The means of randomly chosen parts scatter around the mean of the whole in a predictable shape. That is why examining only a part tells you not just about the whole, but also how large your error is likely to be.
Ecological surveys are built on learning about the whole from a part
Field ecologists rarely count every tree in a forest or capture every monkey in a region. Ecological field surveys are designed as methods for examining only part of a population and inferring the whole from it.
Quadrat sampling
Place several frames of a fixed size, count only the plants inside them, and estimate the density across the whole area.
Mark–recapture
Mark captured individuals and release them, then estimate the total population size from the proportion of marked individuals in a later capture.
Line transect sampling
Walk along a set line, record each individual found and its distance from the line, and estimate population density.
All of these methods share one condition: the surveyor must not choose where to place the frames or which line to walk for their own convenience. If you survey only places where animals are easy to find, no number of sites will reveal the whole. Let us see why this condition matters, using real data.
First, look at the “whole”: body mass of 265 primate species
Here the “whole” is the 265 primate species with a recorded body mass. The lightest is a mouse lemur at just 31 grams; the heaviest is the eastern gorilla at 149 kilograms.
The horizontal axis is cut at 40,000 grams. Large species such as gorillas and orangutans lie beyond it. Data: PanTHERIA (Jones et al. 2009).
The shape is far from symmetric. Most species are small, and a handful of large species pull the mean upward. That is why the mean is roughly twice the median.
Can we pick just a few species from this “skewed whole” and still estimate its mean?
- Population
- The whole you actually want to know about. Here, the 265 species.
- Sample
- The part you actually examine. Here, the n species you pick.
- Sample mean
- The mean calculated from the sample alone. It changes every time you pick again.
- Repetitions
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- Mean of sample means
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- Gap from population mean
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- Spread of sample means
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- Theoretical spread
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The histogram’s vertical axis is a count. The navy curve is the shape expected in theory when species are picked at random.
What is happening: the central limit theorem
However skewed the whole may be, if you repeatedly collect the means of randomly chosen samples, their distribution approaches a bell shape centered on the mean of the whole. This is the central limit theorem.
The width of the bell, that is, the spread of the sample means, can be calculated in advance. It is called the standard error: the spread of the whole divided by the square root of the number you pick.
standard error = spread of the whole ÷ √n
| Species picked, n | Standard error (theoretical) | Relative to the mean |
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Quadrupling n halves the spread. To double the precision, you need four times as much data. The more you add, the smaller the gain.
Common misconceptions
“Pick enough species and the sample itself becomes bell-shaped”
What approaches a bell shape is the distribution of many sample means. The contents of any single sample remain as skewed as the whole. Look at the “last sample” in the panel above: you should see a heavy species or two standing out among mostly light ones.
“With 30 observations, you can treat it as a normal distribution”
If the whole is reasonably symmetric, 30 is a good rule of thumb. When the data are as strongly skewed as these body masses, the distribution still has a long right tail at n=30. How many you need depends on how skewed the whole is.
“Collect enough data and bias disappears”
It does not. Switch the panel above to the biased selection, and the mean stays off target even after 1,000 repetitions or with n set to 100. What is more, the 265 species treated as the “whole” on this page are only those of the 376 primate species whose body mass has been measured. Well-studied species are more likely to have records, so strictly speaking the mean of the 265 species is “the mean of well-studied primates.” What was never measured does not show up in the numbers.
“If many people give the same answer, it must be more reliable”
If each person looks at the others’ answers and conforms, adding more people adds almost no information. Both the central limit theorem and the standard error formula assume that each observation is selected independently.
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, the 265 with a recorded adult body mass are used. Families are shown by their scientific names.
Samples in the hands-on panel are drawn by returning each chosen species before choosing the next (sampling with replacement).