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Statistics

Normal PDF Calculator

Probability density at x. Free online Normal PDF Calculator. Calculate normal pdf online — fast, accurate, mobile-friendly, no signup needed.

f(x)
0.398942

Derivation

  1. ├── 01Givenx = 0, mu = 0, sigma = 1
  2. ├── 02Formula1 / (n × √(2 × π)) × exp(-.5 × ((t-a) / n)^(2))
  3. └── 03Compute f(x)0.398942
Did you know?

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§01What is

Understanding the Normal PDF Calculator

The Normal PDF Calculator computes f(x) from 3 inputs: x, μ, σ. Probability density at x.

Statistics is how we make sense of noisy real-world data. Whether you’re analysing survey results, sports scores, or business metrics, a statistics calculator gives you the exact formula-based answer so you can focus on the interpretation. The Normal PDF Calculator sits in that toolkit — it probability density at x. Enter your numbers above and the result updates instantly; every step of the math is shown in the Derivation panel so you can see exactly how the answer was reached.

§02The Formula

How it’s calculated

1 / (n × √(2 × π)) × exp(-.5 × ((t-a) / n)^(2))

Where

x
x
mu
μ
sigma
σ
§03Practical Example

Step-by-step walkthrough

Scenario

Apply the formula to a realistic set of inputs: x = 0, μ = 0, σ = 1.

  1. 01Start by noting the input — x: 0.
  2. 02Start by noting the input — μ: 0.
  3. 03Start by noting the input — σ: 1.
  4. 04Substitute these values into the formula: 1 / (n × √(2 × π)) × exp(-.5 × ((t-a) / n)^(2))
  5. 05Compute f(x): the calculator returns 0.398942.
  6. 06Cross-check the answer by opening the Derivation panel above — every line of math is shown so you can follow the computation end-to-end.
§04Variants

Common Normal PDF Problems

The formula gets rearranged depending on which variable you need. Here are the patterns you’ll run into in the real world — find the one that matches your problem and follow the worked steps.

01 · PATTERN

σ halved

sigma = 0.5 (from 1)

Keep every other input at its default and halve the σ. See how f(x) responds.

  1. 01New σ: 0.5
  2. 02Baseline f(x): 0.398942
  3. 03New f(x): 0.797885
  4. 04f(x) increases by 100% → use this sensitivity to plan for real-world variation.
02 · PATTERN

σ doubled

sigma = 2 (from 1)

Keep every other input at its default and double the σ. See how f(x) responds.

  1. 01New σ: 2
  2. 02Baseline f(x): 0.398942
  3. 03New f(x): 0.199471
  4. 04f(x) decreases by 50% → use this sensitivity to plan for real-world variation.
§05FAQ

Frequently asked questions

Yes. The calculator implements the standard formula as documented and returns exact floating-point results. No approximations are used unless noted in the formula.
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