68-95-99.7 Rule, please explain

I'm reviewing the 68-95-99.7 rule in my statistic math course and don't get it. Can someone explain it to me?

The question is:
The heights of a large group of people are assumed to be normally distributed. Their mean height is μ = 66.5 inches, and the standard deviation is σ = 2.4 inches.

What percentage of people fall between 59.3 inches and 73.7 inches?

I'm not asking you to solve it for me, can someone simply show me to how?

Thank you!

The rule for a normal distribution says that for a sample with a mean of μ and a standard deviation of σ, we can expect the following:

68% of the data will lie within one standard deviation of the mean:
µ - σ < x < µ + σ

95% will lie within TWO standard deviations:
µ - 2σ < x < µ + 2σ

and 99.7% will lie within THREE standard deviations:
µ - 3σ < x < µ + 3σ

In your example, with a mean of 66.5 and a standard deviation of 2.4, you can calculate the following:
68%:
66.5 - 2.4 < x < 66.5 + 2.4

95%:
66.5 - 2·(2.4) < x < 66.5 + 2·(2.4)

99.7%:
66.5 - 3·(2.4) < x < 66.5 + 3·(2.4)

Whichever range of values matches your given range of 59.3 < x < 73.7 will have the corresponding percentage. (I leave it to you to work that part out)

When the range does not fit neatly into the 68-95-99.7 rule, you need to use other methods to determine the expected percentage.

Hope this helps!

The 68-95-99.7 applies to the standard belt curve. You take the mean "μ" and add/subtract the standard deviation 1 time and it will provide you the point in which 68% of the data falls. Add/subtract the standard deviation 2 times and it will provide you the point in which 95% of the data falls.

You answer:

59.3 is 3 standard deviations away from the mean 66.5.
73.7 is 3 standard deviations away from the mean 66.5.

99.7% of the data lies 59.3inches and 73.7 inches.