Chebyshev’s Inequality

Chebyshev’s inequality is a probability theorem used to characterize the dispersion or spread of data away from the mean. It was developed by a Russian mathematician called Pafnuty Chebyshev. The theorem states that:

For any set of observations, whether sample or population data and regardless of the shape of the distribution, the percentage of the observations that lie within k standard deviations of the mean is at least \(1 – \cfrac {1}{k^2}\) for all \(k > 1\).

Simply put, this means that you can use the formula to determine the percentage of observations lying inside or outside a given number of standard deviations. Remember that the standard deviation tells us how far values are from the arithmetic mean. For example, two-thirds of the observations fall within one standard deviation on either side of the mean in a normal distribution. However, Chebyshev’s inequality goes slightly against the 68-95-99.7 rule commonly applied to the normal distribution.

Chebyshev’s Inequality Formula

$$ P = 1 – \cfrac {1}{k^2} $$

Where

P is the percentage of observations

K is the number of standard deviations

Example: Chebyshev’s Inequality

Suppose we wish to find the percentage of observations lying within two standard deviations of the mean:

k = 2

Hence,

$$ \begin{align*} P & = 1 – \cfrac {1}{2^2} \\ & = 0.75 = 75\% \\ \end{align*} $$

Question

Using Chebyshev’s inequality, calculate the percentage of observations that would fall outside 3 standard deviations of the mean.

1. 11%
2. 89%
3. 90%

The correct answer is B.

Working:

note that the question asks for the percentage that would fall outside 3 standard deviations. Therefore:

$$ P = 1 – \cfrac {1}{3^2} = 89\% $$

But we are interested in 1 – p , i.e., the outside observations.

Therefore,

$$ 1 – 0.89 = 0.11 \text{ or } 11\% $$