Use this normal curve calculator to quickly compute probabilities, z-scores, and percentile cutoffs for a normal distribution. Enter your mean and standard deviation, choose a calculation type, and get an instant result.
What is the normal curve?
The normal curve (or normal distribution) is a bell-shaped probability distribution commonly used in statistics, finance, psychology, quality control, and many other fields. It is symmetric around the mean, and the spread is controlled by the standard deviation.
If your data are approximately normal, this calculator helps you answer practical questions like:
- What percent of values are less than a cutoff?
- What percent are greater than a threshold?
- What is the probability of landing between two values?
- What score corresponds to a target percentile?
How to use this normal curve calculator
1) Enter distribution parameters
Set the mean (μ) and standard deviation (σ) for your distribution. For a standard normal distribution, use μ = 0 and σ = 1.
2) Choose a calculation type
- Left Tail: Computes P(X ≤ x)
- Right Tail: Computes P(X ≥ x)
- Between: Computes P(a ≤ X ≤ b)
- Outside: Computes P(X ≤ a or X ≥ b)
- Z-Score from x: Computes z = (x - μ)/σ
- x from Percentile: Finds the value with a chosen percentile rank
3) Click Calculate
The tool displays both the decimal probability and percentage interpretation, along with helpful details such as converted z-scores.
Quick interpretation guide
For a normal distribution:
- About 68% of values fall within 1 standard deviation of the mean (μ ± 1σ).
- About 95% fall within 2 standard deviations (μ ± 2σ).
- About 99.7% fall within 3 standard deviations (μ ± 3σ).
This is often called the 68–95–99.7 rule and is a useful reasonableness check for your results.
Examples
Example 1: Test scores
Suppose scores are normal with mean 70 and standard deviation 10. To find the proportion scoring 85 or below, choose Left Tail, enter x = 85, and calculate.
Example 2: Manufacturing tolerance
If part diameters follow N(50, 2), and acceptable values are between 47 and 53, choose Between with a = 47 and b = 53 to estimate yield within tolerance.
Example 3: Percentile cutoff
If you need the 90th percentile cutoff for a normal process, choose x from Percentile, enter 90, and the calculator returns the corresponding value.
Common mistakes to avoid
- Using a standard deviation of 0 (invalid for any normal distribution).
- Confusing left-tail and right-tail probabilities.
- Entering percentile as a decimal (0.95) when the input expects a percent (95).
- Using normal methods for highly skewed data without checking assumptions.
Final note
This calculator is ideal for quick probability work, classroom practice, and sanity checks. For advanced statistical modeling, always pair numerical output with context, diagnostics, and domain knowledge.