Wiki Page, Math 13 - Perez Sandbox : Module 12 - Introduction to Normal Probability Distributions (2b of 5)

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2026-01-17T21:15:13-08:00

Earlier we stated that for all normal probability curves, the area within 1 standard deviation of the mean will equal 0.68. What is the probability of values falling outside this interval? This is the question we will investigate now.

We will start with an example about normal probability curves in general. Then we will provide some specific examples.

EXAMPLE

In a normal probability distribution, what is the probability that an X-value is more than one standard deviation above the mean?

The probability is 0.16. Here is how we know this ...

Look at the normal probability curve.

The area under the probability curve always equals 1.

The central area associated with X-values within one standard deviation of the mean is 0.68.

Therefore, the area outside of this region equals 1 − 0.68 = 0.32. This area is associated with X-values more than one standard deviation from the mean, both above and below. This is the area in the two tails.

We want the probability that X-values are more than one standard deviation above the mean. This is the area in the tail on the right, which is half of 0.32. This is how we got 0.16 (1/2 of 0.32).

EXAMPLE
Birth weights have a normal probability distribution with a mean of 120 ounces and a standard deviation of 20 ounces.

What is the probability that a randomly selected infant weighs less than 100 ounces?

Answer: 16%

Here is how we reasoned through this using the normal probability curve with mean of 120 and standard deviation of 20.

EXAMPLE

Birth weights have a normal probability distribution with a mean of 120 ounces and a standard deviation of 20 ounces.

What is the probability that a randomly selected infant weighs more than 100 ounces?

Answer: 16% + 68% = 84%

Here is how we reasoned through this using the normal probability curve with mean of 120 and standard deviation of 20.

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