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Chebyshev's Inequality: Definition, Formula & Examples

Instructor: Bob Bruner

Bob is a software professional with 24 years in the industry. He has a bachelor's degree in Geology, and also has extensive experience in the Oil and Gas industry.

Chebyshev's inequality is used to measure the dispersion of data for any distribution. In this lesson, we look at the formula for Chebyshev's inequality and provide examples of its use.

How Scattered Are Your Measurements?

When you sample data, it is useful to know how dispersible, or spread out, the measurements are. For example, suppose you have been tracking your lunch expenses and have spent, on average, $10 a day. You would probably be interested in knowing if you consistently spent close to that amount, or if you had a few very large expenditures that skewed the overall average.

The dispersion of data that is normally distributed, as described by a bell curve, can be described using the 68-95-99.7 rule, which states that 68% of the data fits within one standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. But if you don't know, or don't expect your data to be normally distributed, can the same statement be made?

Chebyshev's Inequality

Chebyshev's inequality, also known as Chebyshev's theorem, makes a fairly broad but useful statement about data dispersion for almost any data distribution. This theorem states that no more than 1 / k2 of the distribution's values will be more than k standard deviations away from the mean. Looked at another way, 1 - (1 / k2) of the distribution's values will lie within k standard deviations of the mean.

While this equation often results in a relatively broad range of values, it is useful in that it only requires knowledge of the mean and standard deviation, both of which are easily calculated from any sample or data population. The theorem provides what might be called a worst-case look at data dispersion within any data distribution.

Chebyshev's Inequality Formula

In order to investigate this theorem, let's first compare the calculations to the 68-95-99.7 rule of thumb for normal distributions. Since those numbers represent the data lying inside the bounds, we use Chebyshev's inequality for data inside the bounds:

Probability = 1 - (1 / k2)

Mathematically, values less than or equal to 1 are not valid for this computation. However, plugging in the k values for 2 and 3 is relatively simple:

P(k=2): 1 - (1 / 22) = 1 - 0.25 = 0.75 (75%)

P(k=3): 1 - (1 / 32) = 1 - 0.11 = 0.89 (89%)

In these cases, Chebyshev's inequality states that at least 75% of the data will fall within 2 standard deviations of the mean, and 89% of the data is expected to fall within 3 standard deviations of the mean. This is less precise than the 95% and 99.7% values that can be used for a known normal distribution; however, Chebshyev's inequality is true for all data distributions, not just a normal distribution.

A Financial Example

Let's compare the Dow Jones and NASDAQ stock market returns over the last 40 years as an example. In general the NASDAQ market lists smaller, supposedly more volatile stocks. We see that this is true looking at the basic statistical measures for the past 40 years:

Dow Jones: Average = 8.77%; Standard Deviation = 14.43%

NASDAQ: Average = 13.40%; Standard Dev = 24.80%

Using Chebyshev's inequality, we can make a further statement about the likelihood of sampling data close to, or far away from, the averages. For example, from the theorem we know that at least 75% of the data will fall within 2 standard deviations of the average. The associated range of the 75% cut-off looks like this:

Dow Jones: -19.66% to 37.64%

NASDAQ: -35.42% to 62.85%

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