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Date: Sat, 10 Jul 1999 11:18:08 EDT
Subject: Standard Deviation
I have seen several answers to this question: If one standard deviation represents 68% of the population, what does two, three, four and five sigma [std deviation] represent? As stated, I have seen several different answers and thus, the impetus for my question.
My name is Anthony
Secondary level question
Hi Anthony,
I am not surprised that you have seen several different answers since the answer depends on the population distribution. The only general result that I know is Chebyshev's Theorem which implies that for any population:
- at least 3/4 of the population is within 2 standard deviations of the mean.
- at least 8/9 of the population is within 3 standard deviations of the mean.
- at least 15/16 of the population is within 4 standard deviations of the mean.
The theorem says that for any population, if k>1 then the proportion of the population that is within k standard deviations of the mean is at least 1-1/k2
Chebyshev's Theorem does not cover the situation of one standard deviation. The 68% that you state in your question comes from the Normal Distribution. If the population distribution is Normal then:
- 68% of the population is within 1 standard deviation of the mean
- 95% of the population is within 2 standard deviations of the mean
- 99% of the population is within 2 1/2 standard deviations of the mean
- 99.7% of the population is within 3 standard deviations of the mean
- 99.9% of the population is within 4 standard deviations of the mean
In many situations if you have a population distribution that is bell shaped and approximately symmetrical then the numbers for the Normal Distribution give a good approximation for that distribution also.
I hope this helps,
Harley
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