Positive z-score
For x=85, mean=70, and SD=10, z=(85−70)/10=1.5.
Calculate a z-score, solve for a raw value, mean, or standard deviation, and convert between z-scores and normal-distribution percentiles.
A z-score expresses how far a value lies from its mean in standard-deviation units. Standardization creates a scale with mean zero and standard deviation one.
Positive scores are above the mean, negative scores are below it, and zero is at the mean. These directions are mathematical, not judgments about whether a value is desirable.
z = (x − μ) / σx = μ + zσμ = x − zσσ = (x − μ) / zFor x=85, mean=70, and SD=10, z=(85−70)/10=1.5.
For x=60, mean=70, and SD=5, z=(60−70)/5=−2.
For z=1.5, mean=70, and SD=10, x=70+1.5×10=85.
The sign indicates whether a value is above or below its mean.
The absolute value is the distance from the mean in standard-deviation units.
A z-score always standardizes a value, but normal percentile interpretation requires a normal-distribution assumption.
The displayed lower and upper tails use the standard normal distribution. They are not universal empirical percentiles for arbitrary datasets.
Inverse percentile mode accepts values strictly between 0 and 1, or 0% and 100%, because the endpoints correspond to infinite z-scores.
Normal probabilities use the Abramowitz–Stegun error-function approximation. Inverse percentiles use the Acklam rational approximation with a refinement step.
It is the signed distance from a mean measured in standard deviations.
Subtract the mean from the raw value and divide by the positive standard deviation.
The value lies above the mean.
The value lies below the mean.
The value equals the mean.
Yes. Z-scores are not restricted to −3 through 3.
Use x=μ+zσ.
Under a standard normal model, its lower-tail percentile is approximately 84.134%.
A normal percentile assumes a normal distribution; arbitrary data may have different empirical percentiles.
No z-score can be calculated by dividing by a zero standard deviation.