# Chebyshev's inequality

### Merke

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{*LOSnr18*}
Define, calculate, and interpret 1) a range and a mean absolute deviation, and 2) the variance and standard deviation of a population and of a sample.

### Methode

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There are two versions of Chebyshev's inequality The first is stated by CFAI; the second is not. So pay close attention to the second version.
According to Chebyshev's inequality,

• the proportion of the observations within k standard deviations of the arithmetic mean is at least 1 - 1/k2 for all k > 1, and.
• the proportion of the observations out of the range of k standard deviations of the arithmetic mean is at most 1/k2 for all k > 1.

So if we don't know the distribution, we can still say that

• the proportion of the observations within one standard deviation of the arithmetic mean

• is at least 1 - 1/12 = 0,

• the proportion of the observations within two standard deviations of the arithmetic mean

• is at least 1 - 1/22 = 1 - 1/4 = 0.75 , and

• the proportion of the observations within three standard deviations of the arithmetic mean

• is at least 1 - 1/32 = 1 – 1/9 = 8/9 = 0.888.