Normal Distribution Calculator Formula

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

Use the Normal Distribution Calculator

Formulas Used

Z-Score

z_score = z

PDF f(x)

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

Deviations from Mean

deviation = abs(z)

Variance

variance = pow(sigma, 2)

Variables

VariableDescriptionDefault
xValue (x)75
muMean (mu)70
sigmaStandard Deviation (sigma)10
zDerived value= (x - mu) / sigmacalculated

How It Works

How to Use the Normal Distribution

PDF Formula

f(x) = (1 / (sigma * sqrt(2*pi))) * e^(-0.5 * ((x - mu)/sigma)^2)

The normal (Gaussian) distribution is the most important continuous distribution in statistics. It is defined by its mean (mu) and standard deviation (sigma). The z-score standardizes any value to units of standard deviations from the mean.

Worked Example

Test scores have mean 70 and SD 10. Find the z-score and PDF for a score of 75.

x = 75mu = 70sigma = 10
  1. 01Z = (75 - 70) / 10 = 0.5
  2. 02The score is 0.5 standard deviations above the mean
  3. 03PDF = (1 / (10 * sqrt(2*pi))) * e^(-0.5 * 0.25)
  4. 04= 0.03989 * 0.8825 = 0.03521
  5. 05Variance = 10^2 = 100

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