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math.normal-distribution-z Calculator
Calculates the normal distribution PDF and CDF at a given z-score, and the probability of a value falling within an interval. The 68-95-99.7 rule (±1, ±2, ±3 standard deviations) is derived from these integrals.
Inputs
Z
Reference formula or conversion factor shown for context.
Mu
Arithmetic average. Sensitive to outliers — if your data has extreme values, the median may be more representative.
Sigma
Reference formula or conversion factor shown for context.
X
Reference formula or conversion factor shown for context.
Results
PDF at z
The value at the specified point or condition.
CDF approx P(Z<=z)
Reference formula or conversion factor shown for context.
two-tail p-value
Probability of observing this result (or more extreme) if the null hypothesis were true. Below 0.05: conventionally significant. Below 0.01: strong evidence. A low p-value does not measure the size or practical importance of the effect.
z-score for raw x
Number of standard deviations a value is from the mean. |z| below 2: within normal range. |z| above 2: in the outer 5% of the distribution. |z| above 3: in the outer 0.3%.
z=1.96 two-tail 5%
Reference formula or conversion factor shown for context.
z=2.576 two-tail 1%
Reference formula or conversion factor shown for context.