This free Z Score Calculator helps you quickly find the Z value (standard score) and the corresponding cumulative probability from the standard normal distribution. Enter a raw score, mean (μ), and standard deviation (σ) to instantly calculate how many standard deviations the value is from the mean.
In statistics, a Z Score (or standard score) tells you how far a data point is from the mean, measured in standard deviations.
Z = 0 → the value is exactly the mean.
Z > 0 → the value is above the mean.
Z < 0 → the value is below the mean.
This makes Z scores essential for comparing values across different normal distributions.
The Z Score formula is:
$$ Z = \frac{X - \mu}{\sigma} $$
If a test score of 85 comes from a distribution with mean 70 and standard deviation 10:
$$ Z = \frac{85 - 70}{10} = \frac{15}{10} = 1.5 $$
This means the score is 1.5 standard deviations above the mean.
Enter your raw score (X).
Enter the mean (μ).
Enter the standard deviation (σ).
Click Calculate.
The calculator shows:
Z score
Φ(z) cumulative probability (area left of z)
Right-tail probability (area greater than z)
Traditionally, Z scores are looked up in a Z table to find cumulative probabilities. The calculator eliminates the need for manual lookup by computing Φ(z) instantly using the normal distribution function.
Example 1 – Exam scores:
X = 85, μ = 70, σ = 10
→ Z = 1.5, Φ(z) = 0.9332 (93.32% of scores are below 85).
Example 2 – Height data:
X = 160 cm, μ = 170 cm, σ = 8
→ Z = -1.25, Φ(z) = 0.1056 (only 10.56% are shorter than 160 cm).
Education: comparing exam scores across subjects.
Research: standardising results across different datasets.
Finance: risk modelling and return normalisation.
Medicine: growth charts, diagnostic testing.
Quality control: identifying outliers and rare events.
Q: What does a Z score of 2 mean?
A: The value is 2 standard deviations above the mean.
Q: What does a negative Z score mean?
A: The value is below the mean.
Q: What is a “high” Z score?
A: Usually, Z > 2 or Z < -2 indicates an unusual value.
Q: How do I convert Z to probability?
A: Use Φ(z), the cumulative distribution function. The calculator provides this automatically.
Q: What is the difference between Z score and percentile?
A: Percentile is the percentage of data below the value. A Z score can be converted into a percentile via the normal distribution.