Normal Distribution Calculator

Normal Distribution is evaluated from Mean, Standard Deviation and Value x₁. The calculation reports P, P and P.

Results

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About the Normal Distribution Calculator

Normal Distribution is treated here as a quantitative relation between Mean, Standard Deviation, Value x₁ and Value x₂ and P, P, P and z₁ = / sigma.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
P(X < x) = Phi((x - mu)/sigma)
P(X > x) = 1 - Phi((x - mu)/sigma)
P(a < X < b) = Phi((b - mu)/sigma) - Phi((a - mu)/sigma)

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

P(X < x) = Phi((x - mu)/sigma)
P(X > x) = 1 - Phi((x - mu)/sigma)
P(a < X < b) = Phi((b - mu)/sigma) - Phi((a - mu)/sigma)

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: IQ Score Probability

Inputs

mean_val: 100 sd_val: 15 x1: 130
P: 0.97725. P: 0.02275. z₁ = / sigma: 2

With Mean = 100, Standard Deviation = 15 and Value x₁ = 130 as the stated inputs, the result is P = 0.97725, P = 0.02275 and z₁ = / sigma = 2. Each value corresponds to the declared output fields.

Example 2: Bolt Diameter Tolerance

Inputs

mean_val: 10 sd_val: 0.05 x1: 9.9 x2: 10.1
P: 0.02275. P: 0.97725. P: 0.9545. z₁ = / sigma: -2

With Mean = 10, Standard Deviation = 0.05, Value x₁ = 9.9 and Value x₂ = 10.1 as the stated inputs, the result is P = 0.02275, P = 0.97725 and P = 0.9545. Each value corresponds to the declared output fields.

Example 3: Height Distribution — US Women

Inputs

mean_val: 64 sd_val: 2.8 x1: 70
P: 0.983938. P: 0.016062. z₁ = / sigma: 2.1429

With Mean = 64, Standard Deviation = 2.8 and Value x₁ = 70 as the stated inputs, the result is P = 0.983938, P = 0.016062 and z₁ = / sigma = 2.1429. Each value corresponds to the declared output fields.

Example 4: Exam Grading — Score in Range

Inputs

mean_val: 72 sd_val: 11 x1: 60 x2: 84
P: 0.137656. P: 0.862344. P: 0.724687. z₁ = / sigma: -1.0909

With Mean = 72, Standard Deviation = 11, Value x₁ = 60 and Value x₂ = 84 as the stated inputs, the result is P = 0.137656, P = 0.862344 and P = 0.724687. Each value corresponds to the declared output fields.

Common Use Cases

  • Find probability that a measurement falls in a range
  • Calculate area under the bell curve
  • Convert between z-scores and probabilities
  • Determine tail probabilities for hypothesis testing