Interactive Z-Table Calculator

Calculate probabilities from Z-scores and vice-versa, with interactive visualizations.

The Story of the Z-Score: The Great Equalizer

Imagine you have two friends, Alice and Bob. Alice scored an 85 on her finance exam, and Bob scored a 75 on his. Who did better? It seems obvious, but what if Alice's class average was 80 with a standard deviation of 5, while Bob's class average was only 65 with a standard deviation of 10?

This is where the Z-score comes in. It's a tool that lets us compare apples and oranges by translating any data point from any normal distribution onto a single, universal scale called the Standard Normal Distribution (which has a mean of 0 and a standard deviation of 1).

How It Works: The Formula

Z = (X - μ) / σ

X is the data point you're interested in (e.g., Alice's score of 85).
μ (mu) is the mean (average) of the distribution (Alice's class average of 80).
σ (sigma) is the standard deviation of the distribution (her class's deviation of 5).

Alice's Z-score is (85 - 80) / 5 = +1.0. She is exactly one standard deviation above her class average. Bob's Z-score is (75 - 65) / 10 = +1.0. He is also exactly one standard deviation above his class average.

Suddenly, we see they performed identically relative to their peers! The Z-score tells us how many standard deviations a point is from the mean.

When Can You Use a Z-Score? (The Assumptions)

Known Population Standard Deviation (σ)

A Z-test assumes you know the true standard deviation of the entire population. This is rare in practice, but is often a given in academic problems or with very large, stable historical datasets. If σ is unknown, you should use a T-Test instead, which estimates σ from your sample.

Normality or Large Sample Size

The data should either be drawn from a normally distributed population or the sample size must be large enough (typically n > 30) for the Central Limit Theorem to apply. This theorem guarantees that the distribution of sample means will be approximately normal, even if the original population is not.

The Real Power of the Z-Score: Finding Probabilities

Knowing a Z-score is great, but its real power comes from using it to find probabilities. Once we have a Z-score, we can determine the probability of observing a value that low, that high, or even more extreme.

This probability is called a p-value, and it is the absolute foundation of hypothesis testing. It's the p-value that tells us if an observation (like a stock's return) is statistically significant or just random noise.

What Are You Calculating? A Quick Guide

Let's use a number line analogy to make this crystal clear.

Test Type	The Question It Asks	Visually on the Curve
Right Tail	"What is the chance of being greater than this?" (> Z)	The area of the far right tail.
Left Tail	"What is the chance of being less than this?" (< Z)	The area of the far left tail.
Two-Tailed	"What is the chance of being this extreme or more, in either direction?"	The area of BOTH tails combined.

The area "in between" (what you use for confidence intervals) is the opposite of the two-tailed test. It represents the probability of a result being "normal" or "not extreme."

The Interactive Sandbox: Z-Score Calculator

Remember Alice's Z-score was +1.0? To find the percentage of students who scored worse than her, use a Z-score of 1.0 and select 'Left Tail'. Try it now!

Select Area

Enter Z-Score

Calculated Probability (P-Value)

---

Z-Scores in Finance: From Noise to Signal

For a trader, the biggest challenge is separating random market "noise" from a meaningful event. The Z-score is a perfect tool for this, acting like a filter that highlights only the most statistically important moments.

Standard Normal (Z) Table

Area under the curve to the left of Z. The table highlights the value closest to your calculated Z-score.

Z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
-3.9	0.0000	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001
-3.8	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001
-3.7	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0001	0.0002
-3.6	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002
-3.5	0.0002	0.0002	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003	0.0003
-3.4	0.0003	0.0003	0.0004	0.0004	0.0004	0.0004	0.0004	0.0004	0.0005	0.0005
-3.3	0.0005	0.0005	0.0005	0.0005	0.0006	0.0006	0.0006	0.0006	0.0006	0.0007
-3.2	0.0007	0.0007	0.0007	0.0008	0.0008	0.0008	0.0008	0.0009	0.0009	0.0009
-3.1	0.0010	0.0010	0.0010	0.0011	0.0011	0.0011	0.0012	0.0012	0.0013	0.0013
-3.0	0.0013	0.0014	0.0014	0.0015	0.0015	0.0016	0.0016	0.0017	0.0018	0.0018
-2.9	0.0019	0.0019	0.0020	0.0021	0.0021	0.0022	0.0023	0.0023	0.0024	0.0025
-2.8	0.0026	0.0026	0.0027	0.0028	0.0029	0.0030	0.0031	0.0032	0.0033	0.0034
-2.7	0.0035	0.0036	0.0037	0.0038	0.0039	0.0040	0.0041	0.0043	0.0044	0.0045
-2.6	0.0047	0.0048	0.0049	0.0051	0.0052	0.0054	0.0055	0.0057	0.0059	0.0060
-2.5	0.0062	0.0064	0.0066	0.0068	0.0069	0.0071	0.0073	0.0075	0.0078	0.0080
-2.4	0.0082	0.0084	0.0087	0.0089	0.0091	0.0094	0.0096	0.0099	0.0102	0.0104
-2.3	0.0107	0.0110	0.0113	0.0116	0.0119	0.0122	0.0125	0.0129	0.0132	0.0136
-2.2	0.0139	0.0143	0.0146	0.0150	0.0154	0.0158	0.0162	0.0166	0.0170	0.0174
-2.1	0.0179	0.0183	0.0188	0.0192	0.0197	0.0202	0.0207	0.0212	0.0217	0.0222
-2.0	0.0228	0.0233	0.0239	0.0244	0.0250	0.0256	0.0262	0.0268	0.0274	0.0281
-1.9	0.0287	0.0294	0.0301	0.0307	0.0314	0.0322	0.0329	0.0336	0.0344	0.0351
-1.8	0.0359	0.0367	0.0375	0.0384	0.0392	0.0401	0.0409	0.0418	0.0427	0.0436
-1.7	0.0446	0.0455	0.0465	0.0475	0.0485	0.0495	0.0505	0.0516	0.0526	0.0537
-1.6	0.0548	0.0559	0.0571	0.0582	0.0594	0.0606	0.0618	0.0630	0.0643	0.0655
-1.5	0.0668	0.0681	0.0694	0.0708	0.0721	0.0735	0.0749	0.0764	0.0778	0.0793
-1.4	0.0808	0.0823	0.0838	0.0853	0.0869	0.0885	0.0901	0.0918	0.0934	0.0951
-1.3	0.0968	0.0985	0.1003	0.1020	0.1038	0.1056	0.1075	0.1093	0.1112	0.1131
-1.2	0.1151	0.1170	0.1190	0.1210	0.1230	0.1251	0.1271	0.1292	0.1314	0.1335
-1.1	0.1357	0.1379	0.1401	0.1423	0.1446	0.1469	0.1492	0.1515	0.1539	0.1562
-1.0	0.1587	0.1611	0.1635	0.1660	0.1685	0.1711	0.1736	0.1762	0.1788	0.1814
-0.9	0.1841	0.1867	0.1894	0.1922	0.1949	0.1977	0.2005	0.2033	0.2061	0.2090
-0.8	0.2119	0.2148	0.2177	0.2206	0.2236	0.2266	0.2296	0.2327	0.2358	0.2389
-0.7	0.2420	0.2451	0.2483	0.2514	0.2546	0.2578	0.2611	0.2643	0.2676	0.2709
-0.6	0.2743	0.2776	0.2810	0.2843	0.2877	0.2912	0.2946	0.2981	0.3015	0.3050
-0.5	0.3085	0.3121	0.3156	0.3192	0.3228	0.3264	0.3300	0.3336	0.3372	0.3409
-0.4	0.3446	0.3483	0.3520	0.3557	0.3594	0.3632	0.3669	0.3707	0.3745	0.3783
-0.3	0.3821	0.3859	0.3897	0.3936	0.3974	0.4013	0.4052	0.4090	0.4129	0.4168
-0.2	0.4207	0.4247	0.4286	0.4325	0.4364	0.4404	0.4443	0.4483	0.4522	0.4562
-0.1	0.4602	0.4641	0.4681	0.4721	0.4761	0.4801	0.4840	0.4880	0.4920	0.4960
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
3.5	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998
3.6	0.9998	0.9998	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999
3.7	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999
3.8	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999

Test Type

The Question It Asks

Visually on the Curve

Right Tail

"What is the chance of being greater than this?" (> Z)

The area of the far right tail.

Left Tail

"What is the chance of being less than this?" (< Z)

The area of the far left tail.

Two-Tailed

"What is the chance of being this extreme or more, in either direction?"

The area of BOTH tails combined.

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

-3.9

0.0000

0.0001

-3.8

0.0001

-3.7

0.0001

0.0002

-3.6

0.0002

-3.5

0.0002

0.0003

-3.4

0.0003

0.0004

0.0005

-3.3

0.0005

0.0006

0.0007

-3.2

0.0007

0.0008

0.0009

-3.1

0.0010

0.0011

0.0012

0.0013

-3.0

0.0013

0.0014

0.0015

0.0016

0.0017

0.0018

-2.9

0.0019

0.0020

0.0021

0.0022

0.0023

0.0024

0.0025

-2.8

0.0026

0.0027

0.0028

0.0029

0.0030

0.0031

0.0032

0.0033

0.0034

-2.7

0.0035

0.0036

0.0037

0.0038

0.0039

0.0040

0.0041

0.0043

0.0044

0.0045

-2.6

0.0047

0.0048

0.0049

0.0051

0.0052

0.0054

0.0055

0.0057

0.0059

0.0060

-2.5

0.0062

0.0064

0.0066

0.0068

0.0069

0.0071

0.0073

0.0075

0.0078

0.0080

-2.4

0.0082

0.0084

0.0087

0.0089

0.0091

0.0094

0.0096

0.0099

0.0102

0.0104

-2.3

0.0107

0.0110

0.0113

0.0116

0.0119

0.0122

0.0125

0.0129

0.0132

0.0136

-2.2

0.0139

0.0143

0.0146

0.0150

0.0154

0.0158

0.0162

0.0166

0.0170

0.0174

-2.1

0.0179

0.0183

0.0188

0.0192

0.0197

0.0202

0.0207

0.0212

0.0217

0.0222

-2.0

0.0228

0.0233

0.0239

0.0244

0.0250

0.0256

0.0262

0.0268

0.0274

0.0281

-1.9

0.0287

0.0294

0.0301

0.0307

0.0314

0.0322

0.0329

0.0336

0.0344

0.0351

-1.8

0.0359

0.0367

0.0375

0.0384

0.0392

0.0401

0.0409

0.0418

0.0427

0.0436

-1.7

0.0446

0.0455

0.0465

0.0475

0.0485

0.0495

0.0505

0.0516

0.0526

0.0537

-1.6

0.0548

0.0559

0.0571

0.0582

0.0594

0.0606

0.0618

0.0630

0.0643

0.0655

-1.5

0.0668

0.0681

0.0694

0.0708

0.0721

0.0735

0.0749

0.0764

0.0778

0.0793

-1.4

0.0808

0.0823

0.0838

0.0853

0.0869

0.0885

0.0901

0.0918

0.0934

0.0951

-1.3

0.0968

0.0985

0.1003

0.1020

0.1038

0.1056

0.1075

0.1093

0.1112

0.1131

-1.2

0.1151

0.1170

0.1190

0.1210

0.1230

0.1251

0.1271

0.1292

0.1314

0.1335

-1.1

0.1357

0.1379

0.1401

0.1423

0.1446

0.1469

0.1492

0.1515

0.1539

0.1562

-1.0

0.1587

0.1611

0.1635

0.1660

0.1685

0.1711

0.1736

0.1762

0.1788

0.1814

-0.9

0.1841

0.1867

0.1894

0.1922

0.1949

0.1977

0.2005

0.2033

0.2061

0.2090

-0.8

0.2119

0.2148

0.2177

0.2206

0.2236

0.2266

0.2296

0.2327

0.2358

0.2389

-0.7

0.2420

0.2451

0.2483

0.2514

0.2546

0.2578

0.2611

0.2643

0.2676

0.2709

-0.6

0.2743

0.2776

0.2810

0.2843

0.2877

0.2912

0.2946

0.2981

0.3015

0.3050

-0.5

0.3085

0.3121

0.3156

0.3192

0.3228

0.3264

0.3300

0.3336

0.3372

0.3409

-0.4

0.3446

0.3483

0.3520

0.3557

0.3594

0.3632

0.3669

0.3707

0.3745

0.3783

-0.3

0.3821

0.3859

0.3897

0.3936

0.3974

0.4013

0.4052

0.4090

0.4129

0.4168

-0.2

0.4207

0.4247

0.4286

0.4325

0.4364

0.4404

0.4443

0.4483

0.4522

0.4562

-0.1

0.4602

0.4641

0.4681

0.4721

0.4761

0.4801

0.4840

0.4880

0.4920

0.4960

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

0.9978

0.9979

0.9980

0.9981

2.9

0.9981

0.9982

0.9983

0.9984

0.9985

0.9986

3.0

0.9987

0.9988

0.9989

0.9990

3.1

0.9990

0.9991

0.9992

0.9993

3.2

0.9993

0.9994

0.9995

3.3

0.9995

0.9996

0.9997

3.4

0.9997

0.9998

3.5

0.9998

3.6

0.9998

0.9999

3.7

0.9999

3.8

0.9999

Interactive Z-Table Calculator

Known Population Standard Deviation (σ)

Normality or Large Sample Size

Example 1: Spotting a Truly Unusual Trading Day

Example 2: Finding "Overstretched" Stocks for Mean Reversion

Interactive Z-Table Calculator

Known Population Standard Deviation (σ)

Normality or Large Sample Size

Example 1: Spotting a Truly Unusual Trading Day

Example 2: Finding "Overstretched" Stocks for Mean Reversion