Statistics Reference

Z-Table (Standard Normal Distribution Table)

A z-table lists cumulative probabilities for the standard normal distribution: for each z-score, it gives the area under the curve to the left of that value, written P(Z < z). The two tables below cover z from −3.49 to 3.49 in steps of 0.01, which is the full range printed in most statistics textbooks.

How to Read a Z-Table

Every z-score is split into two parts: the row gives the units and tenths digits, and the column gives the hundredths digit. The cell where they meet is the cumulative probability.

Example: look up z = 1.96

Split the z-score: 1.96 = 1.9 (row) + 0.06 (column).
In the positive table, find the row labeled 1.9.
Move across to the column labeled 0.06.
Read the cell: 0.9750. So P(Z < 1.96) = 0.9750, meaning 97.5% of standard normal values fall below 1.96.

Negative z-scores work the same way in the negative table: for z = −1.96, use row −1.9 and column 0.06, which gives 0.0250. In both tables the column digit increases the magnitude of z, so row −1.9 with column 0.06 means z = −1.96, not −1.84.

Negative Z-Table

Cumulative probabilities P(Z < z) for z from −3.49 to −0.00. The row gives the integer and tenths digits; the column adds the hundredths digit to the magnitude. All values are below 0.5 because more than half of the distribution lies above any negative z-score.

Negative z-table: cumulative standard normal probabilities for z from -3.49 to 0
z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
−3.4	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0002
−3.3	0.0005	0.0005	0.0005	0.0004	0.0004	0.0004	0.0004	0.0004	0.0004	0.0003
−3.2	0.0007	0.0007	0.0006	0.0006	0.0006	0.0006	0.0006	0.0005	0.0005	0.0005
−3.1	0.0010	0.0009	0.0009	0.0009	0.0008	0.0008	0.0008	0.0008	0.0007	0.0007
−3.0	0.0013	0.0013	0.0013	0.0012	0.0012	0.0011	0.0011	0.0011	0.0010	0.0010
−2.9	0.0019	0.0018	0.0018	0.0017	0.0016	0.0016	0.0015	0.0015	0.0014	0.0014
−2.8	0.0026	0.0025	0.0024	0.0023	0.0023	0.0022	0.0021	0.0021	0.0020	0.0019
−2.7	0.0035	0.0034	0.0033	0.0032	0.0031	0.0030	0.0029	0.0028	0.0027	0.0026
−2.6	0.0047	0.0045	0.0044	0.0043	0.0041	0.0040	0.0039	0.0038	0.0037	0.0036
−2.5	0.0062	0.0060	0.0059	0.0057	0.0055	0.0054	0.0052	0.0051	0.0049	0.0048
−2.4	0.0082	0.0080	0.0078	0.0075	0.0073	0.0071	0.0069	0.0068	0.0066	0.0064
−2.3	0.0107	0.0104	0.0102	0.0099	0.0096	0.0094	0.0091	0.0089	0.0087	0.0084
−2.2	0.0139	0.0136	0.0132	0.0129	0.0125	0.0122	0.0119	0.0116	0.0113	0.0110
−2.1	0.0179	0.0174	0.0170	0.0166	0.0162	0.0158	0.0154	0.0150	0.0146	0.0143
−2.0	0.0228	0.0222	0.0217	0.0212	0.0207	0.0202	0.0197	0.0192	0.0188	0.0183
−1.9	0.0287	0.0281	0.0274	0.0268	0.0262	0.0256	0.0250	0.0244	0.0239	0.0233
−1.8	0.0359	0.0351	0.0344	0.0336	0.0329	0.0322	0.0314	0.0307	0.0301	0.0294
−1.7	0.0446	0.0436	0.0427	0.0418	0.0409	0.0401	0.0392	0.0384	0.0375	0.0367
−1.6	0.0548	0.0537	0.0526	0.0516	0.0505	0.0495	0.0485	0.0475	0.0465	0.0455
−1.5	0.0668	0.0655	0.0643	0.0630	0.0618	0.0606	0.0594	0.0582	0.0571	0.0559
−1.4	0.0808	0.0793	0.0778	0.0764	0.0749	0.0735	0.0721	0.0708	0.0694	0.0681
−1.3	0.0968	0.0951	0.0934	0.0918	0.0901	0.0885	0.0869	0.0853	0.0838	0.0823
−1.2	0.1151	0.1131	0.1112	0.1093	0.1075	0.1056	0.1038	0.1020	0.1003	0.0985
−1.1	0.1357	0.1335	0.1314	0.1292	0.1271	0.1251	0.1230	0.1210	0.1190	0.1170
−1.0	0.1587	0.1562	0.1539	0.1515	0.1492	0.1469	0.1446	0.1423	0.1401	0.1379
−0.9	0.1841	0.1814	0.1788	0.1762	0.1736	0.1711	0.1685	0.1660	0.1635	0.1611
−0.8	0.2119	0.2090	0.2061	0.2033	0.2005	0.1977	0.1949	0.1922	0.1894	0.1867
−0.7	0.2420	0.2389	0.2358	0.2327	0.2296	0.2266	0.2236	0.2206	0.2177	0.2148
−0.6	0.2743	0.2709	0.2676	0.2643	0.2611	0.2578	0.2546	0.2514	0.2483	0.2451
−0.5	0.3085	0.3050	0.3015	0.2981	0.2946	0.2912	0.2877	0.2843	0.2810	0.2776
−0.4	0.3446	0.3409	0.3372	0.3336	0.3300	0.3264	0.3228	0.3192	0.3156	0.3121
−0.3	0.3821	0.3783	0.3745	0.3707	0.3669	0.3632	0.3594	0.3557	0.3520	0.3483
−0.2	0.4207	0.4168	0.4129	0.4090	0.4052	0.4013	0.3974	0.3936	0.3897	0.3859
−0.1	0.4602	0.4562	0.4522	0.4483	0.4443	0.4404	0.4364	0.4325	0.4286	0.4247
−0.0	0.5000	0.4960	0.4920	0.4880	0.4840	0.4801	0.4761	0.4721	0.4681	0.4641

Positive Z-Table

Cumulative probabilities P(Z < z) for z from 0.00 to 3.49. All values are at least 0.5 because at least half of the distribution lies below any non-negative z-score.

Positive z-table: cumulative standard normal probabilities for z from 0 to 3.49
z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
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

Worked Examples

Left-tail probability: P(Z < 1.5)

Row 1.5, column 0.00 in the positive table gives 0.9332. About 93.3% of standard normal values fall below z = 1.5.

Right-tail probability: P(Z > 1.5)

The table is cumulative from the left, so subtract from 1: P(Z > 1.5) = 1 − 0.9332 = 0.0668. Right-tail values are what hypothesis tests use for upper-tail p-values.

Area between two z-scores: P(−1 < Z < 1)

Subtract the smaller cumulative value from the larger: P(Z < 1) − P(Z < −1) = 0.8413 − 0.1587 = 0.6826, or about 0.6827 before the 4-decimal rounding of each entry. This is the familiar 68% of the empirical rule: roughly 68% of values fall within one standard deviation of the mean.

Common Critical Values

A few z-scores appear constantly in confidence intervals and two-tailed hypothesis tests. These values come from the same distribution as the tables above.

90% confidence

±1.645

two-tailed α = 0.10

95% confidence

±1.960

two-tailed α = 0.05

99% confidence

±2.576

two-tailed α = 0.01

For example, the 95% value follows from the table: a two-tailed α of 0.05 leaves 0.025 in each tail, and the z-score with cumulative probability 0.9750 is 1.96.

When to Use z Instead of t

The z-table applies when a statistic can be treated as standard normal: the population standard deviation is known, the sample size is large, or you are standardizing a single value from a known normal distribution. If you first need to convert a raw value into a z-score, the z-score calculator handles the (x − μ) / σ step.

When the population standard deviation is estimated from a small sample, the sample statistic follows a t-distribution instead, and the wider critical values in the t-table are the right reference. For critical values at any significance level or degrees of freedom without table lookup, use the critical value calculator.

Frequently Asked Questions

What does the value in a z-table represent?

Each cell is the cumulative probability P(Z < z): the area under the standard normal curve from negative infinity up to that z-score. For example, the cell for z = 1.96 reads 0.9750, meaning 97.50% of values in a standard normal distribution fall below 1.96. Equivalently, the value is the percentile rank of the z-score expressed as a decimal.

Is this a left-tail or right-tail z-table?

This is a left-tail (cumulative) table. Every cell gives the area to the left of z. To get a right-tail probability, subtract the table value from 1: P(Z > z) = 1 - P(Z < z). For example, P(Z > 1.5) = 1 - 0.9332 = 0.0668. Some textbooks instead print cumulative-from-mean tables, which show the area between 0 and z; those values equal this table's values minus 0.5 for positive z.

How do I find the area between two z-scores?

Look up the cumulative probability for each z-score and subtract the smaller from the larger: P(a < Z < b) = P(Z < b) - P(Z < a). For example, P(-1 < Z < 1) = 0.8413 - 0.1587 = 0.6826, slightly off the exact value of about 0.6827 only because each table entry is rounded to 4 decimals before subtracting.

Why is a separate negative z-table needed?

A cumulative table needs a row for every z-score, and negative z-scores have their own cumulative probabilities (all below 0.5). The negative table saves you from doing the symmetry arithmetic P(Z < -z) = 1 - P(Z < z) by hand every time. If you only have a positive table, that identity gives the same answer: P(Z < -1.96) = 1 - 0.9750 = 0.0250.

When should I use a t-table instead of a z-table?

Use the z-table when the population standard deviation is known or the sample is large enough for the normal approximation to hold. Use a t-table when you estimate the standard deviation from a small sample, because the extra estimation uncertainty gives the t-distribution heavier tails and larger critical values. As degrees of freedom grow, t critical values converge to the z values, so the two tables agree for large samples.

What if my z-score is beyond the range of the table?

This table covers z from -3.49 to 3.49. Beyond that range the tail probability is smaller than 0.0003, so for most practical work you can treat P(Z < z) as approximately 0 for z below -3.49 and approximately 1 for z above 3.49. If you need more precision in the extreme tails, compute the exact probability with software instead of a printed table.