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%

**63** is the lower half of 99.7% that would be 99.7/2 = 49.85%

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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