Calculator Inputs
Compute a percentile from a value, or recover a value from a percentile.
Distribution Plot
The graph shades the selected probability region after each calculation.
Formula used
For a normal random variable with mean μ and standard deviation σ, the standardized score is:
z = (x − μ) / σ
The left-tail percentile is the standard normal cumulative distribution value:
P(X ≤ x) = Φ(z)
The right-tail probability is 1 − Φ(z). The central area within ±|z| is 2Φ(|z|) − 1. The two-tail outside area is 2(1 − Φ(|z|)).
When reversing from percentile to value, first find the inverse standard normal score z = Φ−1(p), then compute:
x = μ + zσ
How to use this calculator
- Choose Value to percentile when you already know the observed score.
- Enter the normal distribution mean and standard deviation.
- Select the probability view that matches your interpretation goal.
- For inverse mode, enter the desired percentile percentage.
- Press Calculate to show the result section above the form.
- Review the percentile, z-score, probabilities, and derived cutoff value.
- Use the CSV or PDF buttons to save the current result.
Example data table
These examples assume a normal distribution with mean 100 and standard deviation 15.
| Case | Input | z-score | Left-tail percentile | Interpretation |
|---|---|---|---|---|
| Lower score | x = 85 | -1.0000 | 15.87% | About 15.87% of values fall at or below 85. |
| Average score | x = 100 | 0.0000 | 50.00% | The mean sits exactly at the fiftieth percentile. |
| Higher score | x = 118 | 1.2000 | 88.49% | Roughly 88.49% of values fall at or below 118. |
| Target percentile | p = 90% | 1.2816 | 90.00% | The matching score is approximately 119.22. |
Frequently asked questions
1. What percentile does this calculator return?
It returns the selected probability view for a normal distribution. In standard left-tail mode, it shows the percentage of values at or below the entered score.
2. What is the difference between percentile and percentage?
A percentage is a proportion out of one hundred. A percentile ranks a value relative to a distribution, showing how much of the population lies below it.
3. Can I find a score from a known percentile?
Yes. Switch to percentile-to-value mode, enter the target percentile, and the calculator uses the inverse normal function to estimate the corresponding score.
4. Why do I need the standard deviation?
The standard deviation controls spread. Without it, the calculator cannot standardize your value, build the z-score, or determine how extreme the score is.
5. What happens if the standard deviation is zero?
A zero or negative standard deviation makes the model invalid. The page blocks calculation and asks for a positive spread value instead.
6. When should I use right-tail probability?
Use right-tail probability when you care about values greater than the observed score, such as exceedance risk, top performers, or unusually large measurements.
7. Does this calculator work for non-normal data?
Not reliably. It assumes the underlying variable follows a normal distribution. Strong skewness or heavy tails can make percentile estimates misleading.
8. Why does the chart shade different regions?
The shaded region matches your selected probability view. It helps you visually connect the numeric answer to the underlying normal curve and tail area.