Convert a value to a z-score and find the left-tail and right-tail probabilities under a normal curve.
| z | – |
| Left-tail P(X ≤ x) | – |
| Right-tail P(X > x) | – |
Use this calculator for a normal (Gaussian) distribution with mean μ and standard deviation σ. It reports how many standard deviations x lies from the mean, the probability of a value at most x, and the probability of a value above x.
The formulas are: z = (x − μ) / σ, left-tail = Φ(z), and right-tail = 1 − Φ(z), where Φ is the standard normal cumulative distribution. Enter x, μ, and σ in the same units. For the familiar 95% cutoff, x = 1.96, μ = 0, and σ = 1 gives z = 1.96, left ≈ 0.975, and right ≈ 0.025.
The CDF uses a standard numerical approximation of the error function, accurate enough for ordinary z-tables. This is not a t distribution; use t for small samples with unknown σ.
Value (x): Observed or cutoff value.
Mean (μ): Centre of the normal curve.
Standard deviation (σ): Spread; must be positive.
z: Standard score, (x − μ) / σ.
Left-tail P(X ≤ x): Cumulative probability up to x.
Right-tail P(X > x): Upper-tail probability, 1 minus the left tail.
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