Z-Score Calculator

A value's z-score, percentile and p-value, the probability between two values, and z from a probability, on a shaded bell curve.

Value (x)

Mean (μ)

Standard deviation (σ)

Decimal places

Across: the value, ticked at the mean and each standard deviation from it; up: the probability density. Shaded: the area below 85.

z-score of 85

1.5

Below 85 (its percentile): 93.32%
Above 85: 6.68%
Within 15 of the mean: 86.64%
Two-tailed p-value: 13.36%

Show working

Enter a value with the mean and standard deviation of its distribution to get its z-score, the share of values below and above it, and the two-tailed p-value. The same page finds the probability between two values, the z (and the values) for a probability, such as the critical value for 95% confidence, and the z-score of every value in a data set.

Reading a z-score

A z-score is how many standard deviations a value lies from the mean: z = 1.5 is one and a half above it, z = −2 two below. It puts values on different scales side by side, such as marks from two tests with different averages and spreads.

For data that follows a normal distribution, the z-score also gives a probability. About 68.3% of values lie within one standard deviation of the mean, 95.4% within two and 99.7% within three: the 68-95-99.7 rule. A z-score beyond ±3 is rare, and a value that far out is worth checking. For data that is not normal, such as incomes, the z-score still measures distance from the mean, but the percentages do not apply.

How to use it

Z-score takes a value, the mean and the standard deviation. Between two values takes two values and gives the share between them. From a probability takes a percentage and where it lies: Below z, Above z, Between −z and z, Beyond ±z; 90%, 95% and 99% are a click away. From data takes a list of numbers and gives each its z-score, using the sample or population standard deviation, with the first 50 listed and all of them copied.

A mean of 0 and a standard deviation of 1 make the distribution the standard normal, where a value is its own z-score; one button sets them. Enter a z-score as the value to get its percentile and p-value. The curve is ticked at the mean and at whole standard deviations from it. Probabilities are shown as percentages, rounded to the decimal places chosen (two to start). The mode, the region, the sample or population choice and the places are remembered on this device. Everything runs in your browser.

How it is worked out

The z-score is the distance from the mean in standard deviations. The area below z under the standard normal curve, written Φ(z), is the share of values below. The tool computes Φ directly with W. J. Cody's rational approximations, as R does, rather than looking it up in a table rounded to four places, and works each tail from its own side. A tail as small as 10⁻¹⁶ keeps its digits, and a share that would round to 100% is written as 100% less what it falls short by.

The inverse, z for a given area, uses Wichura's algorithm AS 241. The working shows z rounded to the decimal places chosen, while Φ is computed from z unrounded, so a share can differ in its last digit from a z-table looked up at the rounded z.

z = (x − μ) ÷ σ

x = μ + z × σ

P(X < x) = Φ(z); P(X > x) = 1 − Φ(z)

P(a < X < b) = Φ(z_b) − Φ(z_a)

Two-tailed p-value = 2 × (1 − Φ(|z|))

Examples

An IQ of 130

IQ scores have a mean of 100 and a standard deviation of 15. A score of 130 has a z-score of 2, and 97.72% of people score lower.

Subtract the mean and divide by the standard deviation. z = (130 − 100) ÷ 15 = 2
The area below z under the standard normal curve is Φ(z). P(X < 130) = Φ(2) ≈ 97.72%
The area above is the rest. P(X > 130) = 1 − Φ(2) ≈ 2.28%
Both tails beyond 30 from the mean, the two-tailed p-value: 2 × (1 − Φ(2)) ≈ 4.55%

The share of values between two marks

With a mean of 70 and a standard deviation of 10, the share between 60 and 80, one standard deviation either side, is ≈ 68.27%; outside them lies 31.73%.

The critical value for 95% confidence

For a 95% confidence interval, 95% lies between −z and z, so Between −z and z with 95% gives ≈ ±1.96, with 2.5% in each tail.

What lies outside is split equally between the two tails. (100% − 95%) ÷ 2 = 2.5%
z is the point with one tail above it. z = −Φ⁻¹(0.025) ≈ 1.96

The z-table

Area below z for negative z: the row gives z to one place, the column its second decimal

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

Area below z for positive z: the row gives z to one place, the column its second decimal

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

Frequently asked questions

z ≈ 1.28: 90% of a normal distribution lies below it. The 1.645 sometimes given for it belongs to the 95th percentile (z ≈ 1.644853627), which is also the one-tailed critical value at 5%. Use From a probability with Below z for any percentile.

The left tail is the share below a value, which is its percentile. The right tail is the share above, the p-value of a one-sided test for a high value. Two tails add the share beyond the same distance on the other side, the p-value of a two-sided test, which asks only whether a value is unusually far from the mean in either direction.

For a value from a known distribution, such as IQ, use the distribution's own figures. From your own data, use the sample standard deviation unless the data is the whole group; the Standard Deviation Calculator shows both.