About the Z-Score calculator
A z-score tells you how many standard deviations a value is above or below the mean. This calculator converts a raw value to a z-score, finds the probability to the left, right and between z-scores on the normal distribution, converts a probability to a z-score, and turns a z-score back into a value.
It replaces a z table with exact values to ten significant digits.
Who should use the Z-Score calculator
- Statistics students.
- Analysts comparing values from different scales.
- Anyone finding a percentile for a normally distributed score.
How to use the Z-Score calculator: what each input means
- Mode
- Raw value to z, z to probabilities, probability to z, or z to raw value.
- Value, mean, standard deviation
- The standard deviation must be greater than zero.
- z-score or probability
- For the reverse modes.
How the Z-Score calculator works: method and formula
The z-score is (x − μ) ÷ σ. Probabilities come from the standard normal cumulative distribution function. The inverse mode finds the z-score for a given left-tail probability using an inverse normal approximation refined to high precision.
- z-score
- z = (x − μ) ÷ σ
- Back to a value
- x = μ + z × σ
How to read your Z-Score calculator result
A z-score of 0 is exactly at the mean, and ±1, ±2 and ±3 correspond to about 68%, 95% and 99.7% of values within that many standard deviations (for normal data). The left-tail probability is the percentile of the value.
Z-Score Calculator worked example
A score of 85 where the mean is 70 and the standard deviation is 10.
- z = (85 − 70) ÷ 10 = 1.5.
- The left-tail probability Φ(1.5) = 0.933193, so about 93.3% of values in a normal distribution lie below this score.
- The right tail is 0.066807.
z = 1.5, so about 93.3% of values lie below it (roughly the 93rd percentile).
Practical uses of the Z-Score calculator
- Finding a percentile for a test score.
- Comparing values from different distributions.
- Flagging unusually large or small values.
Assumptions and limitations of this calculator
- Probabilities assume the data follow a normal distribution.
- In probability mode, z-scores beyond ±40 are rejected as outside the useful range.