Z-Score Calculator
Standardize a raw score with z = (x − μ) / σ, or solve for the raw score or the standard deviation. Each run shows the deviation from the mean and whether x sits above, below, or at μ.
85 is 1.5 SD above the mean
- A z-score counts how many standard deviations x sits from the mean: z = (x − μ) / σ.
- Deviation x − μ = 85 − 70 = 15.
- z = 15 / 10 = 1.5. The score is above the mean.
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All calculations run locally in your browser. Your entered values are not sent to an external service.
How to Use the Z-Score Calculator
Find z standardizes a known score. Find x undoes that and returns the raw value. Find σ recovers the spread when you already know z.
- Choose Find z, Find x, or Find σ.
- Enter finite numbers. The standard deviation, when it is an input, must be greater than zero.
- Read z or x or σ, the signed deviation, and whether the score is above, below, or at the mean.
Computing s or σ from a list of numbers belongs on the Standard Deviation Calculator. The mean of a list belongs on the Mean, Median, Mode Calculator. Reading Φ(z) as a percentile belongs on the Percentile Calculator. This page assumes μ and σ are already known and standardizes one x.
The Z-Score Formula
Subtracting μ centers the score on the mean. Dividing by σ rescales that gap into “how many standard deviations.” The sign of z is the sign of (x − μ): positive means above the mean, negative means below, zero means x = μ.
The same algebra runs backwards. If a homework statement says “two standard deviations above a mean of 100 with σ = 15,” that is x = 100 + 2 × 15 = 130. Solving for σ needs a nonzero z: if z is 0 then x already equals μ and any positive σ would fit, so the spread is not unique.
Z-Score Calculator Examples
A test score of 85
μ = 70, σ = 10. z = (85 − 70) / 10 = 1.5. Fifteen points is one and a half spreads.
A score at the mean
x = 70, μ = 70, σ = 10. The deviation is 0, so z = 0. Being “average” on this scale is z = 0, not z = 1.
A score below the mean
x = 55 gives z = (55 − 70) / 10 = −1.5. The size |z| = 1.5 matches the first example; only the direction changed.
Find x from z = 2
μ = 100, σ = 15. x = 100 + 2 × 15 = 130. A “two-sigma” score on that scale is 130.
Find σ from z = 1.5
x = 85, μ = 70. σ = 15 / 1.5 = 10. That recovers the first example’s spread.
σ = 0 is rejected
If every value were the mean, there is no spread to divide by. Compute s from a list on the Standard Deviation Calculator first; if that list has s = 0, a z-score is not defined.
The Empirical Rule, Without a Table
For a mound-shaped data set, about 68% of values lie within 1 SD of the mean (|z| < 1), about 95% within 2 SD, and about 99.7% within 3 SD. That is a sketch, not a p-value. The exact left-tail area Φ(z) and the matching percentile belong on the Percentile Calculator.
Common Z-Score Mistakes
Subtracting in the wrong order
(μ − x) / σ flips the sign. 85 vs 70 must be +1.5, not −1.5.
Dividing by the mean instead of σ
The denominator is the spread, not μ. 15 / 70 is not a z-score.
Treating z = 1 as “the mean”
The mean is z = 0. z = 1 is one standard deviation above μ.
Using a sample list here
This page wants one x and a known μ and σ. Paste a column of numbers into the Standard Deviation Calculator to get s first.
Reading z as a probability
z is a signed count of standard deviations. A tail probability needs a table or a normal-curve tool, which this is not.
Z-Score Calculator FAQs
What is a z-score?
z = (x − μ) / σ. It says how many standard deviations the raw score sits from the mean, with a sign for direction.
What is the z-score of 85 when μ = 70 and σ = 10?
1.5, because (85 − 70) / 10 = 15 / 10.
Can a z-score be negative?
Yes. Any x below the mean has a negative z. |z| is the distance; the sign is the side.
How do you find x from z?
x = μ + zσ. For z = 2, μ = 100, σ = 15, x = 130.
Does this look up a z-table or a percentile?
No. It only evaluates z = (x − μ) / σ. Use the Percentile Calculator to read 100 × Φ(z), and the Standard Deviation Calculator when you still need σ from a list.
Are my numbers sent to a server?
No. The calculation runs locally in your browser.
Related Calculators
Mean, Median, Mode CalculatorThe mean of a list, plus a population σ side metric.
Weighted Average CalculatorA mean when the values should not count equally.
Percentile CalculatorTurn z into 100 × Φ(z), or recover z from a percentile.
Explore More Statistics Tools
Continue with averages, spread, and everyday statistics tools in the SolverCalc collection.
Reviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
Sources
Disclaimer
This calculator is for general educational use. It evaluates z = (x − μ) / σ only. It is not a z-table, a p-value tool, or a substitute for a statistics package.