Range and Interquartile Range

Not all spread is best captured by the standard deviation. When data are skewed or contain outliers, two simpler measures are clearer and more honest: the range and the interquartile range (IQR). The range is the simplest possible measure of spread: the largest value minus the smallest. It is easy to understand but fragile, because it depends entirely on the two most extreme values, exactly the points most likely to be errors or outliers. A single mistyped lab value can blow the range wide open, so the range is a quick description, not a robust one. The interquartile range fixes this by focusing on the middle of the data. If you line up all values in order and cut them into four equal parts, the three cut points are the quartiles. The first quartile (Q1, the 25th percentile) has a quarter of the data below it; the second quartile is the median (50th percentile); the third quartile (Q3, the 75th percentile) has three quarters below it. The IQR is simply Q3 minus Q1, the span covered by the middle 50% of patients. Because it ignores the smallest and largest quarter of values, the IQR is resistant to outliers, just like the median. This is why median and IQR are reported together for skewed variables: median (IQR) is the standard, honest summary. The IQR is also the engine behind the box plot, one of the most useful graphs in clinical research. The box spans Q1 to Q3 (the IQR), a line inside marks the median, and 'whiskers' extend to the rest of the data, with points beyond them flagged as outliers. A glance at a box plot tells you the center, the spread, the skew (is the median off-center in the box?), and the outliers all at once, and box plots placed side by side make comparing groups almost effortless. The practical rule mirrors the previous lessons: symmetric data are summarized by mean and SD; skewed data by median and IQR. Reporting a five-number summary, minimum, Q1, median, Q3, maximum, captures the shape of almost any distribution without assuming it is normal. For a clinician, the IQR answers a natural question: forget the extremes, where do the typical middle-of-the-road patients actually fall? That is often exactly what you want to know.