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Standard Normal Probability Calculator

Work out standard normal probability instantly with clear inputs, formula shown and shareable results.

About Standard Normal Probability Calculator

The standard normal distribution has mean 0 and standard deviation 1. Its cumulative distribution Φ(z) gives the area to the left of z, and every other region follows by symmetry and subtraction.

Formulas

Standard normal CDF

Φ(z) = ½[1 + erf(z/√2)]

Symmetry

Φ(−z) = 1 − Φ(z)

Worked example

With the default values below, Standard Normal Probability Calculator returns Probability for the chosen region of 0.9750021739. Change any input above and every figure here is recalculated in your browser.

Inputs used

z score
1.96
Which regionone of Left of z, Right of z, Between −z and z, Outside ±z
Left of z

Results

Probability for the chosen region
0.9750021739
P(Z < z)
0.9750021739
P(Z > z)
0.0249978261
Density φ(z)
0.0584409443

Step by step

  1. 1Values used
  2. 2Standard normal CDF
  3. 3Symmetry
  4. 4Probability for the chosen region= 0.9750021739
  5. 5P(Z < z)= 0.9750021739
  6. 6P(Z > z)= 0.0249978261
  7. 7Density φ(z)= 0.0584409443

Assumptions and limits

  • The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is P(Z < 1.96)?

About 0.975, which is why ±1.96 brackets the central 95% of a normal distribution.

How do I use it for non-standard normals?

Standardise first: z = (x − μ)/σ, then read the same table.

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