Standardize a raw score, convert between a Z-score and normal-curve probability, or find the probability between two Z-scores. Choose the mode that matches the information you have: a raw score, a probability, or two standardized scores.
How to use this calculator
In Z-score, enter a raw value, its population mean, and a positive standard deviation, all in the same measurement units. The result is its distance from the mean in standard deviations, not a percentage.
In Converter, enter exactly one Z-score or normal probability. Clear the example Z-score before filling a probability field, and enter 0.05 for 5%. Probability inputs must be strictly between 0 and 1, with the zero-to-Z center area below 0.5. Symmetric probabilities determine the magnitude of Z; the converter returns the nonnegative value when its sign cannot be recovered.
In Range, enter two Z-scores with the left score smaller than the right. The result is the standard-normal area between them. A raw-score standardization itself does not require normal data; interpreting its normal probabilities does require that model.
Examples
Default raw score
For x = 5, μ = 3, and σ = 2, the Z-score is 1. The probability to the left is 0.84134.
Between Z = −1 and Z = 0
The standard-normal area is Φ(0) − Φ(−1) = 0.34134.
Formula
A raw score x is standardized by subtracting the population mean μ and dividing by the population standard deviation σ:
z = (x − μ) / σ
Converter and range modes use the standard normal CDF Φ. The probability between
Z1 and Z2 is Φ(Z2) − Φ(Z1).
The NIST standard-normal table lists area from 0 to Z for nonnegative Z. Add 0.5 to that area to get the cumulative probability to the left; do not mistake the table's center area for a percentile.
Standardization limits
A z-score reports distance from the entered mean in units of the entered standard deviation. Standard deviation must be positive. Converting a z-score to a normal-tail probability additionally assumes a normal model; a standardized value alone does not prove that assumption or identify an outlier without context.