What Chebyshev´s theorem is
Chebyshev’s theorem, also known as the Bienaymé-Chebyshev theorem, states that for any set of numbers, the proportion of values that lie within k standard deviations of the mean is at least 1-1/k2. This theorem is named after Russian mathematician Pafnuty Chebyshev who proved the theorem in 1867.
Steps for Chebyshev’s Theorem:
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Calculate the mean of the data set.
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Calculate the standard deviation of the data set.
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Choose a value for k, which is the number of standard deviations from the mean.
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Calculate the lower and upper bounds of the data set using the formula: Lower Bound = Mean – (k * Standard Deviation); Upper Bound = Mean + (k * Standard Deviation).
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Count the number of values in the data set that are between the lower and upper bounds.
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Calculate the proportion of values within k standard deviations of the mean using the formula: Proportion of Values = Number of Values in Range / Total Number of Values.
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Compare the result to 1-1/k2. If the result is greater than or equal to 1-1/k2, then Chebyshev’s theorem holds true.
Examples
- Chebyshev’s theorem can be used to calculate the probability that a random variable is within k standard deviations of the mean.
- Chebyshev’s theorem can be used to determine the minimum sample size required to ensure that the sample mean is within a certain number of standard deviations of the population mean.
- Chebyshev’s theorem can be used to estimate the upper and lower bounds of a probability distribution.