Calculate Percentile from Mean & Standard Deviation
Calculate the z-score and percentile for an observed score, or work backwards from a z-score.
Enter Your Data (x, μ, σ)
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How to Use This Calculator
This **percentile calculator** helps you understand where a specific score stands in a normal distribution. Here are the steps to get started:
- **Standard Mode:** Enter the **Observed Score ($x$)**, the **Mean ($\mu$)**, and the **Standard Deviation ($\sigma$)** of your dataset.
- **Z-score Mode:** Alternatively, you can click the **"Swap: Enter z instead"** button to directly input a z-score if you already know it.
- **Choose an Area:** Select the type of percentile area you want to find: **Left-tail** (percentage below your score), **Right-tail** (percentage above your score), or **Two-tailed** (area in both tails).
- **Set Precision:** Choose the number of **decimal places** for your result.
- **Calculate:** Click the **"Calculate Percentile"** button to view the results, including the z-score, percentile, and a visual representation on the bell curve.
How it Works: The Formulas
The calculator uses the standard normal distribution model to find a percentile from a given score. First, it converts your observed score ($x$) into a z-score using the following formula:
$z=\dfrac{x-\mu}{\sigma}$
Once the z-score is calculated, the percentile is determined by finding the cumulative probability of that z-score using the standard normal Cumulative Distribution Function ($\Phi$).
- **Left-tail Percentile:** The probability of a score being less than $z$ is $\Phi(z) \times 100\%$.
- **Right-tail Percentile:** The probability of a score being greater than $z$ is $(1 - \Phi(z)) \times 100\%$.
- **Two-tailed Area:** This represents the area in both tails beyond the z-score, calculated as $2 \cdot \min(\Phi(z), 1 - \Phi(z)) \times 100\%$.
This calculator uses an accurate, erf-based approximation for the standard normal CDF to ensure precise results.
Frequently Asked Questions (FAQs)
What is a percentile?
A percentile is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations fall. For example, the 20th percentile is the value (or score) below which 20% of the observations may be found.
What is a z-score?
A z-score, also known as a standard score, indicates how many standard deviations an element is from the mean. A z-score of 0 means the score is the same as the mean, a positive z-score means it's above the mean, and a negative z-score means it's below the mean.
When should I use the two-tailed option?
The two-tailed option is typically used for hypothesis testing to find the probability of a value being "extreme" in either direction (significantly higher or significantly lower than the mean). For example, finding the probability that a value is more than 2 standard deviations away from the mean, in either direction.