68% Area

68% of the area under the bell curve is within ONE standard deviation of the mean.

68% Area

95% Area

95% of the area under the bell curve is within TWO standard deviations of the mean.

95% Area

50% areaThe bell curve is symmetric.

This means 50% of the area is to the right of the mean, 34% between m and +m s, and 47.5% between m and -2m s.

68/2 = 34% area

95/2 = 47.5% area

.135 .135

.34 .34

.025 .025

We could separate the bell curve into six “chunks”, with areas shown below.

Again, the area within ONE s.d. of the mean is .34 + .34 = .68

Area within TWO s.d.’s of the mean is .135+.34+.34+.135 = .95

Now we can find any normal probability involving the mean plus or minus one or two standard deviations.

Example: Suppose that

X ~ N(6,9)

What is P(3<X<12)?

6 9 1230

m = 6s2 = 9 → s = 3

Here is the picture that should pop up in your head:

The probability of being between one s.d. below the mean and two s.d.’s above is: .34 + .34 + .135

= .815