6.23: The Bell Curve

The normal probability distribution, often depicted as a symmetrical, bell-shaped curve, is fundamental in statistics and the study of natural phenomena. This pattern, famously described by mathematician Carl Friedrich Gauss, shows how data points are distributed around a central mean, with most values near the average and fewer observations occurring as they deviate further from it.

This pattern applies to many human characteristics beyond intelligence, such as height. For example, if you measured the heights of a large group of women, most would fall within the average range, while fewer would represent extreme heights. The normal curve is mathematically defined, smooth, and continuous, peaking at the mean value and tapering symmetrically as it approaches the extremes. However, the curve's tails never touch the X-axis, representing that extreme, rare values are still possible.

Features of the Normal Distribution

In the context of intelligence testing, many tests are designed to follow a normal distribution, with a mean score of 100 and a standard deviation (SD) of 15. The SD measures how much scores deviate from the mean. In this distribution, approximately 68% of scores fall within one SD of the mean, ranging from 85 to 115. These scores are considered to represent the average intelligence range for the population.

Application in Intelligence Testing

Beyond the average range, more extreme scores are found. For instance, scores that fall more than two SDs above or below the mean represent about 2.2% of the population on each side. A score of 130 or higher, two SDs above the mean, indicates superior intelligence. Conversely, a score of 70 or lower, two SDs below the mean, may indicate intellectual disability. This symmetrical distribution of intelligence scores provides a statistical framework for understanding variations in cognitive ability within a population.