How to use the Z-Score Calculator
- 1. Pick a modeChoose one value, a list of values, or a z-score to probability.
- 2. Enter the numbersType the value, mean and SD, or paste your list.
- 3. Read the resultSee z, the percentile and the one- and two-tailed areas.
The z-score formula
z = (x − μ) ÷ σ. A z-score counts how many standard deviations a value is above (positive) or below (negative) the mean. That lets you compare scores from tests with different scales.
Worked example: a student scores 78 on a test with mean 65 and SD 8. z = (78 − 65) ÷ 8 = 1.625. If scores are roughly normal, Φ(1.625) = 0.9479, so the student is at about the 94.79% percentile, ahead of nearly 95 in 100 classmates.
Where z-scores are used
- Comparing tests: 78 on a test with mean 65, SD 8 (z 1.63) beats 85 on a test with mean 80, SD 6 (z 0.83).
- Outliers: |z| above 3 is a common flag, though the IQR method is more robust.
- Machine learning: standardising features to z-scores puts them on the same scale for models like KNN, SVM and logistic regression.
- Hypothesis tests: a two-tailed area below 0.05 means |z| is above 1.96.
For a list, choose sample SD (n − 1) for data that is a subset of a bigger group, or population SD to match scikit-learn's StandardScaler. Percentiles assume a roughly normal shape; for skewed data, use the actual percentile calculator instead. Scale whole columns with the normalization and standardization calculator.
Frequently asked questions
Can a z-score be negative?
Yes. A negative z means the value is below the mean. z = −1 is one SD below.
What z-score is the top 5%?
About 1.645. Use the normal distribution calculator's percentile mode for any cut-off.
Is the Z-Score Calculator free?
Yes, the Z-Score Calculator is free. The first time you see your first result, you are asked for your name and WhatsApp number once. After that, this and every other Uncodemy tool opens straight away without the form.