Measures of Central Tendency
Three ways to name the “typical” value — and the shape of the data decides which one to trust.
The single value that best represents a dataset. The mean is the arithmetic average (best for symmetric data, but sensitive to outliers); the median is the middle value (robust, best for skewed data); the mode is the most frequent value. Always pair a centre with a measure of spread — SD/variance go with the mean, IQR/range with the median.
On skewed data the tail drags the mean outward, while the median stays near the typical value.
A right-skewed distribution — most values low, a long tail of high values. The three measures separate: the mode sits at the peak, the median just to its right, and the mean is dragged furthest out toward the tail by the outliers.
When the three lines coincide the data are symmetric and the mean is fine. When they fan apart the data are skewed: the further the mean drifts from the median, the worse the skew, and the more the median should be your headline number. For a right skew the order is mode < median < mean.
Quoting the wrong centre misleads. ED length of stay, waiting times and costs are right-skewed by a few very long stays, so a mean overstates the typical patient’s experience. Spotting which measure a paper used — and whether it fits the data’s shape — is a basic appraisal reflex.
Mean= average · symmetric data · pair with SDMedian= middle value · skewed/outliers · pair with IQRMode= most frequent · rarely useful for continuous data
Everyday ED example — length of stay is the classic right-skewed metric: most patients are seen and discharged quickly, but a long tail of admitted or “boarding” patients stretches the top end. The honest summary is the median (IQR); a quoted “mean LOS” is inflated by those long-stay outliers and flatters the department. If a paper reports a mean ± SD for length of stay, cost or waiting time, suspect the distribution was never checked — the median almost certainly tells a different story.
- Using the mean on skewed or outlier-laden data — the median is the truer “typical” value.
- Reporting a centre with no measure of spread (SD/variance for the mean, IQR/range for the median).
- Treating the mode as meaningful for continuous data — it rarely is.
Quick check
A billionaire walks into a bar — what happens to the mean versus the median income of the drinkers?
Answer: The mean leaps up (one huge outlier drags the average), while the median barely moves — the median is robust to outliers, which is exactly why it suits skewed data.
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