How do I calculate a z-score?

Use the formula z = (x − μ) / σ, where x is your raw score, μ is the population mean, and σ is the standard deviation. For example, if your score is 85, the mean is 70, and the SD is 10: z = (85 − 70) / 10 = 1.5.

How do I convert a z-score to a percentile?

Use the standard normal distribution table (z-table) or the CDF function. A z-score of 1.5 corresponds to the 93rd percentile, meaning 93.32% of values fall below it. Our calculator does this conversion automatically.

What does a z-score tell you?

A z-score tells you how many standard deviations a data point is from the mean. A z-score of +2 means the value is 2 standard deviations above the mean. It allows you to compare values from different datasets by standardizing them.

What is a good z-score?

It depends on context. In exam scoring, a z-score above 1.0 (top 16%) is good; above 2.0 (top 2.3%) is excellent. In quality control, z-scores outside ±3 indicate outliers (only 0.3% of normal distribution).

What is the difference between z-score and percentile?

A z-score is a raw standardized score (number of standard deviations from the mean). A percentile tells you what percentage of the population scores below you. A z-score of 1.0 equals the 84th percentile; a z-score of −1.0 equals the 16th percentile.

How do I find z-score from percentile?

Use the inverse normal distribution (probit function). For example, the 95th percentile has a z-score of 1.645. The 99th percentile has a z-score of 2.326. Our calculator includes a reverse lookup from percentile to z-score.

Z-Score Calculator

Our free z-score calculator computes the standard score (z-score) from a raw value, population mean, and standard deviation. It also converts z-scores to percentiles and probabilities using the standard normal distribution. Use it for statistics, exam scores, quality control, and hypothesis testing.

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calendar_today Updated: April 2026

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Raw Value (x)

Mean (μ)

Standard Deviation (σ)

Normal Distribution

−4σ−2σμ+2σ+4σ

analytics Z-Score

1.50

1.5 SD above the mean

Percentile Rank 93.32%

P(X < value) 0.9332

P(X > value) 0.0668

Within ±|z| range 86.64%

68-95-99.7 Rule

z = ±1.068.27%

z = ±1.9695.00% (95% CI)

z = ±2.095.45%

z = ±3.099.73%

Input Values

Calculation Mode

Z-Score from Raw Value

Percentile from Z-Score

auto_awesome AI Analysis

Z-Score Result: z = 1.50

Percentile: 93.32nd — higher than 93.32% of the population
P(X < 85): 0.9332 (93.32%)
P(X > 85): 0.0668 (6.68%)

A z-score of 1.5 is 1.5 standard deviations above the mean — well above average.

lightbulb Tips

•z = (x − μ) / σ — distance in std deviations
•68% of data lies within z = ±1
•95% within z = ±1.96 (95% CI)
•|z| > 3 is considered an extreme outlier

bar_chart Z-Score Reference

Empirical Rule

z = ±1.00 68.27%

z = ±1.96 95.00% (CI)

z = ±2.00 95.45%

z = ±2.58 99.00% (CI)

z = ±3.00 99.73%

Classification

|z| < 1.0 Near average

|z| 2–3 Unusual

|z| > 3 Extreme outlier

How to Use This Calculator

numbers

Enter Your Value

Input your raw score or observation (x).

calculate

Enter Mean & SD

Enter the population mean (μ) and standard deviation (σ).

analytics

Get Z-Score

See your z-score, percentile rank, and probability.

info

Interpret Results

Understand what your z-score means relative to the distribution.

The Formula

The z-score measures how many standard deviations a value is from the mean. A z-score of 0 means the value equals the mean. Positive z-scores are above the mean; negative z-scores are below. The percentile is the cumulative probability P(Z < z) from the standard normal distribution.

z = (x − μ) / σ

lightbulb Variables Explained

z Z-score (standard score)
x Raw value (the observation)
μ Population mean
σ Population standard deviation

tips_and_updates Pro Tips

z = 0 means you're exactly at the mean

z = ±1 covers ~68% of a normal distribution

z = ±2 covers ~95% of a normal distribution

z = ±3 covers ~99.7% of a normal distribution (the 68-95-99.7 rule)

A z-score above 2.0 means you're in the top ~2.3% of the distribution

Use negative z-scores when the value is below the mean

Z-Score Calculator — Standard Score, Percentile & Probability

Calculate z-scores, convert to percentiles, and find probability values using our free z-score calculator. Works with exam scores, quality control data, scientific measurements, and any normally distributed dataset.

How to Calculate a Z-Score

The z-score formula is z = (x − μ) / σ, where x is your raw value, μ is the mean, and σ is the standard deviation. Enter these three values into the calculator to get the z-score instantly, along with the corresponding percentile and probability.

Z-Score to Percentile Conversion

Convert any z-score to a percentile rank using the cumulative standard normal distribution. A z-score of 0 = 50th percentile (the mean). A z-score of 1.0 = 84th percentile. A z-score of 1.96 = 97.5th percentile (used in 95% confidence intervals).

Normal Distribution & Z-Scores

Z-scores follow the standard normal distribution (mean=0, SD=1). The 68-95-99.7 rule states that 68% of data falls within z=±1, 95% within z=±2, and 99.7% within z=±3. Our calculator visualizes your z-score on the normal distribution curve.

Frequently Asked Questions

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