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CreatureCast

As ‘Normal’ as Rabbits’ Weights and Dragons’ Wings

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The normal distribution crops up many places in nature. The central limit theorem explains how it provides a near-universal expectation for averages of measurements.

Statistics are used to describe the world. The average height of women in the United States, for example, is 63.8 inches. Statistics are also used to test our ideas. If your friend told you he is 130 pounds, and you also know that 98 percent of healthy adult males weigh more than this, you might have reason to be concerned about his health. This is because you would expect few men sampled at random from the healthy population to weigh this little.

Many common statistical tests are possible because we can compare new observations to expected patterns, as above. But what is the expected pattern? If our expectation is wrong, we could reach incorrect conclusions.

Many real-world observations can be approximated by, and tested against, the same expected pattern: the normal distribution. In this familiar symmetric bell-shaped pattern, most observations are close to average, and there are fewer observations further from the average. The size of flowers, the physiological response to a drug, the breaking force in a batch of steel cables — these and other observations often fit a normal distribution.

There are, however, many important things we would like to measure and test that do not follow a normal distribution. Household income doesn’t — high values are much further from the average than low values are.

But even when raw data does not fit a normal distribution, there is often a normal distribution lurking within it. This makes it possible to still use the normal distribution to test ideas about non-normal data. This hidden normal distribution is revealed by collecting samples of multiple observations, and calculating the average for each sample. As the number of observations in each sample increases, the distribution of these averages becomes more and more similar to the normal distribution. The existence of this hidden normal distribution is known as the Central Limit Theorem.

In this episode of CreatureCast, Shuyi Chiou, an animator, presents the Central Limit Theorem. And, for the record, dragons do not exist. The song is Franz Danzi’s “Wind Quintet Op. 67 No. 3 In E-Flat Major, 4 Allegretto,” performed by the Soni Ventorum Wind Quintet. Further work by Ms. Chiou can be found at her Web site, shuyichiou.blogspot.com.

More animations and images can be found at https://www.nytimes.com/video/creature-cast and creaturecast.org, a project supported in part by the National Science Foundation.

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