Summary Statistics for One Quantitative Variable

One quantitative variable gets two families of summaries. For center, the mean $\bar{x}$ balances all the values while the median simply takes the middle one, and the difference matters the moment the data are skewed: an outlier or a long tail drags the mean and barely touches the median, which is why the median is called resistant. For spread, the standard deviation $s$ reads as the typical distance of values from their mean, and the IQR, $Q_3 - Q_1$, is the width of the middle 50% of the ordered data. The 1.5 IQR rule ($1.5 \times \text{IQR}$ beyond a quartile) and the 2 SD rule ($\bar{x} \pm 2s$) turn "unusually far out" into a computed fence.

The traps are specific. Treating the mean as if it resisted outliers, or expecting mean and median to agree in skewed data, picks the wrong summary and defends it. Misreading the standard deviation as a range, as a bound every value must obey, or as a number that is large or small without units, replaces its actual meaning: typical distance from the mean. And the IQR gets mangled mechanically: reported as an interval instead of a distance, quartiles computed with the median counted into both halves, or fences placed at $Q_3 + 1.5$ with the IQR forgotten entirely.