A z-score (standard score) measures how many standard deviations a value is from the mean of a distribution. Formula: z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. A z-score of 0 means the value equals the mean. A z-score of +1 means one standard deviation above the mean. A z-score of -2 means two standard deviations below the mean.

What does a z-score of 1.96 mean?

A z-score of 1.96 means the value is 1.96 standard deviations above the mean. In a standard normal distribution, 97.5% of data falls below Z = 1.96 and 2.5% falls above. This value is critical in statistics: a 95% confidence interval is defined as the mean ± 1.96 standard deviations. A two-tailed test uses Z = ±1.96 for 95% confidence (5% significance level).

How do I calculate a z-score from a raw value?

To calculate a z-score from a raw value: (1) subtract the mean (μ) from the value (x), (2) divide by the standard deviation (σ). Example: a student scores 85 on a test with mean 70 and std dev 15. Z = (85 - 70) / 15 = 1.0. This means the student scored exactly 1 standard deviation above the mean, at the 84.13th percentile. A negative z-score means the value is below the mean.

What is the 68-95-99.7 rule?

The empirical rule (68-95-99.7 rule) states that for a normal distribution: 68.27% of data falls within ±1 standard deviation of the mean (Z between -1 and +1), 95.45% falls within ±2 standard deviations (Z between -2 and +2), and 99.73% falls within ±3 standard deviations. Values beyond ±3 standard deviations are considered extreme outliers and occur in less than 0.27% of a normally distributed population.

What z-score is considered an outlier?

There is no single universal threshold, but common conventions are: Z > 2 or Z 3 or Z 6 or Z 2.5 as the outlier threshold. Context matters: medical tests often flag values beyond 2 standard deviations.

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Z-Score Calculator

A z-score measures standard deviations from the mean: z = (x - μ) / σ. Z = 0 means at the mean (50th percentile). Z = 1.0 = 84.13th percentile. Z = 1.645 = 95th percentile. Z = 1.96 = 97.5th percentile (used in 95% confidence intervals). Z = 2.0 = 97.72nd percentile. Z = 3.0 = 99.87th percentile. The empirical rule: ±1σ = 68.27%, ±2σ = 95.45%, ±3σ = 99.73% of normally distributed data.

Calculate z-scores, percentiles, and normal distribution probabilities. Convert between z-scores and percentiles. Find the percentage of data below, above, or between values in a standard normal distribution.

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Mode

Presets

Input

Option A: Enter Z-score directly

— or —

Option B: Calculate from raw value

Value (x)

Mean (μ)

Std Dev (σ)

Enter a z-score or raw value (with mean and standard deviation) to calculate percentile and probability.

How to Use

1
Choose your mode
Select "Z-Score → Percentile" to find what percentile a z-score corresponds to, or "Percentile → Z-Score" to find the z-score for a given percentile.
2
Enter your value
For Z-Score mode: type the z-score directly (e.g. 1.96), or enter raw value, mean, and standard deviation to calculate the z-score automatically.
3
Review the results
See the z-score, percentile rank, % of data below (left tail), % above (right tail), and % within ±|Z| of the mean.
4
Copy results
Click Copy to get all statistics as formatted text. Use presets for common critical values: 1.645 (90th percentile), 1.96 (95th), 2.576 (99th).

Frequently Asked Questions

What is a z-score?: A z-score (standard score) measures how many standard deviations a value is from the mean of a distribution. Formula: z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. A z-score of 0 means the value equals the mean. A z-score of +1 means one standard deviation above the mean. A z-score of -2 means two standard deviations below the mean.
How do I convert a z-score to a percentile?: To convert a z-score to a percentile, look up the cumulative probability in the standard normal distribution table (CDF). Z = 0 = 50th percentile (mean), Z = 1.0 = 84.13th percentile, Z = 1.645 = 95th percentile, Z = 1.96 = 97.5th percentile, Z = 2.0 = 97.72nd percentile, Z = 2.576 = 99.5th percentile, Z = 3.0 = 99.87th percentile. This calculator computes the CDF automatically for any z-score.
What does a z-score of 1.96 mean?: A z-score of 1.96 means the value is 1.96 standard deviations above the mean. In a standard normal distribution, 97.5% of data falls below Z = 1.96 and 2.5% falls above. This value is critical in statistics: a 95% confidence interval is defined as the mean ± 1.96 standard deviations. A two-tailed test uses Z = ±1.96 for 95% confidence (5% significance level).
How do I calculate a z-score from a raw value?: To calculate a z-score from a raw value: (1) subtract the mean (μ) from the value (x), (2) divide by the standard deviation (σ). Example: a student scores 85 on a test with mean 70 and std dev 15. Z = (85 - 70) / 15 = 1.0. This means the student scored exactly 1 standard deviation above the mean, at the 84.13th percentile. A negative z-score means the value is below the mean.
What is the 68-95-99.7 rule?: The empirical rule (68-95-99.7 rule) states that for a normal distribution: 68.27% of data falls within ±1 standard deviation of the mean (Z between -1 and +1), 95.45% falls within ±2 standard deviations (Z between -2 and +2), and 99.73% falls within ±3 standard deviations. Values beyond ±3 standard deviations are considered extreme outliers and occur in less than 0.27% of a normally distributed population.
What z-score is considered an outlier?: There is no single universal threshold, but common conventions are: Z > 2 or Z < -2 (mild outlier, 4.55% of data in a normal distribution), Z > 3 or Z < -3 (extreme outlier, 0.27% of data). In quality control (Six Sigma), 6 standard deviations (Z > 6 or Z < -6) is the target — only 3.4 defects per million. Some fields use Z > 2.5 as the outlier threshold. Context matters: medical tests often flag values beyond 2 standard deviations.

Z-Score to Percentile Reference Table

Z-Score	Percentile	% Below	% Above	Common Use
-3.00	0.13th	0.13%	99.87%	Extreme lower outlier
-2.00	2.28th	2.28%	97.72%	Lower outlier threshold
-1.645	5.00th	5.00%	95.00%	90% CI lower bound
-1.00	15.87th	15.87%	84.13%	1 std dev below
0.00	50.00th	50.00%	50.00%	Mean
1.00	84.13th	84.13%	15.87%	1 std dev above
1.282	90.00th	90.00%	10.00%	80% CI one-tail
1.645	95.00th	95.00%	5.00%	90% CI upper bound
1.960	97.50th	97.50%	2.50%	95% CI upper bound
2.00	97.72nd	97.72%	2.28%	2 std devs above
2.326	99.00th	99.00%	1.00%	98% CI one-tail
2.576	99.50th	99.50%	0.50%	99% CI upper bound
3.00	99.87th	99.87%	0.13%	Extreme upper outlier
3.291	99.95th	99.95%	0.05%	99.9% CI upper bound