What Is a Z-Score?
A z-score (also called a standard score or z-value) tells you how many standard deviations a raw value lies above or below the mean. The formula is:
z = (x − μ) / σ
Positive z means the value is above the mean; negative means below; zero means exactly at the mean. Z-scores are unit-free — the same z represents the same relative position regardless of the original measurement scale.
Calculator Modes
The calculator supports eleven connected calculations: z from x/μ/σ, x from z, μ from z, σ from z, probability from z, z from probability, z from percentile, central ±z, probability between two z-values, probability outside two z-values, and a direct Z-table lookup mode. Each mode applies its own inverse or forward formula — the calculator does not run a single generic operation.
Z-Score Formula and Reverse Forms
Forward: z = (x − μ) / σ.
Reverse forms are useful when you know z and want the raw value, mean, or SD:
x = μ + zσμ = x − zσσ = (x − μ) / z(only defined when z ≠ 0)
With the defaults (x = 85, μ = 70, σ = 10), z = 1.5 — the value is 1.5 standard deviations above the mean.
Interpreting a Z-Score
| z | Meaning |
|---|---|
| −2 | 2 SD below the mean |
| −1 | 1 SD below the mean |
| 0 | Exactly at the mean |
| +1 | 1 SD above the mean |
| +2 | 2 SD above the mean |
The sign tells you direction. The absolute value tells you how far — 2.0 is farther than 0.5 in either direction.
Percentile and Probability
Under the standard normal model Z ∼ N(0, 1), the left-tail probability is Φ(z) and the percentile is 100·Φ(z). The right-tail probability is 1 − Φ(z). The two-tail area for a nonzero z is 2·[1 − Φ(|z|)].
Reference values the calculator can produce directly:
- z = 0 → 50th percentile
- z = 1.00 → ≈ 84.13th percentile
- z = 1.645 → ≈ 95th percentile
- z = 1.96 → ≈ 97.5th percentile
- z = 2.326 → ≈ 99th percentile
These are distribution probabilities, not hypothesis-test p-values. A z-score of 1.96 does not mean "significant" outside the context of a specified test.
Reading a Z Table
The PHP page renders a full positive Z table (rows 0.0 to 3.9, columns 0.00 to 0.09) showing cumulative left-tail probabilities Φ(z). To look up z = 1.96, find the row for 1.9 and the column for 0.06; the cell shows approximately 0.97500, which corresponds to the 97.5th percentile.
Negative z values use symmetry: Φ(−z) = 1 − Φ(z). So Φ(−1.96) = 1 − 0.97500 ≈ 0.02500. The calculator computes exact Φ(z) for any signed z in the probability and percentile modes; the printed table is the traditional reference.
Different textbooks use different Z-table conventions. The three most common are: cumulative left-tail P(Z ≤ z), area between the mean and z, and right-tail P(Z ≥ z). The same z produces different numbers in each — check which convention a table uses before reading it.
Reverse Calculations
Given a percentile or cumulative probability p, the z-score is the inverse normal quantile: z = Φ⁻¹(p). For a right-tail probability p, the inverse is z = Φ⁻¹(1 − p). For a central interval that contains probability p, the critical value is z_c = Φ⁻¹((1 + p) / 2), symmetric ±z_c. Common central values: 90% → ±1.6449, 95% → ±1.9600, 99% → ±2.5758.
Probability Between and Outside
For two z-values z₁ < z₂:
- Probability between: P(z₁ < Z < z₂) = Φ(z₂) − Φ(z₁)
- Probability outside: P(Z < z₁) + P(Z > z₂) = Φ(z₁) + [1 − Φ(z₂)]
These are often used in tolerance-interval work and in comparing tail areas against a central reference region.
Z-Score vs Standard Deviation
Standard deviation measures the spread of the distribution. Z-score measures the position of a single value relative to the mean, in standard-deviation units. A large σ means values are more spread out; a large z means a particular value is far from the mean regardless of how spread the distribution is.
Z-Score vs Percentile
A z-score is a position measured in standard deviations from the mean. A percentile is the percentage of the distribution that lies at or below the value. They are related by Φ(z) — one is a standardized distance, the other is a cumulative proportion — but they are not the same measure.
Z-Score vs Z-Test
A z-score is a standardized value. A z-test is a hypothesis-testing procedure that uses a z statistic under a specified null hypothesis. This calculator computes z-scores, percentiles, and normal-distribution probabilities; it does not perform hypothesis tests. Use a dedicated z-test tool for hypothesis testing.
Assumptions and Limitations
The arithmetic z = (x − μ)/σ requires only σ > 0. It does not assume normality.
Converting z to a percentile or probability via Φ(z) assumes the standardized value follows a standard normal distribution. That is often reasonable when the underlying data are approximately normal, or when the z-score is derived from a sample mean with a sufficiently large n via the Central Limit Theorem — but it is not automatic. Applying Φ(z) to a value from a heavily skewed or multimodal distribution produces a percentile that does not describe the data.
Common Mistakes
- Subtracting μ from x in the wrong order — z uses (x − μ), not (μ − x).
- Dividing by the variance σ² instead of the standard deviation σ.
- Forgetting the sign of z when the value is below the mean.
- Reading a Z table without checking which convention it uses (left-tail vs mean-to-z vs right-tail).
- Using Φ(z) to get a percentile from data that clearly are not approximately normal.
- Confusing a z-score with a hypothesis-test z statistic.
- Treating 1.96 or 1.645 as universally significant without reference to the test and its assumptions.