Complete Z-score Guide & Reference
1. What is a Z-score?
A z-score (also called a standard score) indicates how many standard deviations an individual
data point is from the mean of a distribution. It standardizes values from any normal distribution
to the standard normal distribution, which has a mean of 0 and a standard deviation of 1.
z = (x − μ) / σ
- z = z-score
- x = raw score (observed value)
- μ (mu) = population mean
- σ (sigma) = population standard deviation
2. Interpreting Z-scores
- Positive z-score: The raw value is above the mean.
- Negative z-score: The raw value is below the mean.
- z = 0: The value equals the mean.
- z = 1: The value is 1 standard deviation above the mean.
- z = -2: The value is 2 standard deviations below the mean.
3. Standard Normal Distribution
The standard normal distribution, denoted N(0, 1), is a special case of the normal distribution
with mean μ = 0 and standard deviation σ = 1. Converting raw scores to z-scores allows us to
use standard normal tables (z-tables) to find probabilities.
The cumulative distribution function (CDF) Φ(z) gives the probability that a standard normal
variable is less than or equal to z. It is computed using the error function (erf).
Φ(z) = (1 / 2) × [ 1 + erf(z / √2) ]
4. Common Probability Relationships
Probability less than z
P(X < z) = Φ(z)
Probability greater than z
P(X > z) = 1 − Φ(z)
Probability between 0 and z
P(0 < X < z) = Φ(z) − 0.5
Probability between -z and z
P(−z < X < z) = 2 × Φ(z) − 1
Probability in two tails
P(X < −z or X > z) = 2 × (1 − Φ(z))
Probability between z₁ and z₂
P(z₁ < X < z₂) = Φ(z₂) − Φ(z₁)
5. Empirical Rule (68-95-99.7 Rule)
For any normal distribution, nearly all data falls within three standard deviations of the mean:
- ~68.27% of data within 1 standard deviation (z = ±1)
- ~95.45% of data within 2 standard deviations (z = ±2)
- ~99.73% of data within 3 standard deviations (z = ±3)
6. How to Use Each Calculator
Section 1 — Z-score from raw score
- Enter the raw score (x), population mean (μ), and standard deviation (σ).
- Click Calculate.
- The z-score and three probability values are displayed.
- Step-by-step derivation is shown below the results.
Section 2 — Z-score / Probability converter
- Enter any one of the six available values (z-score or any probability).
- Click Calculate.
- All remaining values are computed and filled in automatically.
- This is equivalent to looking up values in a standard z-table.
Section 3 — Probability between two z-scores
- Enter the left bound z₁ and right bound z₂.
- Click Calculate.
- The area between the two z-scores is displayed, along with tail probabilities.
- A diagram shows the shaded area under the curve.
7. Worked Examples
Example 1 — Computing z-score
Given x = 5, μ = 3, σ = 2
- z = (5 − 3) / 2 = 2 / 2 = 1
- P(x < 5) = Φ(1) ≈ 0.84134
- P(x > 5) = 1 − 0.84134 = 0.15866
- P(3 < x < 5) = 0.84134 − 0.5 = 0.34134
Example 2 — Converting z = 2
- P(x < 2) = Φ(2) ≈ 0.97725
- P(x > 2) = 1 − 0.97725 = 0.02275
- P(0 < x < 2) = 0.97725 − 0.5 = 0.47725
- P(−2 < x < 2) = 2 × 0.97725 − 1 = 0.9545
- P(two tails) = 2 × 0.02275 = 0.0455
Example 3 — Between z₁ = -1 and z₂ = 0
- Φ(-1) ≈ 0.15866
- Φ(0) = 0.5
- P(-1 < x < 0) = 0.5 − 0.15866 = 0.34134
8. Real-World Applications
- Education: Standardized test scores (SAT, IQ), grading on a curve
- Statistics & Research: Hypothesis testing, p-values, confidence intervals
- Finance: Risk measurement, Sharpe ratio, Value at Risk (VaR)
- Quality Control: Six Sigma, process capability analysis
- Medicine: Growth charts, lab reference ranges, clinical diagnostics
- Psychology: Personality test scores, norm-referenced assessments
- Data Science: Feature standardization, outlier detection
- Manufacturing: Tolerance analysis, defect rate estimation
9. Important Notes
- Z-scores assume the underlying distribution is approximately normal.
- Standard deviation must be positive; a value of zero produces undefined results.
- This calculator uses the population standard deviation formula.
- Results are computed using the Abramowitz & Stegun approximation of the error function, with maximum error ~1.5×10⁻⁷.
- For very extreme z-scores (|z| > 6), probabilities approach 0 or 1 but remain computable.
- The converter works with any single input; if multiple inputs are provided, z-score takes priority.
- Z-tables in textbooks may vary slightly due to rounding; this calculator provides higher precision.
- For small samples, consider using the t-distribution instead of z-scores.
10. Related Statistical Concepts
- Standard Normal Distribution: N(0, 1), the basis for all z-score calculations
- Error Function (erf): Mathematical function used to compute normal CDF
- Z-table: Tabulated values of Φ(z) for common z-scores
- p-value: Probability of observing results at least as extreme under the null hypothesis
- Confidence Interval: Range likely containing the true population parameter
- T-score / T-distribution: Used for small samples with unknown population standard deviation
- Percentile: Value below which a given percentage of observations fall
- Standardization / Feature Scaling: Transforming data to have mean 0 and std dev 1
- Central Limit Theorem: Why z-scores are so widely applicable
11. References
1. DeGroot, Morris H. and Schervish, Mark J. Probability and Statistics. Pearson Education. 2012.
2. Wackerly, Dennis D., Mendenhall, William, and Scheaffer, Richard L. Mathematical Statistics with Applications. Cengage Learning. 2014.
3. Triola, Mario F. Elementary Statistics. Pearson Education. 2019.
4. Abramowitz, Milton and Stegun, Irene A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications. 1964.
5. Moore, David S., McCabe, George P., and Craig, Bruce A. Introduction to the Practice of Statistics. W.H. Freeman. 2017.
6. NIST/SEMATECH. e-Handbook of Statistical Methods. National Institute of Standards and Technology. 2012.