**Between 63 and 98, the total distribution is 97.5%**

**Between 24 and 36, the total distribution is 87.9%**

I would like to know how to go about getting the answer to these questions.

1. Empirical Rule: Mean is 84 with a standard deviation of 7. Approximately how much is between 63 and 98?

2. Empirical Rule: Mean is 27 with a standard deviation of 3. Approximately how much of the distribution is between 24 and 36?

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Marked as Best Answer

For a standard bell-shape curve, the distribution:

a) From the Mean to 1 sdev = 36.1%

b) between 1sdev and 2sdev = 13.6%

c) between 2sdev and 3sdev = 2.1%

D) between 3sdev and 4sdev = 0.1%

1) Given the Mean is 84 and the sdev=7

98 is 2 sdev away from 84 and the distribution is:

34.1% (between the mean and 1sdev) + 13.6% (between 1sdev and 2sdev)

63 is 3 sdev away from 84 and the distribution is:

34.1% + 13.6% + 2.1%

2) Given the Mean is 27 and the sdev = 3

36 is 3 sdev away from 27 and its distribution is 36.1% + 13.6% + 2.1%

24 is 1 sdev away from 27 and its distribution is 36.1%

Grigori S. | Certified Physics and Math Teacher G.S.Certified Physics and Math Teacher G.S.

Thomas L. | Mathematics TutorMathematics Tutor

Say you survey a bunch of people and get their numerical opinions on 1 thing. You then do some statistics and you find the mean and standard deviation.

The empirical rule says that 99.7% of your results will within 3 standard deviations above or 3 standard deviations below your mean.

Say your mean is 20 and standard deviation is 2. Then 99.7% of your results will fall between

20 - 2 - 2 - 2 = 14 and 20 +2+2+2 = 26. So between 14 and 26.

The empirical rule also says that 95% of your results will be with 2 standard deviation above or 2 standard deviations below your mean.

And the Emp. rule says that 68% of the results will be within 1 standard deviation of your mean.

Since this is a "normal distribution" the % results will also be symmetrical. Meaning half of the 68% will be below the mean and half of the 68% will be above the mean. The same goes for the 95% and the 99.7%.

So let's get to the problems:

1. if your mean is 84 and standard deviation is 7 then:

84 - 7 = 77 and 84 + 7 = 91 68% of your results will be between 77 and 91. 34% above up to 91 and 34% below down to 77.

84-7-7 = 70 and 84+7+7 = **98. ** 95% of your results will be between 70 and
**98**. 47.5% above and 47.5% below.

84-7-7-7 = **63** and 84+7+7+7 = 105. 99.7% of your results will be between
**63** and 105. 49.85% above and 48.85% below.

Remember how I said this is symmetrical.

Since **98** is the upper half of 95% that would be 95/2 = 47.5%

Add them together to get teh upper and the lower of your mean and you get 47.5+49.85 = 97.35%

2) 27 - 3 = 24, so 24 is 1 standard deviation below.( 68%/2 because of lower half)

27+3+3+3 = 36, so 36 is 3 standard deviations above.( 99.7%/2 because of upper half)

34% + 49.85% = 83.85%

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