# Z-Score Calculator

Free z-score calculator. Calculate the z-score, raw value, mean, or standard deviation using the formula z = (x - μ) / σ.

## What this calculates

Calculate the z-score (standard score) that tells you how many standard deviations a data point is from the mean. Solve for z-score, raw value, mean, or standard deviation using z = (x - μ) / σ.

## Inputs

- **Solve For** — options: Z-Score, Raw Value (x), Mean (μ), Standard Deviation (σ) — Select which variable to calculate.
- **Raw Value (x)** — The individual data point or observation.
- **Mean (μ)** — The mean (average) of the population or distribution.
- **Standard Deviation (σ)** — min 0 — The standard deviation of the population or distribution.
- **Z-Score** — The number of standard deviations from the mean.

## Outputs

- **Result** — The calculated value.
- **Variable** — formatted as text — Which variable was calculated.
- **Formula** — formatted as text — Step-by-step calculation.
- **Interpretation** — formatted as text — What the z-score means in context.

## Details

The z-score is one of the most important concepts in statistics. It standardizes values from different distributions, allowing direct comparison.

The Formula: z = (x - μ) / σ

- z = z-score (number of standard deviations from the mean)

- x = raw value (the data point)

- μ = population mean

- σ = population standard deviation

Interpreting Z-Scores

- z = 0: Value is at the mean

- z = 1: Value is 1 standard deviation above the mean

- z = -1: Value is 1 standard deviation below the mean

- z = 2: Value is 2 standard deviations above the mean

The Empirical Rule (for normal distributions)

- ~68% of data falls within z = -1 to z = +1

- ~95% falls within z = -2 to z = +2

- ~99.7% falls within z = -3 to z = +3

Common Application

IQ scores have μ = 100 and σ = 15. An IQ of 130 has z = (130 - 100) / 15 = 2.0, meaning it is 2 standard deviations above the mean, placing it at approximately the 97.7th percentile.

## Frequently Asked Questions

**Q: What does a z-score tell you?**

A: A z-score tells you how many standard deviations a value is from the mean. A z-score of 1.5 means the value is 1.5 standard deviations above the mean. A z-score of -2 means the value is 2 standard deviations below the mean. This allows comparison across different scales.

**Q: What is a 'good' z-score?**

A: It depends on the context. For test scores, a positive z-score means above average. In quality control, z-scores close to 0 are desirable (indicating the product is close to the target). In general, z-scores beyond +/-3 are considered extreme outliers in a normal distribution.

**Q: Can z-scores be used to compare different distributions?**

A: Yes, that is one of the main purposes of z-scores. For example, you can compare a student's score on two different exams by converting both to z-scores. If one exam has mean 75, SD 10, and the student scored 85 (z=1.0), and another has mean 60, SD 5, and the student scored 70 (z=2.0), the student performed relatively better on the second exam.

**Q: What is the relationship between z-score and percentile?**

A: For a normal distribution, each z-score corresponds to a specific percentile. z = 0 is the 50th percentile, z = 1 is approximately the 84th percentile, z = 2 is approximately the 97.7th percentile, and z = -1 is approximately the 16th percentile. A z-table or normal distribution calculator can give exact percentiles.

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Source: https://vastcalc.com/calculators/statistics/z-score
Category: Statistics
Last updated: 2026-04-21