How do you decide whether to headline standard deviation or IQR for a skewed metric?

## The decision is about shape, audience and downstream use Standard deviation and the interquartile range both answer "how spread out is this", but they answer for different parts of the distribution and carry different assumptions. **Standard deviation** is the root-mean-square distance from the mean. It uses every observation, it is the currency the rest of statistical practice is written in, and it inherits both the mean's efficiency on well-behaved data and the mean's fragility on skewed or contaminated data. **Interquartile range** is `Q3 - Q1`, the width of the middle 50%. It is positional, so the size of the tail does not enter, and it describes the typical band rather than an average distance. ## Rule one: match the spread to the centre A report that pairs a mean with an IQR, or a median with a standard deviation, is mixing two frames and will confuse a careful reader. Mean and standard deviation go together because both are moment-based and both are computed around the same centre. Median and IQR go together because both are positional. Pick the frame first, then take the matching pair. ## Rule two: let the shape decide the frame On approximately symmetric, light-tailed data, the two frames tell the same story and the standard deviation is preferable: it is more efficient, it uses all the data, and it feeds anything downstream. The IQR is about 1.35 standard deviations on such data, so nothing is hidden by choosing either. On strongly skewed data — latency, revenue per customer, session duration, file size, income — the frames diverge. The mean sits above the bulk of the distribution, the standard deviation is inflated by the tail, and the familiar reading of one standard deviation around the mean covers a range containing few actual observations and possibly impossible values, such as negative durations. Here the median and IQR describe what a typical unit experiences, and the standard deviation describes an average distance from a centre that is itself unrepresentative. ## Rule three: be honest about the interpretation you are implying Quoting a mean plus or minus a standard deviation invites the reader to apply the roughly 68% and 95% intuitions, which are properties of a bell-shaped distribution, not of the statistic. If the data is not shaped that way, quoting it that way is a claim you cannot support. The distribution-free fallback, Chebyshev's inequality, guarantees only that at least `1 - 1/k^2` of values lie within k standard deviations for k > 1 — at least 75% within two — which is true of everything and therefore says very little. If the strong reading is what you want your audience to take away, you owe them evidence that the shape supports it. ## Rule four: ask who consumes the number and for what - **A business or executive audience** generally wants the band a typical unit falls in. "Half of orders fall between 24 and 61 dollars" needs no statistical training and cannot be misread. - **Reliability and service-level work** wants the tail, not the middle at all, because the commitment is written on a high percentile rather than on any measure of central spread. - **Downstream statistical machinery** wants standard deviations, because that is the input those methods take. This is often the strongest argument for computing and storing the standard deviation even when it is not the number you publish. - **Quality and process monitoring** frequently wants both: a robust band for the day-to-day and a moment-based one whose sudden divergence from the robust band is itself the alarm. ## The organisational layer The reason this is a lead's question rather than an analyst's is that the cost of inconsistency exceeds the cost of choosing slightly wrong. If one dashboard reports mean and standard deviation, another median and IQR, and a third switches when the data gets ugly, then quarter-over-quarter comparisons stop meaning anything and every review reopens the same argument. The durable answer is a written definition for each headline metric, covering which centre and which spread are published, at what precision, whether a tail percentile travels with them, and what happens when the distribution's shape changes enough to invalidate the choice. Where storage is cheap, compute several summaries and publish the pair the definition names — that way changing the presentation later does not require recomputing history. Publishing both frames side by side is also legitimate and often the best option, provided the definition says which one is the headline, because a reader offered two spreads with no guidance will pick whichever supports their prior. ## Common mistakes Defending the standard deviation purely because it uses all the data, without asking whether the mean it is built on describes anything. Switching the reported measure mid-series because the new one looks better, without versioning the metric definition. Publishing an IQR while narrating it as if it were a standard deviation. And treating the choice as a matter of analyst preference rather than as part of the metric's contract with the people who read it.