Question

Which of the following normal curves looks most like the curves mue = 10 and a = 5

Answer 1

Answer:

the area within 1 standard deviation of the mean will equal 0.68. From this fact, we can see that the area outside of this region equals 1 − 0.68 = 0.32. And since normal curves are symmetric, this outside area of 0.32 is evenly divided between the two outer tails. So the area of each tail = 0.16.

\displaystyle \mathrm{area\; of\; each\; tail}=\frac{1}{2}(1-\mathrm{central\; area})=\frac{1}{2}(1-.68)=\frac{1}{2}(.32)=.16areaofeachtail=

2

1

(1−centralarea)=

2

1

(1−.68)=

2

1

(.32)=.16

Normal curve showing outer tail areas in gray

The outer tail areas allow us to answer related probability questions:

Question: What is the probability that a normal random variable is more than 1 standard deviation from its mean?

Answer: 0.32

Question: What is the probability that a normal random variable is more than 1 standard deviation larger than its mean?

Answer: 0.16

Before leaving this example, we highlight one more geometric fact about normal curves. Look at the arrows pointing at the normal curve in the following figure.

Normal curve with inflection points marked

At these points, the curve changes the direction of its bend and goes from bending upward to bending downward, or vice versa. A point like this on a curve is called an inflection point. Every normal curve has inflection points at exactly 1 standard deviation on each side of the mean.