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sci.normal-distribution-z-score Calculator
Calculates normal distribution z-score, probability density, CDF P(X < x), and percentiles from mean and standard deviation. 68-95-99.7 rule: 68% of values within ±1σ, 95% within ±2σ, 99.7% within ±3σ — quality control limits (3σ) yield 2,700 defects per million; 6σ yields 3.4 DPMO.
Inputs
X Value
Reference formula or conversion factor shown for context.
Mu Mean
Arithmetic average. Sensitive to outliers — if your data has extreme values, the median may be more representative.
Sigma Std
Reference formula or conversion factor shown for context.
Confidence Pct
How certain you want to be that the interval contains the true value. 95% is standard. Going to 99% widens the interval.
Results
z-score
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%.
P(X < x) — area below
The computed two-dimensional surface area.
P(X > x) — area above
The computed two-dimensional surface area.
z = (x-μ)/σ
Reference formula or conversion factor shown for context.
95% CI half-width
The computed width.
68-95-99.7 rule
Reference formula or conversion factor shown for context.