📊 Select Calculation Mode
📌 Enter the individual value (X), the population mean (μ), and standard deviation (σ) to get the z-score, percentile, and probability.
X
Enter a valid value.
μ
Enter a valid mean.
σ
Enter σ > 0.
📌 Enter a z-score to get the one-tailed and two-tailed p-values, cumulative probability Φ(z), and percentile rank.
z
Enter a valid z-score.
Two-tailed is most common for hypothesis testing
📌 Enter a p-value (0 to 1) to find the corresponding critical z-score. Used for finding critical values for hypothesis tests and confidence intervals.
p
e.g. 0.05 for 5% significance level
Enter p between 0.0001 and 0.9999.
📌 Find the original raw score (X) corresponding to a z-score. Used for setting cutoff scores, grade thresholds, and percentile boundaries.
z
Enter a valid z-score.
μ
Enter a valid mean.
σ
Enter σ > 0.
📌 One-sample z-test: compare your sample mean to a known population mean when σ is known. Requires n ≥ 30 (or normal population).
μ
Enter the null hypothesis mean.
σ
Enter σ > 0.
x̅
Enter a valid sample mean.
n
Enter n ≥ 1.
Z-Score
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⚠️ Disclaimer: Results are for educational and informational purposes. Verify critical statistical calculations with a qualified statistician.
Sources & Methodology
Normal CDF computed via Abramowitz & Stegun rational approximation (maximum error < 7.5 × 10&sup8;). Inverse normal via Beasley-Springer-Moro algorithm. All values validated against NIST tables.
NIST/SEMATECH e-Handbook of Statistical Methods
Section 1.3.6 Measures of Shape — reference for z-score computation, standard normal distribution tables, and hypothesis testing procedures including one-sample z-test methodology.
Abramowitz, M. & Stegun, I.A. (1964) — Handbook of Mathematical Functions
Section 26.2 — Rational approximation for the standard normal CDF used in this calculator. The approximation achieves maximum error less than 7.5 × 10&sup8;, sufficient for all statistical applications.
Moore, McCabe & Craig — Introduction to the Practice of Statistics (9th ed.)
Reference for z-test procedure, critical value selection, and hypothesis testing decision framework including one-tailed and two-tailed test interpretations.
Computation:
Z-score: z = (X − μ) / σ | Raw score: X = μ + zσ Normal CDF Φ(z) via Abramowitz & Stegun (1964) rational approximation One-tail p = 1 − Φ(|z|) | Two-tail p = 2(1 − Φ(|z|)) Z-test: z = (x̅ − μ⊂0;) / (σ / √n) Inverse normal (p to z) via Beasley-Springer-Moro algorithm accurate to 4 decimal places.
Z-score: z = (X − μ) / σ | Raw score: X = μ + zσ Normal CDF Φ(z) via Abramowitz & Stegun (1964) rational approximation One-tail p = 1 − Φ(|z|) | Two-tail p = 2(1 − Φ(|z|)) Z-test: z = (x̅ − μ⊂0;) / (σ / √n) Inverse normal (p to z) via Beasley-Springer-Moro algorithm accurate to 4 decimal places.
Z-Score — Complete Guide with All Formulas and Examples
The z-score (also called a standard score) is the single most important concept linking raw data to probability in statistics. It answers the question: how unusual is this value? By expressing a value as a number of standard deviations from the mean, z-scores allow comparison across completely different distributions, conversion to probabilities using the normal table, and form the backbone of hypothesis testing, confidence intervals, and quality control.
The Z-Score Formula Explained
The z-score formula standardizes any value from any normal distribution onto the standard normal scale (mean = 0, standard deviation = 1). This is called standardization or z-transformation.
z = (X − μ) / σ
Where: X = individual value, μ = population mean, σ = population standard deviation
Worked example: IQ scores have mean 100, std dev 15. IQ of 130:
z = (130 − 100) / 15 = 30/15 = 2.0
Interpretation: IQ 130 is exactly 2 standard deviations above average. Percentile: 97.7th.
Worked example: IQ scores have mean 100, std dev 15. IQ of 130:
z = (130 − 100) / 15 = 30/15 = 2.0
Interpretation: IQ 130 is exactly 2 standard deviations above average. Percentile: 97.7th.
How to Interpret Any Z-Score
| Z-Score | Percentile | P(Z ≤ z) | Meaning |
|---|---|---|---|
| −3.0 | 0.13% | 0.0013 | Extremely below average (1 in 741) |
| −2.0 | 2.3% | 0.0228 | Well below average |
| −1.0 | 15.9% | 0.1587 | Below average |
| 0.0 | 50.0% | 0.5000 | Exactly at the mean |
| +1.0 | 84.1% | 0.8413 | Above average |
| +1.645 | 95.0% | 0.9500 | 90% CI critical value (one-sided) |
| +1.960 | 97.5% | 0.9750 | 95% CI critical value (two-sided) |
| +2.0 | 97.7% | 0.9772 | Well above average |
| +2.576 | 99.5% | 0.9950 | 99% CI critical value (two-sided) |
| +3.0 | 99.87% | 0.9987 | Extremely above average (1 in 741) |
Converting Z-Score to P-Value (and Back)
The p-value from a z-score represents the probability of getting a value as extreme as the observed z-score if the null hypothesis is true. The conversion uses the standard normal cumulative distribution function Φ(z).
Left-tail p-value: P(Z ≤ z) = Φ(z)
Right-tail p-value: P(Z ≥ z) = 1 − Φ(z)
Two-tail p-value: P(|Z| ≥ |z|) = 2(1 − Φ(|z|))
Example: z = 2.5
Left-tail: Φ(2.5) = 0.9938 = 99.38%
Right-tail: 1 − 0.9938 = 0.0062 = 0.62%
Two-tail: 2 × 0.0062 = 0.0124 = 1.24%
Left-tail: Φ(2.5) = 0.9938 = 99.38%
Right-tail: 1 − 0.9938 = 0.0062 = 0.62%
Two-tail: 2 × 0.0062 = 0.0124 = 1.24%
Finding the Raw Score from a Z-Score
Rearrange the z-score formula to solve for X. This is the inverse z-score calculation — used to find the value that corresponds to a given percentile or significance threshold.
X = μ + z × σ
Example: What IQ score is at the 90th percentile?
z at 90th percentile = 1.282 (from inverse normal table)
X = 100 + 1.282 × 15 = 100 + 19.23 = 119.23
Interpretation: An IQ of 119 is approximately at the 90th percentile.
z at 90th percentile = 1.282 (from inverse normal table)
X = 100 + 1.282 × 15 = 100 + 19.23 = 119.23
Interpretation: An IQ of 119 is approximately at the 90th percentile.
One-Sample Z-Test — Hypothesis Testing with Z-Scores
The one-sample z-test uses z-scores to test whether a sample mean is significantly different from a hypothesized population mean. It requires: (1) the population standard deviation σ is known, and (2) the sample is large (n ≥ 30) or the population is normally distributed.
z = (x̅ − μ⊂0;) / (σ / √n)
Example: A factory claims average output is 500 units (μ⊂0;=500, σ=100). A sample of 25 production runs gives x̅=520. Test at α=0.05 (two-tailed).
z = (520 − 500) / (100/√25) = 20/20 = 1.0
Two-tailed p = 2×(1−Φ(1.0)) = 2×0.1587 = 0.317
Decision: p = 0.317 > 0.05 → Fail to reject H⊂0;. Insufficient evidence that mean differs from 500.
z = (520 − 500) / (100/√25) = 20/20 = 1.0
Two-tailed p = 2×(1−Φ(1.0)) = 2×0.1587 = 0.317
Decision: p = 0.317 > 0.05 → Fail to reject H⊂0;. Insufficient evidence that mean differs from 500.
Z-Score vs T-Score — When to Use Each
Both z-scores and t-scores measure how many standard deviations a value is from the mean, but they apply to different situations:
- Use z-score: Population standard deviation (σ) is known, OR sample is large (n ≥ 30 by CLT)
- Use t-score: Population standard deviation is unknown (estimated from sample s), especially for small samples
- For large n: t-distribution approaches normal distribution, so z and t give nearly identical results
Z-Scores in Real-World Applications
- Education: Grading on a curve, SAT/GRE score interpretation, percentile ranks in standardized tests
- Finance: Altman Z-score for bankruptcy prediction; portfolio risk (Sharpe ratio = z-score of excess return)
- Medicine: Bone density T-scores (which are actually z-scores); z-scores for pediatric growth charts
- Quality control: Six Sigma process capability (target: ±6σ from specification limits)
- A/B testing: Conversion rate significance testing using z-test for proportions
💡 Critical values to memorize: z = 1.645 → 5% one-tailed, 90% CI. z = 1.960 → 5% two-tailed, 95% CI. z = 2.326 → 1% one-tailed, 98% CI. z = 2.576 → 1% two-tailed, 99% CI. These four z-scores appear constantly throughout inferential statistics and are worth committing to memory.
Frequently Asked Questions
Z-score = (X − μ) / σ. It measures how many standard deviations a value is from the mean. z = 0 means the value equals the mean. z = 2.0 means 2 standard deviations above the mean. z = −1.5 means 1.5 standard deviations below. Z-scores allow comparison of values from completely different distributions on the same scale and enable conversion to percentiles and p-values.
z = 0: value equals the mean (50th percentile). z = 1.0: 84th percentile (better than 84% of the distribution). z = 2.0: 97.7th percentile. z = −1.0: 16th percentile. z = −2.0: 2.3rd percentile. The empirical rule: 68% of z-scores are between −1 and +1, 95% between −1.96 and +1.96, 99.7% between −3 and +3. Z-scores beyond ±3 are very rare (<0.27% probability).
Use the standard normal CDF Φ(z). One-tailed (right): p = 1 − Φ(z). One-tailed (left): p = Φ(z). Two-tailed: p = 2(1 − Φ(|z|)). Example: z = 1.96 → two-tailed p = 2×(1−0.975) = 0.05. This is why z = ±1.96 is the boundary for statistical significance at the 5% level (two-tailed). Use this calculator for instant conversion in any direction.
X = μ + z × σ. Example: class mean 70, std dev 10, z = 1.5: X = 70 + 1.5 × 10 = 85. This is the 93.3rd percentile score. Used to find: the score that corresponds to a given percentile, the cutoff for the top 5% of a distribution, the threshold for a statistically significant result. For IQ (mean=100, σ=15): IQ at the 95th percentile = 100 + 1.645 × 15 = 124.7.
At the conventional 5% significance level: |z| > 1.96 (two-tailed) or z > 1.645 (one-tailed right). At 1% level: |z| > 2.576 (two-tailed). At 10% level: |z| > 1.645 (two-tailed) or z > 1.282 (one-tailed). Significance means the observed z-score is in the critical region — unlikely enough to reject the null hypothesis. Statistical significance does not equal practical significance; always consider effect size too.
A one-sample z-test compares a sample mean to a known population mean when σ is known. Test statistic: z = (x̅ − μ⊂0;) / (σ/√n). Use it when: (1) population standard deviation is known, AND (2) n ≥ 30 or population is normal. Use a t-test when σ is unknown (far more common). Use a z-test for proportions when testing p vs a known proportion: z = (p̂ − p⊂0;) / √(p⊂0;(1−p⊂0;)/n).
Both measure standard deviation distance from the mean, but use different reference distributions. Z-scores use the standard normal N(0,1). T-scores use the t-distribution with heavier tails (more probability in the extremes). Use z when σ is known (or n ≥ 30 by CLT). Use t when σ is unknown and estimated by sample s. As n increases, t-distribution converges to normal — for n > 100, the difference is negligible. The t critical value at df=∞ exactly equals the z critical value.
90th percentile: z = 1.282. 95th percentile: z = 1.645. 97.5th percentile: z = 1.960 (used for 95% two-sided CI). 99th percentile: z = 2.326. 99.5th percentile: z = 2.576 (used for 99% two-sided CI). 99.9th percentile: z = 3.090. For below-mean: 10th percentile = −1.282, 5th = −1.645, 1st = −2.326. The negative value is just the mirror image of the positive percentile above 50.
A negative z-score means the value is below the population mean. z = −1.5 means 1.5 standard deviations below the mean, corresponding to the 6.7th percentile. Negative z-scores are equally valid as positive ones — they simply indicate the value is on the left (below-average) side of the normal distribution. Example: body temperature of 96.8°F with mean 98.6 and σ = 0.7: z = (96.8−98.6)/0.7 = −2.57, meaning it is 2.57 std devs below normal.
Confidence intervals use z-scores (called critical values z*) to define the interval boundaries. 90% CI: mean ± 1.645 × SEM. 95% CI: mean ± 1.960 × SEM. 99% CI: mean ± 2.576 × SEM. The z* value is the inverse normal of (1 − α/2). These critical values come directly from the standard normal distribution and represent the z-scores that contain the middle 90%, 95%, or 99% of the distribution.
Z-score grading: compute z = (score − class mean) / class std dev for each student. Then assign grades: A for z > 1, B for 0 < z ≤ 1, C for −1 < z ≤ 0, D for z ≤ −1. Alternatively, convert to a new scale: new score = target mean + target std dev × z. Example to scale to mean 75, std dev 10: new score = 75 + 10 × z. This preserves relative performance while adjusting the grade distribution to a desired center and spread.
For a sample mean, use the standard error (SEM = σ/√n) instead of σ: z = (x̅ − μ) / (σ/√n). Example: population mean 500, σ = 100, sample of n=25, x̅ = 520: z = (520−500)/(100/5) = 20/20 = 1.0. This accounts for the CLT: sample means vary less than individual observations, by factor √n. This z-score is used directly in the one-sample z-test.
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