Statistics
Normal Distribution Calculator
Convert raw scores to z-scores and compute tail or interval probabilities for any normal distribution.
Calculate z-scores and tail probabilities for the normal distribution.
Normal CDF
Φ(z) = ½ [1 + erf(z / √2)]
The calculator leverages an error-function approximation to deliver accurate probabilities and densities.
How to use
- Enter the mean μ and standard deviation σ.
- Provide one or two values depending on whether you want a single tail or a between probability.
- Read the z-score, PDF value, and probability.
Example
Input: μ = 0, σ = 1, mode = P(X ≤ a), a = 1.2
Output: z ≈ 1.2, P ≈ 0.884
Student-friendly breakdown
This walkthrough emphasizes the most searched ideas for Normal Distribution Calculator: normal distribution calculator, z score calculator, normal cdf calculator, probability between two z scores. Start with the formula above, then follow the guided steps to double-check your work. For quick revision, highlight the givens, plug into the equation, and finish by verifying your units.
Need more support? Use the links below to open the long-form guide, browse additional examples, or hop into adjacent calculators within the same topic — each one is a quick way to double-check your work or handle a related question without starting from scratch.
Deep dive & study plan
Normal Distribution Calculator: Evaluates z-scores and normal probabilities. It's built around normal distribution, z-score, Gaussian, so you can go from a raw question to a checked answer without switching tools.
The math behind it: The calculator leverages an error-function approximation to deliver accurate probabilities and densities. The core relationship is Φ(z) = ½ [1 + erf(z / √2)], shown above the calculator so you can see exactly how your inputs turn into the result.
To use it well: (1) Enter the mean μ and standard deviation σ. (2) Provide one or two values depending on whether you want a single tail or a between probability. (3) Read the z-score, PDF value, and probability. Keep your units consistent as you go, and re-run a case you already know the answer to — it's the fastest way to catch a typo before it throws off a result you're relying on.
Worked example: entering μ = 0, σ = 1, mode = P(X ≤ a), a = 1.2 returns z ≈ 1.2, P ≈ 0.884. Try swapping in your own numbers next, especially a case you're unsure about, before you use this for something that matters.
Quick retention checklist
- Speak the formula aloud (or annotate it) so the relationships stick.
- Write each step in your own words and compare with the numbered list above.
- Swap in new numbers for the Example to make sure the calculator (and your logic) handles edge cases.
- Check at least one related calculator below — it's the fastest way to confirm your numbers still line up from a different angle.
FAQ & notes
Can it compute two-tailed probabilities?
Use the between mode with symmetric bounds (e.g., ±z) to obtain central or tail probabilities.
What happens if σ = 0?
A zero or negative standard deviation is invalid for a normal distribution, so the calculator flags the input.
What formula does the Normal Distribution Calculator use?
The calculator leverages an error-function approximation to deliver accurate probabilities and densities.
How do I use the Normal Distribution Calculator?
Enter the mean μ and standard deviation σ. Provide one or two values depending on whether you want a single tail or a between probability. Read the z-score, PDF value, and probability.