Applying the Empirical Rule

Oct 13, 2025 by miodeya.com 147 views

Introduction

The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical concept that helps us understand the distribution of a normal distribution. It states that for a normal distribution, about 68% of the data points fall within one standard deviation of the mean, about 95% fall within two standard deviations, and about 99.7% fall within three standard deviations. In this article, we will use the Empirical Rule to answer a question about the interest rate for financing a used car.

The Problem

The interest rate for financing a used car is on average 9% with a standard deviation of 3%. Assuming that interest rates for used car loans are normally distributed, approximately what percentage of interest rates fall within the following ranges:

Within one standard deviation of the mean (i.e., between 6% and 12%)
Within two standard deviations of the mean (i.e., between 3% and 15%)
Within three standard deviations of the mean (i.e., between 0% and 18%)

Applying the Empirical Rule

To answer these questions, we will use the Empirical Rule. The rule states that about 68% of the data points fall within one standard deviation of the mean, about 95% fall within two standard deviations, and about 99.7% fall within three standard deviations.

Within One Standard Deviation of the Mean

One standard deviation of the mean is equal to the standard deviation (3%) multiplied by the number of standard deviations (1). Therefore, one standard deviation of the mean is equal to 3%.

To find the range of interest rates within one standard deviation of the mean, we add and subtract one standard deviation from the mean.

Lower limit: 9% - 3% = 6%
Upper limit: 9% + 3% = 12%

Therefore, approximately 68% of interest rates fall within the range of 6% to 12%.

Within Two Standard Deviations of the Mean

Two standard deviations of the mean is equal to the standard deviation (3%) multiplied by the number of standard deviations (2). Therefore, two standard deviations of the mean is equal to 6%.

To find the range of interest rates within two standard deviations of the mean, we add and subtract two standard deviations from the mean.

Lower limit: 9% - 6% = 3%
Upper limit: 9% + 6% = 15%

Therefore, approximately 95% of interest rates fall within the range of 3% to 15%.

Within Three Standard Deviations of the Mean

Three standard deviations of the mean is equal to the standard deviation (3%) multiplied by the number of standard deviations (3). Therefore, three standard deviations of the mean is equal to 9%.

To find the range of interest rates within three standard deviations of the mean, we add and subtract three standard deviations from the mean.

Lower limit: 9% - 9% = 0%
Upper limit: 9% + 9% = 18%

Therefore, approximately 99.7% of interest rates fall within the range of 0% to 18%.

Conclusion

In this article, we used the Empirical Rule to answer a question about the interest rate for financing a used car. We found that approximately 68% of interest rates fall within the range of 6% to 12%, approximately 95% fall within the range of 3% to 15%, and approximately 99.7% fall within the range of 0% to 18%. The Empirical Rule is a useful tool for understanding the distribution of a normal distribution and can be applied to a wide range of problems in statistics and data analysis.

References

Moore, D. S., & McCabe, G. P. (2017). Introduction to the practice of statistics. W.H. Freeman and Company.
Ross, S. M. (2017). Introduction to probability models. Academic Press.

Introduction

The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical concept that helps us understand the distribution of a normal distribution. In our previous article, we used the Empirical Rule to answer a question about the interest rate for financing a used car. In this article, we will answer some frequently asked questions about the Empirical Rule.

Q: What is the Empirical Rule?

A: The Empirical Rule is a statistical concept that states that for a normal distribution, about 68% of the data points fall within one standard deviation of the mean, about 95% fall within two standard deviations, and about 99.7% fall within three standard deviations.

Q: What is the purpose of the Empirical Rule?

A: The purpose of the Empirical Rule is to help us understand the distribution of a normal distribution. It provides a way to estimate the percentage of data points that fall within a certain range of the mean.

Q: How do I apply the Empirical Rule?

A: To apply the Empirical Rule, you need to know the mean and standard deviation of the data. You can then use the following steps:

Calculate one standard deviation of the mean by multiplying the standard deviation by 1.
Calculate two standard deviations of the mean by multiplying the standard deviation by 2.
Calculate three standard deviations of the mean by multiplying the standard deviation by 3.
Add and subtract the calculated standard deviations from the mean to find the range of data points.

Q: What are the limitations of the Empirical Rule?

A: The Empirical Rule is a simplification of the normal distribution and is not always accurate. It assumes that the data is normally distributed, which may not always be the case. Additionally, the Empirical Rule does not take into account the shape of the distribution, which can affect the accuracy of the estimates.

Q: Can I use the Empirical Rule for non-normal distributions?

A: No, the Empirical Rule is specifically designed for normal distributions. If the data is not normally distributed, you may need to use other statistical methods to understand the distribution.

Q: How do I know if my data is normally distributed?

A: There are several ways to check if your data is normally distributed, including:

Using a histogram or density plot to visualize the data.
Calculating the skewness and kurtosis of the data.
Using statistical tests such as the Shapiro-Wilk test or the Anderson-Darling test.

Q: What are some real-world applications of the Empirical Rule?

A: The Empirical Rule has many real-world applications, including:

Finance: Understanding the distribution of stock prices or interest rates.
Engineering: Understanding the distribution of measurements or errors.
Medicine: Understanding the distribution of patient outcomes or disease severity.

Conclusion

In this article, we answered some frequently asked questions about the Empirical Rule. We hope that this article has provided you with a better understanding of the Empirical Rule and its applications.

References

Moore, D. S., & McCabe, G. P. (2017). Introduction to the practice of statistics. W.H. Freeman and Company.
Ross, S. M. (2017). Introduction to probability models. Academic Press.