Student Question

# Explain Chebyshev's theorem and what is it good for?  Would you be correct if you said Chebyshev's theorem applies to everything from butterflies to the orbits of planets?

Suppose you have some distribution of values. Chebyshev's theorem says that "most" of the values are "close" to the mean of the distribution.

By "close" we mean that a random variable is within k (k>1) standard deviations of the mean.

To describe "most": Assume we want to know the percentage of the distribution that lies within k (k>1) standard deviations of the mean. (This is equivalent to knowing the probability that a random variable is within k standard deviations of the mean.) The theorem states that the percentage of the distribution within k standard deviations of the mean is at least `1-1/k^2` .

For example, to find the percentage of any distribution within 2 standard deviations of the mean we apply the theorem to find that at least `1-1/2^2=3/4` lies within 2 standard deviations of the mean.

In contrast to the empirical rule which requires the distribution to be approximately "normal", Chebyshev's theorem applies to any distribution no matter how skewed. You need only know the mean and the standard deviation.

Since the theorem applies to any distribution, it is fair to say that it applies to everything from butterflies to planetary orbits.