Equations · Probability and statistics
Chebyshev's inequality
A guarantee for any distribution: few values can lie far from the mean.
P(|X − μ| ≥ kσ) ≤ 1/k²
Aethel says
the probability that the absolute value of X minus mu, is greater than or equal to k sigma, is less than or equal to 1 over k squared
What it says
Whatever the shape of the distribution, provided it has a mean and a variance, the probability of landing k or more standard deviations from the mean is at most one over k squared.
So at least 75% of values lie within two standard deviations of the mean, and at least 89% within three. For a normal distribution the true figures are much higher; the bound is for when nothing else is known.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| |X − μ| | the absolute value of X minus mu | the distance of X from the mean |
| k | k | a number of standard deviations, greater than 0 |
| ≥, ≤ | is greater than or equal to, is less than or equal to | at least, at most |
Where you meet it
Probability courses and proofs of the law of large numbers.
Questions
How do you say Chebyshev's inequality out loud?
P(|X − μ| ≥ kσ) ≤ 1/k² is read: “the probability that the absolute value of X minus mu, is greater than or equal to k sigma, is less than or equal to 1 over k squared.” That is exactly what Aethel says when it meets the formula in a book.
What does |X − μ| mean in Chebyshev's inequality?
|X − μ| (said “the absolute value of X minus mu”) is the distance of X from the mean.
What does k mean in Chebyshev's inequality?
k (said “k”) is a number of standard deviations, greater than 0.
Can I listen to the equations in my own textbook like this?
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