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Descriptive Statistics - Chebyshev's Theorem and Empirical rule

Chebyshev's Theorem

Chebyshev's Theorem enables us to state that a proportion of data values must be within a specified number of standard deviation of the mean. The advantage of this theorem is that the theorem applies to any data set regardless of the shape of the distribution of the data.

Here is the Chebyshev's rule:

At least $(1- \dfrac{1}{z^2})$ of the data values must be within $z$ standard deviation of the mean where $z$ is any value greater than 1.

For z =2, 3, and 4, these are what the theorem states:

  • z = 2, at least 75% of the data values must be within 2 standard deviations of the mean.
  • z = 3, at least 89% of the data values must be within 3 standard deviations of the mean.
  • z = 4, at least 94% of the data values must be within 4 standard deviations of the mean.

Empirical Rule

Unlike Chebyshev's theorem, empirical rule can only be applied to data or observations that are believed to approximate symmetric bell-shaped also known as a normal distribution.

For data having a bell-shaped distribution:
  • Approximately 68% of data values will be within one standard deviation of the mean.
  • Approximately 95% of data values will be within two standard deviations of the mean.
  • Almost all of the data values will be within three standard deviations of the mean.

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