Convert raw scores into standard Z-scores, calculate normal distribution tail probabilities and percentiles, evaluate areas between two scores, and perform inverse Z-score calculations.
The score 115 is 1.00 standard deviations above the mean (μ = 100, σ = 15).
Measures the exact number of standard deviations the data point x lies above or below the mean.
The proportion of a normal population scoring below this value.
The proportion of observations scoring higher than x.
Symmetric probability mass centered within ±z standard deviations.
A Z-score (standard score) measures the exact number of standard deviations by which a raw value x lies above or below the mean μ of a distribution. Converting raw scores into standard Z-scores maps them directly onto the standard normal distribution (Z ~ N(0, 1)).
A Z-score of 0 is precisely equal to the mean (50th percentile). z = +1.0 is ~84.13th percentile, z = +2.0 is ~97.72th percentile, and z = +3.0 is ~99.87th percentile.
Z-scores allow fair comparison between completely different metrics (such as comparing a student’s score on an SAT test vs an ACT test).
Find the Z-score and percentile for an IQ test score of 120 (where μ = 100 and σ = 15).
Yes! A negative Z-score indicates that the observed value falls below the mean of the distribution.
Common scientific practice treats observations with |z| > 2 as notable/uncommon (top/bottom 5%) and |z| > 3 as extreme statistical outliers (top/bottom 0.3%).
Understanding your calculation
A z-score equals (value − mean) / standard deviation and describes distance from the mean in standard-deviation units.
A value of 120 with mean 100 and standard deviation 10 has a z-score of 2.
The standard deviation must be positive. Interpreting a z-score as a normal-distribution percentile requires a suitable distributional assumption.