Z-score Calculator
Calculate the z-score of a data point relative to a dataset's mean and standard deviation.
About Z-score Calculator
The z-score standardizes a value by expressing how many standard deviations it lies from the mean: z = (x − μ) / σ. A z-score of 0 means the value equals the mean; positive z-scores are above the mean, negative below. Assuming an approximately normal distribution, the z-score maps directly to a percentile via the standard normal cumulative distribution function (CDF). Z-scores are widely used in ML for feature standardization and for flagging statistical outliers (commonly |z| > 2 or |z| > 3).
Formula
z = (x - mu) / sigma
- x
- Observed value
- mu
- Dataset mean
- sigma
- Dataset standard deviation
Worked example
With the default values below, Z-score Calculator returns Z-score of 1.5000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Value (x)
- 85
- Dataset Mean (μ)
- 70
- Dataset Std Deviation (σ)minimum 0
- 10
Results
- Z-score
- 1.5000
- Percentile
- 6.68 th percentile
- Interpretation
- Within typical range (|z| ≤ 2)
Step by step
- 1z = (x − μ) ÷ σ= 1.5000
- 2Percentile (from standard normal CDF)= 6.68th percentile
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
What z-score threshold indicates an outlier?
There's no universal cutoff, but |z| > 2 is often used to flag 'unusual' values (roughly the outer 5% of a normal distribution) and |z| > 3 to flag 'extreme' outliers (roughly the outer 0.3%).
Does the z-score-to-percentile conversion assume normality?
Yes — the percentile shown uses the standard normal CDF, which is only an accurate percentile estimate if the underlying data is approximately normally distributed; for heavily skewed data, the z-score is still valid as a standardized distance measure, but the percentile interpretation may be inaccurate.
How is z-score used in feature scaling?
Standardization transforms every value in a feature to its z-score, giving the transformed feature a mean of 0 and standard deviation of 1 — this is a standard preprocessing step before training many ML models, especially those sensitive to feature scale like SVMs, k-NN, and neural networks.
Can z-score be negative?
Yes — any value below the mean produces a negative z-score; the sign simply indicates direction (below vs. above the mean), while the magnitude indicates distance in standard deviations.