Mean and standard deviation or median and quartiles?

Skewed distributed data

Figure 2 shows a right-skewed distribution. Such distributions may originate from measurements which cannot be negative, for example plasma concentration. In a right-skewed distribution, the mean will be larger than the median, and the standard deviation is not related to specific percentiles, as was the case for the normal distribution.

What are relevant measures of central tendency and variability if data are not normally distributed? The mathematical expressions for mean and standard deviation are defined regardless of whether data are normally distributed. Now, consider a fictitious data set, from (1): Assume we have recorded the length of stay in hospital for 13 patients with a given diagnosis: (respectively 3, 9, 10, 10, 10, 12, 13, 14, 18, 21, 27, 38, and 62 days). The average is 19 days, and the median is 13 days. The standard deviation is 15.8 days, and the quartiles are 10 days and 24 days. If we intend to estimate cost or need for personnel, the mean is more relevant than the median. If we want to state a 'typical' length of stay for a single patient, the median may be more relevant.

Some authors report only the interquartile range, which is 24–10=14 days in this example, instead of the quartiles. This is less informative than reporting the quartiles, which in combination with the median also indicate the skewness of the distribution. In our example, the median of 13 days is closer to the lower quartile of 10 days than the upper quartile of 24 days, indicating a right-skewed distribution, similar to the one illustrated in Figure 2. In some contexts, it may be useful to report the minimum and maximum values instead of, or in addition to, the quartiles. But one should bear in mind that unlike the interquartile range, the distance between minimum and maximum is expected to increase with increasing sample size.