Introduction to Normal Distributions

Standardized test scores often follow a predictable pattern—understanding this helps interpret performance fairly. In a histogram, bars show score ranges, and their widths are called bins. As the number of data points increases and the bins narrow, the bars begin to smooth out. Dividing the number of occurrences by the total count and bin width changes the vertical axis to probability density. This change produces a continuous curve modeled by the normal distribution.

The curve is centered around the mean, the average value. This is where the curve peaks and is balanced evenly on both sides.

The standard deviation measures how much the scores vary from the mean.

A small standard deviation means scores are tightly clustered; a larger one means they are more widely scattered.

The curve’s shape follows a probability density function that uses the mean, the standard deviation, and an exponential expression.

The area under this curve across a score range represents probability.

When the curve is integrated across all possible values, it adds up the probabilities of every possible outcome. The result is one, or 100 percent, because all possibilities are accounted for.