Online CalculatorOnline Calculators
Menu

z-Score & Percentile Calculator

Compare any score to its peer group in a snap. The z-Score tells you how many standard deviations a value sits above or below the mean, and the paired percentile translates that position into “how rare or common” it is. Use this for exams, product metrics, lab values, or quality checks-anytime you need apples-to-apples comparisons across different scales.

This z-Score & Percentile Calculator lets you normalize raw numbers or reverse the process to find the value that hits a target percentile. The goal is clear interpretation: quickly identify outliers, set data-driven cutoffs, and communicate results in plain language so decisions-admissions, thresholds, alerts-are grounded in statistics, not guesswork.

Find z-scores, percentiles, and probabilities under the normal curve - with step-by-step math.

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

Your inputs

z-Score & Percentile 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.

Explore more calculators related to Statistics