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Chebyshevs Theorem and The Empirical Rule

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Introduction

Chebyshev’s Theorem and The Empirical Rule

Suppose we ask 1000 people what their age is. If this is a representative sample then there will be very few people of 1-2 years old just as there will not be many 95 year olds. Most will have an age somewhere in their 30’s or 40’s. A list of the number of people of a certain age may look like this:

Age

Number of people

0

1

1

2

2

3

3

8

..

..

..

..

30

45

31

48

..

..

..

..

60

32

61

30

..

..

..

..

80

6

81

3

Next, we can turn this list into a scatter diagram with age on the horizontal axis and the number of people of a certain age on the vertical axis.

From the statistical point of view a scatter diagram may have two shapes.  

It may be shaped or at least looks approximately like a 'bell curve', which looks like this:

A 'bell curve' is perfectly symmetrical with respect to a vertical line through its peak and is sometimes called a "Gauss curve" or a "normal curve".

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Middle

 years which simplifies to a range of 22 to 58 years.At least 93.75% of all the ages will lie in the range of image20.png.
In our case this means that at least 93.75% of the people will have an age in the range of
image21.png years which simplifies to a range of 16 to 64 years.At least 96% of all the ages will lie in the range of image22.png.
In our case this means that at least 96% of the people will have an age in the range of
image23.png years which simplifies to a range of 10 to 70 years.At least 97.2% of all the ages will lie in the range of image12.png.
In our case this means that at least 97.2% of the people will have an age in the range of
image01.png years which simplifies to a range of 4 to 76 years.

How can we calculate these percentages? To calculate the 75%, the 88.9%, the 93.75%, etc, we look at the number of standard deviations in the respective intervals. The 75% goes together with 'mean ± 1 standard deviation', the 88.9% with 'mean ± 2 standard deviations', the 93.

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Conclusion

image17.png. The same is true for the difference between the mean and the lower limit of this interval. According to the table above this coincides with 96%.

The Empirical Rule

When the data values seem to have a normal distribution, or approximately so, we can use a much easier theorem than Chebyshev’s.

The "empirical rule" states that in cases where the distribution is normal, the following statements are true:

  • Approximately 68% of the data values will fall within 1 standard deviation of the mean.
  • Approximately 95% of the data values will fall within 2 standard deviations of the mean.
  • Approximately 99.7% of the data values will fall within 3 standard deviations of the mean.

Example 3:

The average salary for graduates entering the actuarial field is $60,000. If the salaries are normally distributed with a standard deviation of $5000, then what percentage of the graduates will have a salary between $50,000 and $70,000?

Solution:

Both $50,000 and $70,000 are $10,000 away from the mean of $60,000. This is two standard deviations away from the mean, so 95% of the graduates will have a salary in this interval.

© 2008 UMUC – European Division

Ron Souverein and Nada Wray

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