Chebyshev's theorem

1. Chebyshev’s Theorem

2. Relations between the Mean and the
Standard Deviation
• The mean is a measure of the centrality of a
set of observations.
• The standard deviation is a measure of their
spread.
• There are two general rules that establish a
relation between these measures and the set
of observations.
• The first is called Chebyshev’s theorem.
• The second is the empirical rule.

3. Chebyshev’s Theorem
• At least three-quarters of the observations in
a set will lie within 2 standard deviations of
the mean.
• At least eight-ninths of the observations in a
set will lie within 3 standard deviations of the
mean.

4. Chebyshev’s Theorem
• In other words,

5. The Empirical Rule
If the distribution of the data is more or less
symmetrical (normal distribution), then:
• Approximately 68% of the observations will be
within 1 standard deviation of the mean.
• Approximately 95% of the observations will be
within 2 standard deviations of the mean.
• A vast majority of the observations (all, or
almost all) will be within 3 standard deviations
of the mean.

6. Problem
• Check the applicability of Chebyshev’s
theorem and the empirical rule for the
graduation percentages of your batch-mates.