Z Table (Standard Normal Table) and How to Read It

A standard normal (z) table gives the area under the curve to the left of a z-score, which equals the cumulative probability. Find the ones and tenths of z on the row, the hundredths on the column, and read the cell. For example, z = 1.96 gives 0.9750.

AP Statistics: Unit 2 (topics 2.11 The Normal Distribution). The standard normal table supports Unit 2 topic 2.11, The Normal Distribution, in the Fall 2026 AP Statistics course.

What the standard normal table shows

The standard normal table, also called a z table, lists the area under the standard normal curve to the left of a given z-score. A z-score (a standardized value) tells you how many standard deviations a value sits above or below the mean. The standard normal distribution has a mean $\mu$ (the population mean) of 0 and a standard deviation $\sigma$ (the population standard deviation) of 1.

The area to the left is a cumulative probability: the chance of landing at or below that z-score. Because the total area under the curve is 1, every table entry falls between 0 and 1. Written as a probability, an entry is $P(Z \le z)$ .

How to read the z table: row plus column

Split the z-score into two parts. The row gives the ones and tenths place (for example, 1.9), and the column gives the hundredths place (for example, 0.06). Find where that row and column meet, and read the area in the cell.

Walk through $z = 1.96$ . Go to the row labeled 1.9, slide across to the column labeled 0.06, and the cell reads 0.9750. That means about 97.50% of the standard normal curve lies to the left of $z = 1.96$ .

The negative z table

Negative z-scores sit to the left of the mean, so they use a separate negative z table. Reading it works the same way: match the row to the ones and tenths, and the column to the hundredths.

Try $z = -1.28$ . Find the row labeled -1.2 and the column labeled 0.08, and the cell reads 0.1003. About 10.03% of the curve lies to the left of $z = -1.28$ . Notice the symmetry: the area left of $-1.28$ equals the area right of $+1.28$ , since $1 - 0.8997 = 0.1003$ .

Left tail, right tail, and area between

The table always gives a left-tail area, $P(Z \le z)$ . To get a right-tail area, $P(Z \ge z)$ , subtract the table value from 1, because the two areas add to the full area of 1.

P(Z \ge z) = 1 - P(Z \le z)

For the area between two z-scores, look up both and subtract the smaller left-tail area from the larger one.

P(a \le Z \le b) = P(Z \le b) - P(Z \le a)

Finding a percentile from the table

A percentile is the value with a set percent of data at or below it, so it maps directly to a left-tail area. To find a raw value's percentile, convert it to a z-score with $z = \frac{x - \mu}{\sigma}$ , read the left-tail area, then multiply by 100.

To go the other way, start from the target area inside the table, read off the matching z-score, then unstandardize with $x = \mu + z\sigma$ . When the exact area is not printed in the table, pick the closest entry, or split the difference between the two nearest z-scores for a value in between.

z table vs calculator

A graphing calculator returns normal areas directly, so you rarely need a printed table outside of practice. A command such as normalcdf gives the area between two z-scores or two raw values without rounding to two decimals.

The z table still matters for two reasons. It builds intuition for where area sits under the curve, and the AP Statistics exam provides a printed normal table you can fall back on. When the two answers differ slightly, the calculator is more precise because the table rounds each z-score to the hundredths.

Percent below a value (left-tail area)

Exam scores are approximately normal with population mean $\mu = 500$ and population standard deviation $\sigma = 100$ . What proportion of scores fall below 640?

Standardize 640: $z = \frac{640 - 500}{100} = \frac{140}{100} = 1.40$ .
Look up $z = 1.40$ in the positive z table: row 1.4, column 0.00, cell 0.9192.
This entry is already a left-tail area, so no further step is needed.

About 0.9192, so roughly 91.92% of scores fall below 640.

Percent above a value (right-tail area)

Using the same distribution ( $\mu = 500$ , $\sigma = 100$ ), what proportion of scores fall above 640?

Standardize: $z = \frac{640 - 500}{100} = 1.40$ .
The table left-tail area for $z = 1.40$ is 0.9192.
Subtract from 1 for the right tail: $1 - 0.9192 = 0.0808$ .

About 0.0808, so roughly 8.08% of scores fall above 640.

Area between two values

Adult heights are approximately normal with mean $\mu = 68$ inches and standard deviation $\sigma = 3$ inches. What proportion of heights fall between 65 and 71 inches?

Standardize both ends: $z_1 = \frac{65 - 68}{3} = -1.00$ and $z_2 = \frac{71 - 68}{3} = 1.00$ .
Read left-tail areas: $P(Z \le 1.00) = 0.8413$ and $P(Z \le -1.00) = 0.1587$ .
Subtract the smaller from the larger: $0.8413 - 0.1587 = 0.6826$ .

About 0.6826, so roughly 68.26% of heights fall between 65 and 71 inches, matching the empirical rule.

Find the value at a given percentile

For the height distribution ( $\mu = 68$ inches, $\sigma = 3$ inches), what height marks the 75th percentile?

The 75th percentile has a left-tail area of 0.7500, so search the table body for 0.7500.
The closest entries are 0.7486 at $z = 0.67$ and 0.7517 at $z = 0.68$ ; 0.7500 is nearer 0.7486, so use $z = 0.67$ .
Unstandardize: $x = \mu + z\sigma = 68 + 0.67 \times 3 = 68 + 2.01 = 70.01$ .

About 70.0 inches marks the 75th percentile.

Frequently asked questions

Does the z table give the area to the left or the right?

A standard normal table gives the area to the left, $P(Z \le z)$ . For a right-tail area, subtract the table value from 1. Some older tables instead show the area from 0 out to z, so check the small diagram printed above the table.

Why is there a separate negative z table?

Negative z-scores sit below the mean, and their left-tail areas are all less than 0.5. Splitting positive and negative values into two tables keeps each one readable. You can also use symmetry, since the area left of $-z$ equals 1 minus the area left of $+z$ .

What z-score gives an area of 0.9750?

A left-tail area of 0.9750 corresponds to $z = 1.96$ . This is the critical value behind a 95% confidence interval, because the middle 95% leaves 0.025 of the area in each tail.

Is a z table on the AP Statistics formula sheet?

Yes. The Fall 2026 AP Statistics exam provides formulas and tables, including a standard normal table, for both the multiple-choice and free-response sections. A graphing calculator is expected too, and most students find it faster and more precise.

Negative z (z from -3.4 to 0)

Each entry is the area under the standard normal curve to the LEFT of z.

z	.00	.01	.02	.03	.04	.05	.06	.07	.08	.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 (z from 0 to 3.4)

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