Checking Whether a Distribution Is Normal

A histogram can look bell-shaped and still hide real departures from normality, especially in the tails, where a small number of points are hard to judge by eye. A normal probability plot fixes this by plotting each data value against the value you'd expect at that rank if the data were perfectly normal. If the data really are normal, the points fall close to a straight line. Curves, bends, or S-shapes mean the data drift away from normal somewhere, usually in the tails or in one direction overall.

Think of the plot as a stress test for the bell shape. The mechanism: each point's position compares what you observed to what normal data would predict at that percentile. Points that match line up; points that don't, veer off. A curve bending up at the high end and down at the low end often signals skew; a curve that bows away from the line at both ends signals heavier or lighter tails than a true normal curve has. Sample size does not fix this. A large n can make a sampling distribution of a statistic approximately normal through the central limit theorem, but that is a claim about the statistic, not about the shape of the raw data, which can stay skewed or lumpy no matter how many observations you collect.

This check matters because later inference procedures, like t-tests and confidence intervals for means, assume either that the population is normal or that the sample is large enough for the central limit theorem to kick in. Checking normality with a histogram and a probability plot is how you decide which justification you actually have before trusting the inference.