Normal & Skewed Distributions

A normal (Gaussian) distribution is symmetric and bell-shaped: mean = median = mode, with about 68 / 95 / 99.7% of values within 1 / 2 / 3 standard deviations. Skewed data are asymmetric: a right (positive) skew has a long right tail, dragging the mean above the median.

Two distributions side by side. In the symmetric normal curve the mean, median and mode sit together. In the right-skewed curve the bulk of values cluster low with a long tail of high values; the median stays at the peak while the mean is pulled out towards the tail.

The shape tells you the right summary. Symmetric → report mean ± SD and use parametric tests. Asymmetric/skewed → report median + IQR and use non-parametric tests. The further the mean drifts from the median, the more skewed the data and the more misleading a mean becomes.

Most ED time variables — length of stay, waiting times, door-to-needle — are right-skewed by a few very long stays. Quoting a mean length of stay overstates the typical patient’s experience, and parametric tests on skewed data can mislead. Checking the distribution first is a basic appraisal discipline.

• Normal: mean = median = mode, 68/95/99.7%
• Right skew: mean > median
• Skewed → median + IQR, non-parametric

Everyday ED example — length of stay and waiting times are textbook right-skewed: most patients leave quickly, but a long tail of admitted or “boarding” patients stretches the upper end. The honest summary is the median (IQR); a “mean LOS” is inflated by those long-stay outliers and flatters the department. If a paper reports a mean ± SD for length of stay, suspect the distribution was never checked — and treat any parametric comparison with caution.

• Using the mean on skewed data — the tail inflates it away from the typical value.
• Assuming normality without checking (a histogram, or mean vs median).
• Forgetting that a log or other transformation can normalise skewed data so parametric tests become valid.