Use this calculator to compute the z-score of a normal distribution.

Raw Score, x

Population Mean, μ

Standard Deviation, σ

Z-Table: Area from Mean (0 to Z)

This table shows the area under the standard normal curve from the mean (z = 0) to any positive z-score. For negative z-scores, use the same value due to symmetry.

z	.00	.01	.02	.03	.04	.05	.06	.07	.08	.09
0.0	0.0000	0.0040	0.0080	0.0120	0.0160	0.0199	0.0239	0.0279	0.0319	0.0359
0.1	0.0398	0.0438	0.0478	0.0517	0.0557	0.0596	0.0636	0.0675	0.0714	0.0753
0.2	0.0793	0.0832	0.0871	0.0910	0.0948	0.0987	0.1026	0.1064	0.1103	0.1141
0.3	0.1179	0.1217	0.1255	0.1293	0.1331	0.1368	0.1406	0.1443	0.1480	0.1517
0.4	0.1554	0.1591	0.1628	0.1664	0.1700	0.1736	0.1772	0.1808	0.1844	0.1879
0.5	0.1915	0.1950	0.1985	0.2019	0.2054	0.2088	0.2123	0.2157	0.2190	0.2224
0.6	0.2257	0.2291	0.2324	0.2357	0.2389	0.2422	0.2454	0.2486	0.2517	0.2549
0.7	0.2580	0.2611	0.2642	0.2673	0.2704	0.2734	0.2764	0.2794	0.2823	0.2852
0.8	0.2881	0.2910	0.2939	0.2967	0.2995	0.3023	0.3051	0.3078	0.3106	0.3133
0.9	0.3159	0.3186	0.3212	0.3238	0.3264	0.3289	0.3315	0.3340	0.3365	0.3389
1.0	0.3413	0.3438	0.3461	0.3485	0.3508	0.3531	0.3554	0.3577	0.3599	0.3621
1.1	0.3643	0.3665	0.3686	0.3708	0.3729	0.3749	0.3770	0.3790	0.3810	0.3830
1.2	0.3849	0.3869	0.3888	0.3907	0.3925	0.3944	0.3962	0.3980	0.3997	0.4015
1.3	0.4032	0.4049	0.4066	0.4082	0.4099	0.4115	0.4131	0.4147	0.4162	0.4177
1.4	0.4192	0.4207	0.4222	0.4236	0.4251	0.4265	0.4279	0.4292	0.4306	0.4319
1.5	0.4332	0.4345	0.4357	0.4370	0.4382	0.4394	0.4406	0.4418	0.4429	0.4441
1.6	0.4452	0.4463	0.4474	0.4484	0.4495	0.4505	0.4515	0.4525	0.4535	0.4545
1.7	0.4554	0.4564	0.4573	0.4582	0.4591	0.4599	0.4608	0.4616	0.4625	0.4633
1.8	0.4641	0.4649	0.4656	0.4664	0.4671	0.4678	0.4686	0.4693	0.4699	0.4706
1.9	0.4713	0.4719	0.4726	0.4732	0.4738	0.4744	0.4750	0.4756	0.4761	0.4767
2.0	0.4772	0.4778	0.4783	0.4788	0.4793	0.4798	0.4803	0.4808	0.4812	0.4817
2.1	0.4821	0.4826	0.4830	0.4834	0.4838	0.4842	0.4846	0.4850	0.4854	0.4857
2.2	0.4861	0.4864	0.4868	0.4871	0.4875	0.4878	0.4881	0.4884	0.4887	0.4890
2.3	0.4893	0.4896	0.4898	0.4901	0.4904	0.4906	0.4909	0.4911	0.4913	0.4916
2.4	0.4918	0.4920	0.4922	0.4925	0.4927	0.4929	0.4931	0.4932	0.4934	0.4936
2.5	0.4938	0.4940	0.4941	0.4943	0.4945	0.4946	0.4948	0.4949	0.4951	0.4952
2.6	0.4953	0.4955	0.4956	0.4957	0.4959	0.4960	0.4961	0.4962	0.4963	0.4964
2.7	0.4965	0.4966	0.4967	0.4968	0.4969	0.4970	0.4971	0.4972	0.4973	0.4974
2.8	0.4974	0.4975	0.4976	0.4977	0.4977	0.4978	0.4979	0.4979	0.4980	0.4981
2.9	0.4981	0.4982	0.4982	0.4983	0.4984	0.4984	0.4985	0.4985	0.4986	0.4986
3.0	0.4987	0.4987	0.4987	0.4988	0.4988	0.4989	0.4989	0.4989	0.4990	0.4990

How to Use This Z-Table

Step 1: Find the row corresponding to the first decimal place of your z-score (e.g., for z = 1.96, find row 1.9)

Step 2: Find the column for the second decimal place (e.g., for z = 1.96, find column .06)

Step 3: The intersection shows the area from mean (z = 0) to your z-score

Example: For z = 1.96, find row 1.9 and column .06. The value 0.4750 means 47.50% of the data falls between the mean and z = 1.96.

Common Z-scores

z = 1.00: 0.3413 (34.13%)
z = 1.645: 0.4500 (45.00%) - 90% CI
z = 1.96: 0.4750 (47.50%) - 95% CI
z = 2.00: 0.4772 (47.72%)
z = 2.576: 0.4950 (49.50%) - 99% CI
z = 3.00: 0.4987 (49.87%)

Converting to Other Probabilities

P(x < z): Add 0.5 to table value
P(x > z): Subtract table value from 0.5
P(-z < x < z): Multiply table value by 2
For negative z: Use absolute value (symmetry)
Example: For z = 1.96, table shows 0.4750. So P(x < 1.96) = 0.5 + 0.4750 = 0.9750 (97.50%)

Understanding Z-scores and Normal Distribution

What is a Z-score?

A z-score (also called a standard score) is a statistical measurement that describes how many standard deviations a value is from the mean of a distribution. Z-scores are fundamental in statistics because they allow us to compare values from different normal distributions and calculate probabilities. By converting raw scores to z-scores, we standardize data to a common scale with a mean of 0 and standard deviation of 1, making comparisons meaningful across different datasets.

The z-score tells us whether a value is typical for a dataset or unusual. A z-score of 0 indicates the value is exactly at the mean, while positive z-scores indicate values above the mean and negative z-scores indicate values below the mean. Z-scores are essential in hypothesis testing, quality control, standardized testing, and any field that uses the normal distribution.

The Z-score Formula

Basic Z-score Formula

z = (x - μ) / σ

Where:

z = the z-score (standard score)
x = the raw score (individual data point)
μ (mu) = the population mean
σ (sigma) = the population standard deviation

Example Calculation

If a student scores 85 on a test where the mean is 75 and standard deviation is 10: z = (85 - 75) / 10 = 1.0. This means the student scored 1 standard deviation above the mean.

Interpreting Z-scores

z ≈ 0: The value is at or very close to the mean (typical/average)

z = ±1: The value is 1 standard deviation from the mean (fairly common, ~68% of data falls within ±1σ)

z = ±2: The value is 2 standard deviations from the mean (uncommon, ~95% of data falls within ±2σ)

z = ±3: The value is 3 standard deviations from the mean (rare, ~99.7% of data falls within ±3σ)

|z| > 3: The value is extremely unusual or may be an outlier

The Normal Distribution and Empirical Rule

The 68-95-99.7 Rule

For normally distributed data, specific percentages of values fall within certain standard deviations of the mean:

68.27% of data falls within ±1 standard deviation (z = -1 to z = +1)
95.45% of data falls within ±2 standard deviations (z = -2 to z = +2)
99.73% of data falls within ±3 standard deviations (z = -3 to z = +3)

This rule is fundamental in quality control, hypothesis testing, and determining confidence intervals. It helps identify outliers and assess whether observations are typical or unusual.

Understanding Probability Values

P(x < Z) - Cumulative Probability

The probability that a randomly selected value is less than the given z-score. This is the area under the normal curve to the left of Z. For example, P(x < 1.96) ≈ 0.975 or 97.5%.

P(x > Z) - Upper Tail Probability

The probability that a value exceeds the z-score. This equals 1 - P(x < Z). Useful for finding "at least" probabilities.

P(0 to Z) - Distance from Mean

The probability between the mean (z=0) and the given z-score. This shows what percentage of data falls in this specific region.

P(-Z < x < Z) - Symmetric Interval

The probability within Z standard deviations on both sides of the mean. Used for confidence intervals (e.g., z = 1.96 gives 95% confidence interval).

Real-World Applications

Education & Testing

Standardizing test scores (SAT, GRE, IQ tests)
Comparing student performance across different tests
Identifying gifted students or those needing support
Setting grade curves and percentile rankings

Business & Finance

Risk assessment and portfolio management
Detecting fraudulent transactions (outliers)
Quality control in manufacturing processes
Sales performance evaluation

Healthcare & Medicine

Growth charts for height and weight percentiles
Laboratory test result interpretation
Clinical trial analysis and hypothesis testing
Identifying abnormal vital signs or lab values

Research & Science

Hypothesis testing and p-values
Confidence interval calculation
Identifying significant differences in experiments
Data normalization and standardization

Z-Tables vs. Calculator

Traditionally, statisticians used printed z-tables to look up probabilities for given z-scores. These tables show the cumulative probability P(x < Z) for standard normal distribution values.

Advantages of this calculator over z-tables:

Instant calculations for any z-score value (not limited to table entries)
Higher precision (6+ decimal places vs. 4 in most tables)
Multiple probability values calculated simultaneously
Bidirectional conversion (z-score to probability and vice versa)
No interpolation needed for values between table entries

Common Mistakes to Avoid

Confusing Population vs. Sample Statistics: This calculator uses population parameters (μ, σ). For sample data, use sample mean (x̄) and sample standard deviation (s) as estimates.
Assuming All Data is Normally Distributed: Z-scores are most meaningful when data follows a normal distribution. Check your data distribution first using histograms or normality tests.
Misinterpreting Negative Z-scores: A negative z-score doesn't mean "bad" - it simply means below the mean. Context matters!
Ignoring Outliers: Extreme z-scores (|z| > 3) may indicate data entry errors, measurement problems, or genuine outliers that need investigation.
Wrong Standard Deviation: Using the wrong σ will give incorrect z-scores. Ensure you're using the appropriate standard deviation for your population or sample.
Forgetting Direction: When calculating P(x > Z), remember this is the upper tail, not the entire distribution.

Practical Example Walkthrough

Scenario: SAT Scores

The SAT has a mean of 1050 and standard deviation of 200. A student scores 1300. How did they perform?

z = (1300 - 1050) / 200 = 1.25

Interpretation: The student scored 1.25 standard deviations above the mean.

Probability: P(x < 1.25) ≈ 0.8944 or 89.44%

Percentile: The student scored better than approximately 89% of all test takers - this is excellent performance!

Best Practices

Always verify your data is approximately normally distributed before using z-scores
Use z-scores to compare measurements from different scales or units
Report both z-scores and probabilities for complete statistical communication
Consider the context: is a z-score of 2 good or bad? It depends on what you're measuring
For hypothesis testing, common critical values are z = ±1.96 (95% confidence) and z = ±2.58 (99% confidence)
When working with percentiles, remember: z = 0 is the 50th percentile (median)
Use the symmetric interval P(-Z < x < Z) for confidence intervals
Document your calculations: show the raw score, mean, standard deviation, and resulting z-score

Z-score Calculator