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

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2026-01-17T21:15:14-08:00

Now we extend this idea to look at the probability of a value falling within 2 standard deviations of the mean or 3 standard deviations of the mean.

If X is a normal quantitative variable with mean and standard deviation , then:

1. The probability that X is within 1 standard deviation of the mean equals approximately 0.68.

2. The probability that X is within 2 standard deviations of the mean equals approximately 0.95.

3. The probability that X is within 3 standard deviations of the mean equals approximately 0.997.

To summarize using probability notation:

These three facts together are called the empirical rule for normal curves.

Comment

Let’s take a moment to think about the empirical rule.

The first statement of the empirical rule help us to identify an interval of likely values of X. It gives us an interval—within 1 standard deviation of the mean—that contains the central 68% of the values.

The second and third statements in the empirical rule help us identify values that are unlikely to occur.

95% of values fall within 2 standard deviations of the mean. It is therefore unlikely for a value to fall more than 2 standard deviations away from the mean. Values more than 2 standard deviations away from the mean in a normal distribution are often called outliers.

99.7% of values fall within 3 standard deviations of the mean. It is therefore extremely unlikely for a value to fall more than 3 standard deviations away from the mean. Values more than 3 standard deviations away from the mean are often called extreme outliers.

EXAMPLE The empirical rule in a context

Suppose that foot length of a randomly chosen adult male is a normally distributed with mean and standard deviation .

Then the empirical rule lets us sketch the probability distribution of X. Use this sketch to answer the following questions.

1) What is the probability that a randomly chosen adult male will have a foot length between 8 and 14 inches?
Answer: 0.95 = 95%
Why? The question asks for a probability, so think area.
The standard deviation is 1.5 inches. 8 is two standard deviations below the mean (11 - 2(1.5) = 8) and 14 is 2 standard deviations above the mean (11 + 2(1.5) = 14). The Empirical Rule says that 95% of the time X will fall within 2 standard deviations of the mean.

2) What is the probability that a randomly chosen adult male will have a foot length greater than 14 inches?
Answer: 0.025 = 2.5%
Why? We want a probability, so think area.
14 is two standard deviations above the mean. The Empirical Rule says that central area associated with X values within two standard deviations of the mean is 0.95.
We want the area associated with X>14.
Find the area in the two tails: 1(total area) - 0.95 = 0.05
We want the area in the right tail: 0.05/2 = 0.025

3) What are the foot measurements for adult males who are extreme outliers?

Answer: less than 6.5 inches or greater than 15.5 inches
Why? Here we are asked for a foot measurement (X-value), not a probability.
Extreme outliers lie more than 3 standard deviations away from the mean.
Find the X-values that mark 3 standard deviations below and above the mean:
11 - 3(1.5) = 6.5 and 11 + 3(1.5) = 15.5.

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