Understanding the 68-95-99.7 Rule: Calculating Percentages of Test Scores within Specific Ranges

Assume that a set of test scores is normally distributed with a mean of 80 and a standard deviation of 5. Use the​ 68-95-99.7 rule to find the following quantities.

The 68-95-99

The 68-95-99.7 rule, also known as the Empirical Rule, is a statistical guideline that helps determine the percentage of data that falls within a certain number of standard deviations from the mean in a normally distributed dataset. The rule states that:

– Approximately 68% of the data falls within one standard deviation of the mean.
– Approximately 95% of the data falls within two standard deviations of the mean.
– Approximately 99.7% of the data falls within three standard deviations of the mean.

In this case, we are given that the test scores are normally distributed with a mean of 80 and a standard deviation of 5. Now let’s use the 68-95-99.7 rule to find the following quantities:

1. What percentage of test scores is between 75 and 85?

To find the percentage of scores between 75 and 85, we need to consider one standard deviation on both sides of the mean.

– Lower limit: 80 (mean) – 1(standard deviation) = 75
– Upper limit: 80 (mean) + 1(standard deviation) = 85

According to the 68-95-99.7 rule, approximately 68% of the data falls within one standard deviation from the mean. So, approximately 68% of test scores are between 75 and 85.

2. What percentage of test scores is between 70 and 90?

To find the percentage of scores between 70 and 90, we need to consider two standard deviations on both sides of the mean.

– Lower limit: 80 (mean) – 2(standard deviations) = 70
– Upper limit: 80 (mean) + 2(standard deviations) = 90

According to the 68-95-99.7 rule, approximately 95% of the data falls within two standard deviations from the mean. So, approximately 95% of test scores are between 70 and 90.

3. What percentage of test scores is between 65 and 95?

To find the percentage of scores between 65 and 95, we need to consider three standard deviations on both sides of the mean.

– Lower limit: 80 (mean) – 3(standard deviations) = 65
– Upper limit: 80 (mean) + 3(standard deviations) = 95

According to the 68-95-99.7 rule, approximately 99.7% of the data falls within three standard deviations from the mean. So, approximately 99.7% of test scores are between 65 and 95.

These calculations illustrate the concept of the Empirical Rule and how it can be applied to find the percentage of data within certain ranges based on a normal distribution with known mean and standard deviation.

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