Standard Normal Calculator Formula

Understand the math behind the standard normal calculator. Each variable explained with a worked example.

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Formulas Used

PDF phi(z)

pdf = (1 / sqrt(2 * pi)) * pow(e, -0.5 * pow(z, 2))

PDF phi(-z)

symmetric_pdf = (1 / sqrt(2 * pi)) * pow(e, -0.5 * pow(z, 2))

z-squared

z_squared = pow(z, 2)

Variables

VariableDescriptionDefault
zZ-Value1.96

How It Works

Standard Normal Distribution

PDF Formula

phi(z) = (1 / sqrt(2*pi)) * e^(-z^2/2)

The standard normal distribution has mean 0 and standard deviation 1. It is the reference distribution for z-scores and hypothesis testing. The PDF is symmetric around z=0, meaning phi(z) = phi(-z). Common critical values: z = 1.645 (90%), 1.96 (95%), 2.576 (99%).

Worked Example

Find the PDF value at z = 1.96 (the 95% critical value).

z = 1.96
  1. 01phi(1.96) = (1 / sqrt(2*pi)) * e^(-1.96^2 / 2)
  2. 02= 0.39894 * e^(-1.9208)
  3. 03= 0.39894 * 0.14634
  4. 04= 0.05844

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