Calculator
Normal distribution plot
The shaded region updates with your selected calculation mode and parameters.
The chart shows the probability density function. Shaded areas represent the requested probability region.
Example data table
Use these sample scenarios to verify behavior and understand typical outputs.
| Scenario | Mean | Standard Deviation | Calculation | Result |
|---|---|---|---|---|
| Standard normal left of 1.00 | 0 | 1 | P(X ≤ 1.00) | 0.841345 |
| Standard normal between -1 and 1 | 0 | 1 | P(-1 ≤ X ≤ 1) | 0.682689 |
| Exam scores above 130 | 100 | 15 | P(X ≥ 130) | 0.022750 |
| Central 95% bounds | 50 | 10 | Symmetric interval | 30.40 to 69.60 |
Formula used
The standardized score is calculated with z = (x - μ) / σ, where μ is the mean and σ is the standard deviation.
The cumulative probability for a normal variable is P(X ≤ x) = Φ((x - μ)/σ). This calculator evaluates Φ using an error-function approximation.
Between-area probability is P(a ≤ X ≤ b) = Φ((b - μ)/σ) - Φ((a - μ)/σ). Outside-area probability is the complement of that interval.
For symmetric central areas, the calculator finds z* from the inverse normal distribution, then returns bounds using μ ± z*σ.
How to use this calculator
- Enter the distribution mean and standard deviation.
- Select left tail, right tail, between, outside, or central mode.
- Provide a single value, interval bounds, or central percentage.
- Choose the number of decimal places you want displayed.
- Click Calculate Area to place the result above the form.
- Review the summary cards, detailed table, and shaded plot.
- Download the results as CSV or PDF when needed.
Frequently asked questions
1. What does the calculator actually measure?
It measures probability areas under a normal curve. You can evaluate left tails, right tails, intervals, outside regions, and symmetric central coverage from a chosen mean and standard deviation.
2. When should I use left-tail mode?
Use left-tail mode when you need the probability of a value being at or below a threshold. It is common in percentile work, quality control, and risk cutoff interpretation.
3. What is the difference between between and outside?
Between mode returns the probability inside two bounds. Outside mode returns the probability beyond those bounds. Together, these two results always add up to one.
4. Why is the z score useful?
The z score shows how many standard deviations a value sits from the mean. It standardizes different distributions, making comparisons and probability lookups much easier.
5. Can I use nonstandard normal distributions?
Yes. Enter any mean and any positive standard deviation. The calculator automatically standardizes the value internally before evaluating the probability area.
6. How does central area mode help?
Central mode is useful when you know the confidence or retained coverage you want. It computes symmetric lower and upper bounds that capture that chosen percentage.
7. Are the results exact?
The results are numerically very close for practical work. This page uses stable approximations for the error function and inverse normal calculations, which are standard in statistical computing.
8. What can I export from this page?
You can export the calculated result summary as CSV and PDF. The example scenarios table also includes a dedicated CSV download button for quick reference sharing.