On a normal Q-Q plot of a raw sample, what do an S-shape and a single bent tail mean?

## How the plot is built A quantile is a value below which a given proportion of a distribution falls: the 0.9 quantile is the point with 90% of the mass beneath it. A normal Q-Q plot pairs the quantiles of your sample with the quantiles a normal distribution would have at the same positions. Concretely: sort the n observations; the i-th smallest is treated as the empirical quantile at roughly the (i - 0.5)/n position; look up the standard normal value at that same position; plot the pair. By convention the sample quantiles go on the vertical axis and the theoretical ones on the horizontal. A reference line is drawn through the points - commonly through the first and third quartile pair - so that a sample whose shape matches a normal falls along it regardless of its mean and standard deviation. The plot is about shape, not about location or scale. ## Reading the four canonical departures **Heavy tails.** If the sample's extremes are more extreme than a normal's, the smallest points are more negative than predicted (below the line) and the largest are more positive than predicted (above the line). The two ends bend in opposite vertical directions, producing the classic S-shape. This is what a t-like distribution with few degrees of freedom, or a contaminated sample, looks like. **Light tails.** The reverse S: the low end sits above the line and the high end below it, because the sample's extremes are milder than a normal's. Bounded or truncated measurements often look like this. **Right skew.** The long tail is on the high side, so the largest values overshoot dramatically and the points at the right end climb well above the line. On the low side the distribution is compressed relative to a normal, so those points also sit above the line. Both ends deviate the same way and the picture reads as a single upward-curving arc, dominated by the right tail. Income, session duration, revenue per customer and waiting times routinely look like this. **Left skew.** The mirror image: both ends below the line, with the lower tail dropping furthest, giving a downward-curving arc. ## What is not a departure Order statistics are noisy. With n = 25 a perfectly normal sample produces visible wandering, worst at the two extreme points where a single draw fixes the position. The habit that separates a competent reader from a nervous one is to look for a systematic, monotone bend rather than local wiggle, and, when uncertain, to compare against a handful of simulated normal samples of the same size to calibrate the eye. Discrete or rounded data produce a staircase of horizontal runs; that is a property of the measurement, not evidence about the underlying shape. A handful of points detached at one end is an outlier signal, and is often more actionable than the overall shape. ## Why plot rather than test A plot answers the question you actually have. It shows the direction of the departure (skew versus tail weight), its magnitude, and whether it is driven by a few points or by the bulk. A formal normality test compresses all of that into a single verdict about a hypothesis nobody believes exactly, and the verdict is driven heavily by sample size. Knowing that the departure is one long right tail suggests a different response - a transformation, a different summary, more caution about the mean - than knowing that both tails are heavy, which is mostly a warning about outlier influence and power. ## Turning the reading into a decision The follow-up question is always: does this shape threaten the specific procedure? For a comparison of means at a healthy sample size, moderate skew is largely absorbed by the central limit theorem and the plot's message is mainly about outliers. For a small sample, the raw shape matters directly, because the central limit theorem has not had room to work. For anything that depends on the tails themselves - a high percentile, a prediction interval, a threshold-crossing probability - the tail behaviour the plot shows is exactly the thing being estimated, and a heavy-tail S-shape is a serious warning that a normal-based answer will understate the extremes. Say all of that in an interview and you have shown the skill being probed: not memorising which curve is which, but connecting a shape to a consequence.