Z-score Calculator: Raw Score to Z-Score & Percentile
Free Z-score calculator: convert raw scores to Z-scores, Z-scores to percentiles, or find the probability between two scores. Formula z = (x−μ)/σ. Standard normal CDF. Shaded distribution. No sign-up. Used in AP Statistics and introductory stats.
What is a Z-score? Standardizing Your Data
- Positive Z-scores:The value is above the mean.
- Negative Z-scores:The value is below the mean.
- Z = 0:The value equals the mean.
- Interpretation:A Z-score of 2 means the value is 2 standard deviations above the mean.
A Z-score (or standard score) tells you how far a data point is from the mean in units of standard deviation. By converting raw values to Z-scores, you put different datasets on the same scale—for example, comparing a student's SAT score (scale 200–800) to their GPA (scale 0–4). This is why how to calculate a z score is central to introductory statistics and data science.
How to Use This Z-score Calculator
Three Modes, One Tool
- Trust:Uses the error function for Φ(z). Logic Trace shows exact formulas with your numbers. Built for AP Statistics and introductory courses.
- Visual:Shaded normal curve: left-tail (percentile) or region between two Z-scores. Red dashed lines mark your Z values.
This Z-score calculator supports three workflows so you can convert raw score to z score, z score to percentile, or find the probability between two scores. Value → Z: Enter raw score (x), mean (μ), and standard deviation (σ) to get the Z-score, P-values, and percentile. Z → Probability: Enter a Z-score to get P(Z < z), P(Z > z), and percentile—no μ or σ needed. Between Two Scores: Enter x₁ and x₂ with μ and σ to get P(Z₁ < Z < Z₂). Calculations use the standard normal CDF (Φ) with high precision; all processing runs in your browser.
The Z-Score Formula and Standard Normal Distribution
Formula and CDF
To convert raw score to z score, use
where x is the raw score, μ is the population mean, and σ is the standard deviation. The result z is in "standard deviation units." Probabilities come from the standard normal distribution (mean 0, SD 1) via the cumulative distribution function Φ(z), which gives
So
(percentile) and
The Logic Trace shows the full substitution and the Theory tab explains what your Z-score means in plain language.
P-Values, Percentile, and the Standard Normal Table
Interpretation and Lookup
The P-value (Z < x) is the area under the standard normal curve to the left of your Z-score—the proportion of the distribution below that value. The percentile is this proportion as a percentage: P(Z < z) × 100. A percentile of 97.72% means your score is higher than 97.72% of values. P-value (Z > x) is the area to the right. P(Z < z) + P(Z > z) = 1. The Theory tab includes a standard normal Z-table (0–3.0) for manual lookup—match your Z to row and column, or use symmetry: P(Z < −z) = 1 − P(Z < z) for negative Z.
When to Use a Z-Score Calculator
Use Cases
Use this tool when you need to find percentile from z score, compare scores across different scales (e.g., test scores vs. grades), verify homework in AP Statistics or intro stats, or check the probability between two bounds. It complements our P-value Calculator (for hypothesis tests) and Confidence Interval Calculator (for Z-values at 90%, 95%, 99%).
Z-score Calculator FAQ
? What is a Z-score?
A Z-score (standard score) measures how many standard deviations a data point is from the mean. It standardizes different datasets so you can compare values—e.g., comparing an SAT score to a GPA. Positive Z-scores are above average; negative Z-scores are below average; Z = 0 is exactly average.
? How do I calculate a Z-score from a raw value?
Use the formula
where x is your raw score, μ is the population mean, and σ is the standard deviation. Enter these in "Calculate Z-score" mode; the tool shows the result, P-values, and percentile.
? What does the percentile mean?
The percentile is the share of the distribution below your Z-score, as a percentage. A percentile of 84% means your score is higher than 84% of values. It equals
.
? What is P-value (Z < x) vs P-value (Z > x)?
is the area to the left of your Z-score (below that value).
is the area to the right (above it). They sum to 1. The shaded region in the chart is the left-tail area (percentile).
? How do I find the probability between two scores?
Use Probability Between Two Scores mode. Enter raw scores x₁ and x₂ with μ and σ. The calculator converts both to Z-scores and uses:
The shaded area between the red lines shows this probability.