Convert between z-scores, raw values and normal distribution probabilities.
A z-score (standard score) measures how many standard deviations a value lies from the mean of its distribution. z = (x − μ) / σ. A z-score of 1.96 means the value is 1.96 standard deviations above the mean, which corresponds to the 97.5th percentile. By converting raw values to z-scores you can compare observations from distributions with different scales.
Z-score
z = (x − μ) / σ
Reverse: raw value
x = μ + z·σ
Cumulative probability
Left-tail probability = Φ(z), the standard normal CDF
It means the value is exactly 2 standard deviations above the mean. In a standard normal distribution roughly 97.7 % of values lie below a z-score of 2 (left-tail), so only about 2.3 % are above it.
Yes. A negative z-score means the raw value is below the mean. z = −1 places the value at the 15.9th percentile — about 84 % of values in the population are larger.
Use a z-score when the population standard deviation is known and/or the sample size is large (n > 30 is a common rule of thumb). Use a t-score for small samples where you only have the sample standard deviation as an estimate.