IQ Percentile Calculator

What is an IQ Percentile?

An IQ percentile represents the percentage of people in a population who score lower than a given IQ score. It helps to understand where an individual's IQ score stands in relation to the general population.

The Formula

To calculate the IQ percentile, we use the standard normal distribution (z-score) and the error function (erf). The formula is:

\[Percentile = \frac{1 + erf(\frac{z}{\sqrt{2}})}{2} \times 100\%\]

Where:

• \(z = \frac{IQ - \mu}{\sigma}\)
• \(\mu\) is the mean IQ score (typically 100)
• \(\sigma\) is the standard deviation of IQ scores (typically 15)
• erf is the error function

Calculation Steps

1. Calculate the z-score using the given IQ score
2. Use the error function to calculate the area under the standard normal curve up to the z-score
3. Convert this area to a percentage to get the percentile

Example Calculation

Let's calculate the percentile for an IQ score of 120:

1. Calculate z-score:
   \[z = \frac{120 - 100}{15} = 1.33\]
2. Calculate percentile:
   \[Percentile = \frac{1 + erf(\frac{1.33}{\sqrt{2}})}{2} \times 100\% \approx 90.82\%\]

Visual Representation

This visual representation shows an IQ score of 120 on the IQ distribution curve. The vertical red line indicates the position of the IQ score, and the percentage (90.82%) represents the area to the left of this line, which is the percentile rank for this IQ score.