersion?

No. The interval assumes events occur independently at a constant rate. If counts vary more than a Poisson model allows, the true interval is wider.

Exact Poisson Confidence Intervals for Counts and Rates

Exact Poisson CI

How to read this table

Worked example

Calculator

Formula and how the values were computed

Frequently asked questions

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