Z-Score Calculator
Convert any value into a z-score (how many standard deviations from the mean), get the percentile (normal CDF), and even reverse-solve for the raw score using z = (x − μ)/σ. Includes steps, quick picks, and a mini bell-curve visual.
Background
A z-score standardizes a value so you can compare results across different scales. If z = 0, you’re exactly at the mean. Positive z is above-average; negative z is below-average. The percentile uses the standard normal distribution to tell you “what % of values fall below this z”.
How to use this calculator
- Choose a mode (solve for z, x, μ, or σ).
- Enter the required values (and keep σ > 0).
- For z-score mode, optionally choose a percentile style (below, above, or two-tailed).
- Click Calculate to get the answer, steps, and a bell curve visual.
How this calculator works
- Z-score standardizes values: z = (x − μ)/σ.
- Percentiles use the standard normal CDF Φ(z).
- Reverse-solve by rearranging the same formula (solve for x, μ, or σ).
Formula & Equation Used
Z-score: z = (x − μ)/σ
Reverse for x: x = μ + zσ
Reverse for μ: μ = x − zσ
Reverse for σ: σ = (x − μ)/z (z ≠ 0)
Standard normal percentile: Φ(z)
Example Problem & Step-by-Step Solution
Example 1 — Exam score percentile (below x)
A student scored x = 84. The class mean is μ = 70 with standard deviation σ = 10.
- Compute z: z = (x − μ)/σ = (84 − 70)/10 = 1.4
- Percentile below: Φ(1.4) ≈ 0.9192
- So the score is about the 92nd percentile (above ~92% of scores).
Example 2 — Probability above x (right tail)
A height is x = 180 cm. The population mean is μ = 170 cm with standard deviation σ = 7 cm. What fraction of people are taller than 180 cm?
- Compute z: z = (180 − 170)/7 ≈ 1.4286
- Left-tail CDF: Φ(1.4286) ≈ 0.9234
- Right-tail probability: P(X ≥ 180) = 1 − Φ(z) ≈ 1 − 0.9234 = 0.0766
- So about 7.66% of people are taller than 180 cm (under a normal model).
Example 3 — Two-tailed “how extreme?” probability
A test statistic comes out to z = 2.0. What is the two-tailed probability P(|Z| ≥ 2.0)?
- Compute the one-tail beyond +2: 1 − Φ(2.0) ≈ 1 − 0.97725 = 0.02275
- Two tails: P(|Z| ≥ 2.0) = 2(1 − Φ(2.0)) ≈ 2 · 0.02275 = 0.0455
- So the two-tailed probability is about 4.55%.
Note: These probabilities assume the normal (bell-curve) model is appropriate for the situation.
Frequently Asked Questions
Q: What does a negative z-score mean?
It means the value is below the mean. For example, z = −1 is one standard deviation below average.
Q: Is the percentile always accurate?
The percentile is based on a normal (bell-curve) model. If your data isn’t roughly normal, treat the percentile as an approximation.
Q: What does “two-tailed” mean?
It’s the probability of being at least as extreme as your z-score in either direction: P(|Z| ≥ |z|).
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