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

## Definition

The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical guideline in a normal distribution which states that approximately 68% of data lies within one standard deviation from the mean, around 95% lies within two standard deviations, and 99.7% falls within three standard deviations. The formula for standard deviation is the square root of the variance, which involves finding the mean, subtracting it from each value, squaring the results, getting the average of these squares, and then finally finding the square root. This rule is particularly useful in finance to predict probabilities of certain outcomes based on historical data.

### Phonetic

The phonetic pronunciation for “Empirical Rule: Definition, Formula, Example, How It’s Used” is as follows:Empirical Rule: /ɛmˈpɪrɪkəl ruːl/ Definition: /ˌdɛfɪˈnɪʃən/Formula: /ˈfɔːrmjuːlə/Example: /ɪgˈzɑːmpəl/How It’s Used: /haʊ ɪts juːzd/Please note that these phonetic pronunciations use the International Phonetic Alphabet (IPA). Different dialects or accents may pronounce words slightly differently.

## Key Takeaways

1. Definition: The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical rule which states that for a normal distribution, nearly all of the data will fall within three standard deviations of the mean. The name “68-95-99.7” comes from the rule’s assertion that around 68% of data falls within the first standard deviation from the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
2. Formula: No specific formula exists for the Empirical Rule since it’s a guideline concerning the proportion of data one can expect to lie within one, two, or three standard deviations from the mean in a normal distribution. However, it can be practically used in related calculations. For example, to calculate the range within which 95% of data lies, for a given mean (µ) and standard deviation (σ), we can use the formula: range = µ + 2σ (upper limit) and µ – 2σ (lower limit).
3. Application and Uses: The Empirical Rule is majorly used in statistics for predicting the outcome and making sense of the data. It’s an essential concept in many fields like finance, social sciences, and quality control. Risk managers often use the rule to evaluate stock market returns and to model potential losses. It can be used to identify outliers and to understand data patterns in quality control and process improvement initiatives, among others.

## Importance

The Empirical Rule, also known as the 68-95-99.7 rule, is a critical concept in business and finance due to its application in statistical analysis and forecasting. This rule is a shorthand to understand the distribution of data within a standard deviation in a normal distribution. In essence, it provides that approximately 68% of data falls within one standard deviation, 95% within two standard deviations, and 99.7% within three standard deviations from the mean. This understanding allows business analysts, financial planners, and strategists to make meaningful predictions, risk assessments, measure variability, and understand data trends, thereby facilitating effective decision-making.

## Explanation

The Empirical Rule, also known as the 68-95-99.7 rule, is a fundamental tool in statistics, serving as a shortcut to estimate the probability of a data point falling within a certain range in a normal distribution. The rule is used in various fields such as business and finance, particularly in risk management and financial forecasting due to its effectiveness. Normal distribution or the ‘bell curve’ is a key feature in finance where factors like asset prices, inflation rates, or the returns on a specific investment usually conform to this distribution.The Empirical Rule is pivotal in financial analysis as it aids in making strategic decisions. It sheds light on where majority of the values lie in a given data set, helping analysts to understand the likelihood of a specific event occurring and making predictions more coherent. For instance, the rule can predict the probability of a particular stock will fall within a certain price range, based on historical data. Financial institutions also use it to calculate the risk level of their investments, ensuring they limit potential losses while maximizing return. Understanding and applying this rule assists businesses in accurate forecasting, minimizing risk, making informed decisions, and ultimately, thriving in their industries.

## Examples

1. Stock Market Analysis: The Empirical Rule can be widely used in the stock market, especially when analysts try to determine a specific company’s stock price behavior. For instance, if a certain stock’s return is normally distributed with an average of 8% and a standard deviation of 2%. According to the Empirical Rule, we can say that the stock’s return will be between 6% (8%-1*2%) and 10%(8%+1*2%) around 68% of the time; between 4% (8%-2*2%) and 12%(8%+2*2%) around 95% of the time; and between 2% (8%-3*2%) and 14%(8%+3*2%) around 99.7% of the time. Trade decisions can be made based on these probabilities.

2. Quality Control in Manufacturing: In manufacturing, the Empirical Rule can be used to monitor the quality control process. Suppose a manufacturing company has found that the weight of its product is normally distributed with an average of 500 grams and a standard deviation of 50 grams. According to the Empirical Rule, approximately 68% of products will weigh between 450 and 550 grams; 95% will weigh between 400 and 600 grams; and 99.7% will weigh between 350 and 650 grams. If products fall outside the last range, they could potentially be considered defective.

3. Credit Score Analysis: Credit scoring companies often use the empirical rule to set credit score thresholds and evaluate the credit risk of borrowers. Suppose that credit scores for a particular company are normally distributed with a mean of 700 and a standard deviation of 50. By using the Empirical Rule, the company knows that about 68% of people have a credit score between 650 and 750, about 95% of people have a credit score between 600 and 800, and about 99.7% of people have a score between 550 and 850. This information can be helpful in determining credit terms and assessing the overall risk of their credit portfolios.

What is the definition of the Empirical Rule?

The Empirical Rule, also known as the 68-95-99.7 rule, is a statistical rule which states that for a normal distribution, nearly all data will fall within three standard deviations of the mean.

What is the formula for the Empirical Rule?

The Empirical Rule does not have a specific formula as it is based on observation and not calculation. However, it indicates that approximately 68% of data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.

Can you provide an example of the Empirical Rule?

Let’s take a set of test scores with a mean of 70 and a standard deviation of 10. According to the Empirical Rule, about 68% of students scored between 60 and 80 (one standard deviation from the mean), about 95% of students scored between 50 and 90 (two standard deviations from the mean), and about 99.7% of students scored between 40 and 100 (three standard deviations from the mean).

How is the Empirical Rule used in finance and business?

In business and finance, the Empirical Rule is often used to predict outcomes and assist in decision making. For example, predicting the stock market trends, market risks, customer behavior or even employee performance. It helps to indicate where most of the observed data is expected to fall and its deviation from the mean.

Can the Empirical Rule be used for all data distributions?

No, the Empirical Rule applies strictly to normal or bell-shaped distributions. It’s not accurate for skewed or bi-modal distributions, as these do not share the same properties.

Why is the Empirical Rule important in statistical analysis?

The Empirical Rule is an important concept in statistics as it gives a quick estimate of the spread of data in a normal distribution. It’s a useful guideline for identifying outliers and for understanding the proportion of values that lie within different intervals in a distribution.

## Related Finance Terms

• Standard Deviation: A measure that quantifies the amount of variation in a set of data values. In relation to the Empirical Rule, about 68% of data falls within one standard deviation from the mean in a normal distribution.
• Normal Distribution: Also known as the Gaussian distribution, it’s a probability function that describes how the values of a variable are distributed. The Empirical Rule applies specifically to a normal distribution.
• Bell Curve: A graphical depiction of normal distribution in statistics. Named for its bell shape, the graph represents how data is dispersed in different standard deviation zones, which is the basis for the Empirical Rule.
• Data Analysis: The process of inspecting, cleansing, transforming, and modeling data to discover useful information and support decision-making. It often involves the application of statistical measures such as the Empirical Rule.
• Outliers: Observed data points that are notably distant from the rest in a sample. According to the Empirical Rule, anything that falls outside three standard deviations can be considered an outlier.