There is more than one way to reduce many values to a single number. The mean, the median, and the geometric mean each represent something different, and which one to use depends on the shape of the data and on what you want to know.
What is the “typical body mass” of a primate?
If you had to describe the body mass of 265 primate species with a single number, how many grams would you give? The plain mean is 5,881 g. Yet the species that sits exactly in the middle when the masses are sorted from lightest to heaviest weighs 3,006 g, only about half the mean.
Neither calculation is wrong. The two numbers describe different aspects of the same data. This page examines what each measure of central tendency and each measure of spread represents, using body mass, group size, and home range size.
- Measure of central tendency
- A single value that stands for the whole dataset, such as the mean, median, or mode.
- Mean (arithmetic mean)
- The sum of all values divided by the number of values.
- Median
- The value exactly in the middle when the values are sorted in increasing order.
- Spread
- How widely the values are scattered around the central value, measured for example by the standard deviation or the interquartile range.
265 species on a single axis
The figure below plots the body mass of each of the 265 species as one dot. Body mass ranges from 31 g to 149,325 g, a span of roughly 4,800-fold, so the horizontal axis is a log scale on which each tick is ten times the previous one.
The vertical position of the dots has no meaning; they are spread out only to reduce overlap. The darker dots are the 7 species of Hominidae.
The mean line sits clearly to the right of where the dots are densest. Of the 265 species, 65 %, or roughly two in three, are lighter than the mean. What pushes the mean up is a small number of large species at the right end: the 7 species of Hominidae make up less than 3 % of the species, yet account for 32 % of the combined body mass of all 265.
The median depends only on rank, not on the size of the values. Among these 265 species, the median is exactly the 3,006 g of Cebus capucinus, a member of the family Cebidae.
The shaded band is the interquartile range, which contains the species from the 25th to the 75th percentile.
- Species
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- Mean
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- Median
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- Geometric mean
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- Standard deviation
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- Interquartile range
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- Below the mean
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Measures of spread, and the geometric mean
A central value alone does not tell you how scattered the values are. The most common measure is the standard deviation: square how far each value is from the mean, average those squares, and take the square root to return to the original units.
Because the standard deviation is anchored to the mean, it is pulled strongly by large values, just as the mean is. The standard deviation of body mass is 13,110 g, more than twice the mean. For skewed data, the interquartile range is therefore also widely used. It is the distance between the value 25 % of the way up the sorted list (the first quartile) and the value 75 % of the way up (the third quartile), and it is the width that contains the middle half of the species.
Another option is the geometric mean. Take the logarithm of each value, average those, and convert back to the original units. For data that vary by multiples, taking logarithms makes the distribution nearly symmetric, and the center of that distribution is the geometric mean.
The skewness of the body mass distribution, a measure of asymmetry that is 0 for a symmetric distribution and positive for one with a long right tail, is 7.60 on the original scale but falls to −0.40 after taking logarithms. The summaries for the three variables are as follows.
| Variable (species) | Mean | Median | Geometric mean | Standard deviation | Interquartile range |
|---|---|---|---|---|---|
| Body mass (265) | 5,881 g | 3,006 g | 2,120 g | 13,110 g | 749–7,078 g |
| Group size (216) | 12.3 | 6.9 | 6.3 | 14.6 | 3.0–16.1 |
| Home range size (157) | 1.70 km² | 0.19 km² | 0.20 km² | 4.91 km² | 0.05–0.88 km² |
Group size is given in individuals.
In all three, the mean is larger than the median, indicating a distribution with a long right tail. For home range, 85 % of species have a range smaller than the mean.
Common misconceptions
“The mean is the value of a common species.”
In data with a long right tail, the mean is larger than most species’ values. For primate body mass, 65 % of species are lighter than the mean, and for group size, 69 % of species live in groups smaller than the mean. The “average species” is not necessarily a typical species in the data.
“Most values fall within the mean ± one standard deviation.”
This rule of thumb assumes a symmetric, bell-shaped distribution. For body mass, the mean minus one standard deviation is −7,229 g, an impossible value. For group size it is −2.3 individuals. Check the shape of the distribution before interpreting the standard deviation.
“Extreme values are outliers, so remove them before summarizing.”
Excluding the 7 species of Hominidae lowers the mean body mass from 5,881 g to 4,127 g, a drop of about 30 %, while the median moves only 7 %, from 3,006 g to 2,797 g. But the body mass of a gorilla is not a measurement error; it is the value of a real species. Whether to exclude such values is decided not by statistical convenience but by what you want to know about.
“For skewed data, always use the median.”
The median does not move no matter how heavy the species at the right end become. That is a strength, but it becomes a weakness for questions about totals or aggregate quantities. The fact that the 7 species of Hominidae account for 32 % of the combined body mass of the 265 species cannot be seen from the median.
Covered in the courses
The course sites are in Japanese and open to anyone. To go further, visit the page for each session.
Data: Jones, K. E. et al. (2009) PanTHERIA: a species-level database of life history, ecology, and geography of extant and recently extinct mammals. Ecology 90: 2648. Of 376 primate species, this page uses the 265 with a recorded adult body mass, the 216 with a recorded group size, and the 157 with a recorded home range size.
The standard deviation treats each set of species as a whole and divides by the count n. Quartiles are obtained by linear interpolation between sorted values.