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z-Score & Percentile Calculator

Enter a value with mean & SD (or a z-score) to get the percentile and tail areas under the standard normal curve—plus clear calculation steps.

Enter x, mean, and SD to compute z & percentile — or switch to z-mode to convert z → percentile.

Your inputs

Results

z-score
2
Percentile (Φ(z))
97.72%
Left tail P(Z ≤ z)
97.72%
Right tail P(Z ≥ z)
2.28%
Two-tail P(|Z| ≥ |z|)
4.55%
Central area
95.45%
PDF ϕ(z)
0.053991
Show calculation steps
  • z = (x − μ) / σ = (130 − 100) / 15 = 2.0000
  • Percentile = Φ(z) = 0.9772 (97.72%)
  • Left tail P(Z ≤ z) = 0.9772
  • Right tail P(Z ≥ z) = 0.0228
  • Two-tail P(|Z| ≥ |z|) = 0.0455
  • Central area P(−|z| ≤ Z ≤ |z|) = 0.9545

Results interpretation

  • Percentile is the proportion of the population at or below your score. 84th ≈ z of +1; 50th ≈ z of 0.
  • Two-tail tests how extreme |z| is on either side—useful for many hypothesis tests.
  • Central area shows how much probability lies between −|z| and |z| (e.g., ~68% within |z| ≤ 1; ~95% within |z| ≤ 2).
  • Who it’s for: students, analysts, and researchers translating scores into probabilities under the normal model.

How to use this calculator

  1. Select your mode: Value → z or z → Percentile.
  2. Enter x, μ, σ for value mode; enter z (and μ, σ if you want x back) for z mode.
  3. Read the percentile and tail probabilities instantly.
  4. Open “Show calculation steps” to see the exact math.
  5. Adjust inputs to test scenarios (higher/lower mean/SD or different z’s).

How this calculator works

Formula & assumptions

For value → z, we compute z = (x − μ) / σ. Then percentile is the standard normal CDF: Φ(z) = ½(1 + erf(z/√2)). Tail probabilities are P(Z ≤ z) = Φ(z), P(Z ≥ z) = 1 − Φ(z), and two-tail P(|Z| ≥ |z|) = 2 · min(Φ(z), 1 − Φ(z)). Central area is 1 − two-tail.

Assumptions: the underlying distribution is normal, inputs are in the same units, and SD is positive. For skewed/heavy-tailed data, normal approximation can misstate probabilities.

Use cases & examples

Example 1 (value → z): Exam score x=130 with μ=100 and σ=15 gives z ≈ 2. The percentile is Φ(z) ≈ 97.7%.

Example 2 (z → percentile): z=−1.2 ⇒ percentile ≈ Φ(−1.2) ≈ 11.5%, right-tail ≈ 88.5%.

Example 3 (two-tail): z=2.0 ⇒ two-tail ≈ 4.55%, central area ≈ 95.45%.

z-Score & Percentile — FAQ

What does a z-score mean?

It’s how many standard deviations your value is above (positive) or below (negative) the mean.

How do I get percentile from z?

Percentile = Φ(z) × 100, where Φ is the standard normal CDF.

What’s the two-tailed probability?

It’s the probability of being at least as extreme as |z| on either side: 2 · min(Φ(z), 1 − Φ(z)).

Can I convert z back to x?

Yes, if you also provide the original mean and SD: x = μ + z·σ.

Does this assume normality?

Yes. If your data are non-normal, z-based probabilities can be misleading.

What precision should I use?

3–4 decimals is plenty for most work; more precision rarely changes conclusions.

z-Score & Percentile: What They Mean, When to Use Them, and How to Explain Results Clearly

Our z-score & percentile calculator translates a raw value into a common scale so you can compare results across tests, time periods, or groups. By converting to z and applying the standard normal CDF Φ(z), you get an immediate estimate of where a score sits in the distribution. This helps with quality control, grading, clinical screening, and research reporting. Because many processes are modeled as normal—or approximately normal via the Central Limit Theorem—z-scores are a compact, powerful way to reason about probability and how unusual a value is.

Quick intuition for common z-scores

z = 0 sits at the mean (50th percentile). z = +1 is about the 84th percentile, while z = −1 is about the 16th. The classic “68-95-99.7 rule” says that roughly 68% of values lie within ±1 SD, 95% within ±2 SD, and 99.7% within ±3 SD. That means a z around ±2 is already unusual (two-tail ≈ 5%), and ±3 is very rare under normality (two-tail ≈ 0.3%).

Reporting that stakeholders understand

When communicating results, pair z-scores with percentiles. Percentiles are intuitive—“you scored better than 84% of people”—while z makes methods explicit and reproducible. Include the tail probability if you need to convey how extreme a result is for significance testing.

Limits of the normal model

Not every dataset is normal. Skew, heavy tails, truncation, and outliers can distort z-based probabilities. For small samples, consider robust or exact methods. Still, z is a solid starting point—especially for aggregated measures (e.g., means) or standardized scores designed to be approximately normal.

Practical tips

  • Check SD > 0: a zero or tiny SD makes z explode.
  • Use consistent units for x, μ, and σ.
  • Beware of multiple comparisons when scanning many z’s.
  • Round sensibly: 2–3 decimals are usually enough.

With these guidelines, you can move smoothly between raw values, z-scores, percentiles, and tail probabilities—and explain the story behind the numbers with confidence.