Z-score Calculator

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

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.

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.

How to Use This Z-score Calculator

Three Modes, One Tool

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.

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.

The Z-Score Formula and Standard Normal Distribution

Formula and CDF

To convert raw score to z score, use

z = \frac{x - \mu}{\sigma}

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

P(Z \leq z)

P(Z < z) = \Phi(z)

(percentile) and

P(Z > z) = 1 - \Phi(z)

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

z = \frac{x - \mu}{\sigma}

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

P(Z < z) \times 100

What is P-value (Z < x) vs P-value (Z > x)?

P(Z < x)

is the area to the left of your Z-score (below that value).

P(Z > x)

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:

P(z_1 < Z < z_2) = \Phi(z_2) - \Phi(z_1)

The shaded area between the red lines shows this probability.

Z-score Calculator

Step-by-Step Calculation

Distribution Visualization

Z-score Calculator: Standardizing Your Data

Quick Workflow

Value → Z

Z → Probability

Between Two Scores

Theory & Z-Table