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Statistical tables

The three tables every statistics exam leans on, plus calculators that skip the tables entirely: exact Φ(z) for any z, Student's t and chi-squared probabilities and percentage points for any degrees of freedom, inverse percentiles and two-sided critical values.

Probability from a value

Value xMean μStd dev σ

Φ(z), i.e. P(Z ≤ z)0.97500

1 − Φ(z), i.e. P(Z > z)0.02500

Two-sided P(|Z| > |z|)0.05000

Density φ(z)0.05844

Value from a probability

Probability p

z with Φ(z) = p1.9600

Two-sided |z| for confidence p2.2414

Enter p as a decimal, so 0.975 gives the familiar 1.9600. The two-sided row reads p as a confidence level, so 0.95 gives 1.9600 too.

Shape

Mean μ 0Std dev σ 1

Common critical values

The values that come up in every hypothesis test and confidence interval. One-sided uses Φ(z) = level; two-sided splits the remaining probability across both tails.

Confidence level	One-sided z	Two-sided z
90%	1.2816	1.6449
95%	1.6449	1.9600
97.5%	1.9600	2.2414
99%	2.3263	2.5758
99.5%	2.5758	2.8070

Standard normal table, Φ(z)

P(Z ≤ z) for z from 0.00 to 3.49. Read the row for the first two digits and the column for the second decimal place, so Φ(1.96) sits at row 1.9, column .06. For negative z use symmetry: Φ(−z) = 1 − Φ(z).

z	.00	.01	.02	.03	.04	.05	.06	.07	.08	.09
0.0	0.5000	0.5040	0.5080	0.5120	0.5160	0.5199	0.5239	0.5279	0.5319	0.5359
0.1	0.5398	0.5438	0.5478	0.5517	0.5557	0.5596	0.5636	0.5675	0.5714	0.5753
0.2	0.5793	0.5832	0.5871	0.5910	0.5948	0.5987	0.6026	0.6064	0.6103	0.6141
0.3	0.6179	0.6217	0.6255	0.6293	0.6331	0.6368	0.6406	0.6443	0.6480	0.6517
0.4	0.6554	0.6591	0.6628	0.6664	0.6700	0.6736	0.6772	0.6808	0.6844	0.6879
0.5	0.6915	0.6950	0.6985	0.7019	0.7054	0.7088	0.7123	0.7157	0.7190	0.7224
0.6	0.7257	0.7291	0.7324	0.7357	0.7389	0.7422	0.7454	0.7486	0.7517	0.7549
0.7	0.7580	0.7611	0.7642	0.7673	0.7704	0.7734	0.7764	0.7794	0.7823	0.7852
0.8	0.7881	0.7910	0.7939	0.7967	0.7995	0.8023	0.8051	0.8078	0.8106	0.8133
0.9	0.8159	0.8186	0.8212	0.8238	0.8264	0.8289	0.8315	0.8340	0.8365	0.8389
1.0	0.8413	0.8438	0.8461	0.8485	0.8508	0.8531	0.8554	0.8577	0.8599	0.8621
1.1	0.8643	0.8665	0.8686	0.8708	0.8729	0.8749	0.8770	0.8790	0.8810	0.8830
1.2	0.8849	0.8869	0.8888	0.8907	0.8925	0.8944	0.8962	0.8980	0.8997	0.9015
1.3	0.9032	0.9049	0.9066	0.9082	0.9099	0.9115	0.9131	0.9147	0.9162	0.9177
1.4	0.9192	0.9207	0.9222	0.9236	0.9251	0.9265	0.9279	0.9292	0.9306	0.9319
1.5	0.9332	0.9345	0.9357	0.9370	0.9382	0.9394	0.9406	0.9418	0.9429	0.9441
1.6	0.9452	0.9463	0.9474	0.9484	0.9495	0.9505	0.9515	0.9525	0.9535	0.9545
1.7	0.9554	0.9564	0.9573	0.9582	0.9591	0.9599	0.9608	0.9616	0.9625	0.9633
1.8	0.9641	0.9649	0.9656	0.9664	0.9671	0.9678	0.9686	0.9693	0.9699	0.9706
1.9	0.9713	0.9719	0.9726	0.9732	0.9738	0.9744	0.9750	0.9756	0.9761	0.9767
2.0	0.9772	0.9778	0.9783	0.9788	0.9793	0.9798	0.9803	0.9808	0.9812	0.9817
2.1	0.9821	0.9826	0.9830	0.9834	0.9838	0.9842	0.9846	0.9850	0.9854	0.9857
2.2	0.9861	0.9864	0.9868	0.9871	0.9875	0.9878	0.9881	0.9884	0.9887	0.9890
2.3	0.9893	0.9896	0.9898	0.9901	0.9904	0.9906	0.9909	0.9911	0.9913	0.9916
2.4	0.9918	0.9920	0.9922	0.9925	0.9927	0.9929	0.9931	0.9932	0.9934	0.9936
2.5	0.9938	0.9940	0.9941	0.9943	0.9945	0.9946	0.9948	0.9949	0.9951	0.9952
2.6	0.9953	0.9955	0.9956	0.9957	0.9959	0.9960	0.9961	0.9962	0.9963	0.9964
2.7	0.9965	0.9966	0.9967	0.9968	0.9969	0.9970	0.9971	0.9972	0.9973	0.9974
2.8	0.9974	0.9975	0.9976	0.9977	0.9977	0.9978	0.9979	0.9979	0.9980	0.9981
2.9	0.9981	0.9982	0.9982	0.9983	0.9984	0.9984	0.9985	0.9985	0.9986	0.9986
3.0	0.9987	0.9987	0.9987	0.9988	0.9988	0.9989	0.9989	0.9989	0.9990	0.9990
3.1	0.9990	0.9991	0.9991	0.9991	0.9992	0.9992	0.9992	0.9992	0.9993	0.9993
3.2	0.9993	0.9993	0.9994	0.9994	0.9994	0.9994	0.9994	0.9995	0.9995	0.9995
3.3	0.9995	0.9995	0.9995	0.9996	0.9996	0.9996	0.9996	0.9996	0.9996	0.9997
3.4	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9998

Student's t

Value tDegrees of freedom νProbability p

P(T ≤ t)0.97499

P(T > t)0.02501

Two-sided P(|T| > |t|)0.05001

Density f(t)0.04239

t with P(T ≤ t) = p2.2281

Shape

ν 3

Student's tStandard normal

Slide ν up and watch the heavy tails tuck in; by ν = 30 the two curves are hard to tell apart.

Student's t critical values

The value t with P(T > t) = α for ν degrees of freedom. For a two-sided test at significance level α, use the column for α/2. As ν grows the values approach the normal ones in the bottom row.

ν	α = 0.1	α = 0.05	α = 0.025	α = 0.01	α = 0.005
1	3.078	6.314	12.706	31.821	63.657
2	1.886	2.920	4.303	6.965	9.925
3	1.638	2.353	3.182	4.541	5.841
4	1.533	2.132	2.776	3.747	4.604
5	1.476	2.015	2.571	3.365	4.032
6	1.440	1.943	2.447	3.143	3.707
7	1.415	1.895	2.365	2.998	3.499
8	1.397	1.860	2.306	2.896	3.355
9	1.383	1.833	2.262	2.821	3.250
10	1.372	1.812	2.228	2.764	3.169
12	1.356	1.782	2.179	2.681	3.055
15	1.341	1.753	2.131	2.602	2.947
20	1.325	1.725	2.086	2.528	2.845
25	1.316	1.708	2.060	2.485	2.787
30	1.310	1.697	2.042	2.457	2.750
40	1.303	1.684	2.021	2.423	2.704
60	1.296	1.671	2.000	2.390	2.660
120	1.289	1.658	1.980	2.358	2.617
∞	1.282	1.645	1.960	2.326	2.576

Chi-squared

Value xDegrees of freedom νProbability p

P(X ≤ x)0.95005

P(X > x)0.04995

Density f(x)0.01547

x with P(X ≤ x) = p18.307

Mean ν, variance 2ν. For a goodness-of-fit test at 5% you want the upper point, so enter p = 0.95.

Shape

ν 4

Chi-squared percentage points

The value x with P(X > x) = p for ν degrees of freedom. The left columns give the lower tail (for two-sided variance intervals); the right columns are the usual test critical values.

ν	p = 0.995	p = 0.99	p = 0.975	p = 0.95	p = 0.05	p = 0.025	p = 0.01	p = 0.005
1	0.000	0.000	0.001	0.004	3.841	5.024	6.635	7.879
2	0.010	0.020	0.051	0.103	5.991	7.378	9.210	10.597
3	0.072	0.115	0.216	0.352	7.815	9.348	11.345	12.838
4	0.207	0.297	0.484	0.711	9.488	11.143	13.277	14.860
5	0.412	0.554	0.831	1.145	11.070	12.833	15.086	16.750
6	0.676	0.872	1.237	1.635	12.592	14.449	16.812	18.548
7	0.989	1.239	1.690	2.167	14.067	16.013	18.475	20.278
8	1.344	1.646	2.180	2.733	15.507	17.535	20.090	21.955
9	1.735	2.088	2.700	3.325	16.919	19.023	21.666	23.589
10	2.156	2.558	3.247	3.940	18.307	20.483	23.209	25.188
12	3.074	3.571	4.404	5.226	21.026	23.337	26.217	28.300
15	4.601	5.229	6.262	7.261	24.996	27.488	30.578	32.801
20	7.434	8.260	9.591	10.851	31.410	34.170	37.566	39.997
25	10.520	11.524	13.120	14.611	37.652	40.646	44.314	46.928
30	13.787	14.953	16.791	18.493	43.773	46.979	50.892	53.672
40	20.707	22.164	24.433	26.509	55.758	59.342	63.691	66.766
50	27.991	29.707	32.357	34.764	67.505	71.420	76.154	79.490

F distribution

Value xNumerator ν₁Denominator ν₂Probability p

P(F ≤ x)0.95017

P(F > x)0.04983

Density f(x)0.04042

x with P(F ≤ x) = p3.326

For a variance-ratio test put the larger sample variance on top; the lower percentage point comes free from F with the degrees of freedom swapped: F(p, ν₁, ν₂) = 1 / F(1 − p, ν₂, ν₁).

Shape

ν₁ 5ν₂ 10

F upper 5% points

The value x with P(F > x) = 0.05, for ν₁ degrees of freedom in the numerator (columns) and ν₂ in the denominator (rows).

ν₂ \ ν₁	1	2	3	4	5	6	8	10	12	24
1	161.4	199.5	215.7	224.6	230.2	234.0	238.9	241.9	243.9	249.1
2	18.51	19.00	19.16	19.25	19.30	19.33	19.37	19.40	19.41	19.45
3	10.13	9.55	9.28	9.12	9.01	8.94	8.85	8.79	8.74	8.64
4	7.71	6.94	6.59	6.39	6.26	6.16	6.04	5.96	5.91	5.77
5	6.61	5.79	5.41	5.19	5.05	4.95	4.82	4.74	4.68	4.53
6	5.99	5.14	4.76	4.53	4.39	4.28	4.15	4.06	4.00	3.84
7	5.59	4.74	4.35	4.12	3.97	3.87	3.73	3.64	3.57	3.41
8	5.32	4.46	4.07	3.84	3.69	3.58	3.44	3.35	3.28	3.12
9	5.12	4.26	3.86	3.63	3.48	3.37	3.23	3.14	3.07	2.90
10	4.96	4.10	3.71	3.48	3.33	3.22	3.07	2.98	2.91	2.74
12	4.75	3.89	3.49	3.26	3.11	3.00	2.85	2.75	2.69	2.51
15	4.54	3.68	3.29	3.06	2.90	2.79	2.64	2.54	2.48	2.29
20	4.35	3.49	3.10	2.87	2.71	2.60	2.45	2.35	2.28	2.08
30	4.17	3.32	2.92	2.69	2.53	2.42	2.27	2.16	2.09	1.89
60	4.00	3.15	2.76	2.53	2.37	2.25	2.10	1.99	1.92	1.70

F upper 1% points

The value x with P(F > x) = 0.01, for ν₁ degrees of freedom in the numerator (columns) and ν₂ in the denominator (rows).

ν₂ \ ν₁	1	2	3	4	5	6	8	10	12	24
1	4052.2	4999.5	5403.4	5624.6	5763.6	5859.0	5981.1	6055.8	6106.3	6234.6
2	98.50	99.00	99.17	99.25	99.30	99.33	99.37	99.40	99.42	99.46
3	34.12	30.82	29.46	28.71	28.24	27.91	27.49	27.23	27.05	26.60
4	21.20	18.00	16.69	15.98	15.52	15.21	14.80	14.55	14.37	13.93
5	16.26	13.27	12.06	11.39	10.97	10.67	10.29	10.05	9.89	9.47
6	13.75	10.92	9.78	9.15	8.75	8.47	8.10	7.87	7.72	7.31
7	12.25	9.55	8.45	7.85	7.46	7.19	6.84	6.62	6.47	6.07
8	11.26	8.65	7.59	7.01	6.63	6.37	6.03	5.81	5.67	5.28
9	10.56	8.02	6.99	6.42	6.06	5.80	5.47	5.26	5.11	4.73
10	10.04	7.56	6.55	5.99	5.64	5.39	5.06	4.85	4.71	4.33
12	9.33	6.93	5.95	5.41	5.06	4.82	4.50	4.30	4.16	3.78
15	8.68	6.36	5.42	4.89	4.56	4.32	4.00	3.80	3.67	3.29
20	8.10	5.85	4.94	4.43	4.10	3.87	3.56	3.37	3.23	2.86
30	7.56	5.39	4.51	4.02	3.70	3.47	3.17	2.98	2.84	2.47
60	7.08	4.98	4.13	3.65	3.34	3.12	2.82	2.63	2.50	2.12

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Values are computed from series and continued-fraction expansions accurate to well beyond the digits shown, matching printed exam tables. In an exam, use the tables you are given.