## Empirical Rule: Definition, Formula, Example, and How It’s Used

### Definition

The empirical rule states that approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and around 99.7% falls within three standard deviations. This rule applies to data that follows a normal distribution.

### Formula

The formula for the empirical rule is:

**68% of the data falls within one standard deviation of the mean**

**95% of the data falls within two standard deviations of the mean**

**99.7% of the data falls within three standard deviations of the mean**

### Example

Let’s say we have a dataset of exam scores. The mean score is 80, and the standard deviation is 5. According to the empirical rule, approximately 68% of the scores will fall between 75 and 85, about 95% will fall between 70 and 90, and around 99.7% will fall between 65 and 95.

### How It’s Used

The empirical rule is used in various fields, including finance, economics, and quality control. In financial analysis, it can be used to analyze stock returns, where the mean represents the average return and the standard deviation represents the volatility. By applying the empirical rule, analysts can assess the likelihood of certain returns falling within a specific range.

Standard Deviations from the Mean | Percentage of Data |
---|---|

1 | 68% |

2 | 95% |

3 | 99.7% |

## What is the Empirical Rule?

This rule is based on the concept of standard deviation, which measures the spread of data around the mean. The Empirical Rule provides a quick and easy way to estimate the percentage of data that falls within a certain range in a normal distribution.

For example, if a dataset follows a normal distribution and has a mean of 50 and a standard deviation of 10, the Empirical Rule tells us that approximately 68% of the data will fall between 40 and 60, approximately 95% will fall between 30 and 70, and approximately 99.7% will fall between 20 and 80.

The Empirical Rule is widely used in various fields, including finance, economics, and social sciences. It helps analysts and researchers understand the distribution of data and make predictions based on the characteristics of a normal distribution.

By applying the Empirical Rule, analysts can identify outliers, assess the likelihood of certain events occurring, and make informed decisions based on the probabilities associated with different ranges of data.

## The Formula for the Empirical Rule

### 68% of the data falls within one standard deviation of the mean

### 95% of the data falls within two standard deviations of the mean

### 99.7% of the data falls within three standard deviations of the mean

The formula for the empirical rule can be expressed as:

Percentage of Data | Number of Standard Deviations |
---|---|

68% | 1 |

95% | 2 |

99.7% | 3 |

This formula allows us to estimate the proportion of data that falls within a certain range of standard deviations from the mean in a normal distribution. It is a useful tool in statistical analysis and can be applied in various fields, including finance.

## Example of Using the Empirical Rule

Let’s say we have a dataset that represents the heights of a group of individuals. We want to analyze this data using the empirical rule to understand the distribution of heights.

First, we need to calculate the mean and standard deviation of the dataset. The mean represents the average height, while the standard deviation measures the spread or variability of the heights.

Next, we can use the empirical rule to make some conclusions about the dataset. The empirical rule states that for a normal distribution:

Percentage of Data | Within | Standard Deviations from the Mean |
---|---|---|

68% | 1 | standard deviation |

95% | 2 | standard deviations |

99.7% | 3 | standard deviations |

Using this information, we can draw some conclusions about our dataset. For example, if the mean height is 170 cm and the standard deviation is 5 cm, we can say that approximately 68% of the individuals have heights between 165 cm and 175 cm (1 standard deviation from the mean). Similarly, about 95% of the individuals have heights between 160 cm and 180 cm (2 standard deviations from the mean), and nearly all individuals (99.7%) have heights between 155 cm and 185 cm (3 standard deviations from the mean).

The empirical rule is a useful tool in financial analysis as well. It can help analysts understand the distribution of financial data, such as stock prices or returns. By applying the empirical rule, analysts can make informed decisions and predictions based on the likelihood of certain outcomes.

## How the Empirical Rule is Used in Financial Analysis

One way the empirical rule is used in financial analysis is to determine the probability of a certain return falling within a certain range. For example, if a stock has an average return of 10% with a standard deviation of 5%, the empirical rule can be used to estimate the probability of the stock’s return falling within one standard deviation of the mean (i.e., between 5% and 15%).

Another way the empirical rule is used is to identify outliers or extreme values in a data set. In financial analysis, outliers can indicate unusual or unexpected events that may have a significant impact on investment performance. By identifying these outliers, analysts can better understand the potential risks and opportunities associated with a particular investment.

The empirical rule is also used to assess the normality of a distribution. In financial analysis, it is often assumed that returns on investments follow a normal distribution. By applying the empirical rule, analysts can determine whether the distribution of returns deviates significantly from a normal distribution. This information can be useful in evaluating the validity of certain financial models and assumptions.

Overall, the empirical rule is a valuable tool in financial analysis as it provides a simple yet powerful way to understand and analyze the distribution of data. By applying this rule, analysts can make more informed decisions about investments and assess the potential risks and rewards associated with different options.

Emily Bibb simplifies finance through bestselling books and articles, bridging complex concepts for everyday understanding. Engaging audiences via social media, she shares insights for financial success. Active in seminars and philanthropy, Bibb aims to create a more financially informed society, driven by her passion for empowering others.