Z-score Calculator
Convert between z-scores, raw values and normal distribution probabilities.
About Z-score Calculator
A z-score (standard score) measures how many standard deviations a value lies from the mean of its distribution. z = (x − μ) / σ. A z-score of 1.96 means the value is 1.96 standard deviations above the mean, which corresponds to the 97.5th percentile. By converting raw values to z-scores you can compare observations from distributions with different scales.
Formulas
Z-score
z = (x − μ) / σ
- x
- Raw value
- μ
- Population mean
- σ
- Population standard deviation
Reverse: raw value
x = μ + z·σ
Cumulative probability
Left-tail probability = Φ(z), the standard normal CDF
Worked example
With the default values below, Z-score Calculator returns Z-score of 0.7000. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Calculateone of Raw value → Z-score, Z-score → Raw value, Z-score → Probability, Probability → Z-score
- Raw value → Z-score
- Raw value (x)
- 72
- Population mean (μ)
- 65
- Population std deviation (σ)minimum 0
- 10
- Z-score
- 1.64
- Tailone of Left tail P(Z ≤ z), Right tail P(Z ≥ z), Two-tailed P(|Z| ≤ z)
- Left tail P(Z ≤ z)
- Cumulative probability (0 – 1)accepts 0–1
- 0.95
Results
- Z-score
- 0.7000
- P(Z ≤ z) Left tail
- 0.758036
- P(Z ≥ z) Right tail
- 0.241964
- PercentilePercentage of values in a normal distribution that fall below this point.
- 75.80 %
Step by step
- 1Z-score formula
- 2Substitute= 0.7000
- 3Left-tail probability= 0.7580
- 4Percentile= 75.80th
Assumptions and limits
- The result is derived only from the 7 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 6 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 does a z-score of 2 mean?
It means the value is exactly 2 standard deviations above the mean. In a standard normal distribution roughly 97.7 % of values lie below a z-score of 2 (left-tail), so only about 2.3 % are above it.
Can a z-score be negative?
Yes. A negative z-score means the raw value is below the mean. z = −1 places the value at the 15.9th percentile — about 84 % of values in the population are larger.
When should I use a z-score rather than a t-score?
Use a z-score when the population standard deviation is known and/or the sample size is large (n > 30 is a common rule of thumb). Use a t-score for small samples where you only have the sample standard deviation as an estimate.