Distributions Concepts

The normal distribution (bell curve) is a symmetric, bell-shaped probability distribution where most data clusters around the mean, with probabilities decreasing symmetrically toward the tails. It is defined by two parameters: the mean and the standard deviation.

"Heights, test scores, measurement errors - many real phenomena cluster around an average with decreasing frequency toward extremes. The bell curve captures this pattern: most values are 'average,' few are extreme."

Why it matters: The normal distribution is the foundation of statistical inference. Many statistical tests assume normality.

The empirical rule (also called the 68-95-99.7 rule) states that for a normal distribution, approximately 68% of data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and roughly 99.7% falls within three standard deviations.

"Most data clusters near the center of a bell curve; the further from the mean, the rarer the value."

Why it matters: The empirical rule provides a quick way to estimate probabilities and understand spread in normal distributions without a z-table. It is the basis for z-scores, quality control limits, and the concept of unusual values in statistics.

A measure of how asymmetric a probability distribution is around its mean — positive skew tails right, negative skew tails left.

"A right-skewed distribution has a long tail to the right (a few very large values); left-skewed has a long tail to the left."

Why it matters: Skewness tells you whether the mean or median is a better measure of center and whether standard statistical methods (which often assume symmetry) are appropriate for your data.