## **The Empirical Rule Explained**

The Empirical Rule, also known as the 68-95-99.7 rule, holds an influential place in the realm of statistics. While statistics might seem daunting, concepts like the Empirical Rule turn data into understandable insights. Understanding the Empirical Rule not only simplifies the interpretation of data, but also facilitates effective decision-making in various fields, including business, economics, social sciences, health sciences, and more. In this blog post, we'll delve into the fundamentals of the Empirical Rule, explore practical examples, and discuss how this rule intertwines with the concept of z-scores and the z-table.

## Unpacking the Empirical Rule

The Empirical Rule is a statistical guideline for data that is normally distributed — data that forms a "bell curve" when plotted on a graph. Normal distribution is symmetrical and has the mean (average), median (middle value), and mode (most frequent value) at its center.

According to the Empirical Rule:

According to the Empirical Rule:

- Approximately 68% of data falls within one standard deviation from the mean.
- About 95% of data lies within two standard deviations from the mean.
- Nearly 99.7% of data resides within three standard deviations from the mean.

## The Empirical Rule in Practice: Real-World Examples

**Example 1: Education**

Consider a large university course where the final exam scores follow a normal distribution. The mean score is 72, and the standard deviation is 15.

By applying the Empirical Rule:

- About 68% of students have scored between 57 and 87 (72 ± 15).
- Around 95% of students have scores between 42 and 102 (72 ± 2*15).
- Nearly all students (99.7%) have scores between 27 and 117 (72 ± 3*15).

**Example 2: Business**

In a manufacturing company, the production cost of a particular item follows a normal distribution, with a mean of $150 and a standard deviation of $20.

According to the Empirical Rule:

- Roughly 68% of items cost between $130 and $170 to produce.
- Approximately 95% cost between $110 and $190.
- Almost all items (99.7%) cost between $90 and $210 to produce.

## Empirical Rule and Z-Scores: An Intertwined Relationship

The Z-score, or standard score, is another fundamental concept in statistics. It measures how many standard deviations an element is from the mean. A positive Z-score indicates the data point is above the mean, while a negative score means it's below the mean.

The Empirical Rule and Z-scores are closely related. Essentially, the Empirical Rule provides the percentage of data within 1, 2, and 3 standard deviations from the mean, which are z-scores of 1, 2, and 3 respectively. For instance, a Z-score of 2 implies that the data point is two standard deviations above the mean, which aligns with the Empirical Rule's statement that 95% of data falls within two standard deviations of the mean in a normal distribution.

The Empirical Rule and Z-scores are closely related. Essentially, the Empirical Rule provides the percentage of data within 1, 2, and 3 standard deviations from the mean, which are z-scores of 1, 2, and 3 respectively. For instance, a Z-score of 2 implies that the data point is two standard deviations above the mean, which aligns with the Empirical Rule's statement that 95% of data falls within two standard deviations of the mean in a normal distribution.

## Z-Table: Linking the Empirical Rule to Probabilities

A Z-table, also known as the standard normal table, is a mathematical table that displays the probability that a normally distributed random variable Z is less than or equal to z, given some value of Z. It provides the area (probability) under the curve to the left of any specific Z-score.

Let's consider the Empirical Rule again. When you know that 68% of values are within one standard deviation (or a Z-score of 1), the Z-table can be used to find the probability of a value being within this range. The Z-table gives you a probability of 0.8413 for Z=1. This means that the probability of a value being less than one standard deviation above the mean is 0.8413, or 84.13%.

But you might ask: isn't this supposed to be 68% according to the Empirical Rule? Remember, the Empirical Rule's 68% covers both sides of the mean, while the Z-table gives you the probability from the mean to a Z-score on one side. To get the probability for both sides, you would subtract the Z-table value for Z=0 (which is 0.5 or 50%) from the value for Z=1, then double the result. That is, (0.8413 - 0.5) * 2 = 0.6826 or 68.26%, which aligns with the Empirical Rule.

Let's consider the Empirical Rule again. When you know that 68% of values are within one standard deviation (or a Z-score of 1), the Z-table can be used to find the probability of a value being within this range. The Z-table gives you a probability of 0.8413 for Z=1. This means that the probability of a value being less than one standard deviation above the mean is 0.8413, or 84.13%.

But you might ask: isn't this supposed to be 68% according to the Empirical Rule? Remember, the Empirical Rule's 68% covers both sides of the mean, while the Z-table gives you the probability from the mean to a Z-score on one side. To get the probability for both sides, you would subtract the Z-table value for Z=0 (which is 0.5 or 50%) from the value for Z=1, then double the result. That is, (0.8413 - 0.5) * 2 = 0.6826 or 68.26%, which aligns with the Empirical Rule.

## Final Thoughts

The Empirical Rule, Z-scores, and the Z-table are powerful tools that bring structure and clarity to the seemingly chaotic world of data. They allow us to make predictions, form strategies, and understand the world in a more quantified manner. Whether you're a student, a business professional, a researcher, or just a curious mind, grasping these concepts opens up a new perspective on how you interpret and handle information.

If you need to run a quick empirical rule calculation for a normal data set check out our Empirical Rule Calculator.

If you need to run a quick empirical rule calculation for a normal data set check out our Empirical Rule Calculator.

## Empirical Rule FAQs

Q1: What is the Empirical Rule? A1: The Empirical Rule states that in a normal distribution, about 68% of data falls within one standard deviation, 95% within two, and 99.7% within three.

Q2: Why is the Empirical Rule important? A2: It's a quick way to understand the distribution of data in statistics, offering insights about data spread and likelihood of certain values.

Q3: Can the Empirical Rule be used for all data sets? A3: It applies only to normally distributed (bell-shaped curve) data sets.

Q4: How does the Empirical Rule relate to standard deviation? A4: It quantifies how data is spread around the mean in terms of standard deviations.

Q5: What's the link between Empirical Rule and z-scores? A5: Z-scores measure how many standard deviations a value is from the mean, closely related to the Empirical Rule.

Q6: Is the Empirical Rule accurate for real-world data?

A6: While it provides an excellent framework for understanding data distribution, it may not be precisely accurate due to possible outliers or non-normal distribution in real-world data.

Q7: Can we use the Empirical Rule for skewed distributions?

A7: No, the Empirical Rule is best applied to symmetric, bell-shaped distributions (normal distributions). For skewed or heavily-tailed distributions, it may not provide accurate estimations.

Q8: What is the connection between the Empirical Rule and the Central Limit Theorem?

A8: Both relate to normal distributions. The Central Limit Theorem states that, with a large enough sample size, the sampling distribution of the mean will approximate a normal distribution, regardless of the shape of the population distribution.

Q9: How does the Empirical Rule relate to confidence intervals?

A9: The Empirical Rule helps to estimate confidence intervals in a normal distribution, with 68%, 95%, and 99.7% confidence intervals corresponding to one, two, and three standard deviations from the mean, respectively.

Q10: How is the Empirical Rule used in quality control?

A10: In quality control, the Empirical Rule can be used to identify whether a process is in control. If data points fall outside the expected range (more than three standard deviations from the mean), it suggests a process anomaly.

Q2: Why is the Empirical Rule important? A2: It's a quick way to understand the distribution of data in statistics, offering insights about data spread and likelihood of certain values.

Q3: Can the Empirical Rule be used for all data sets? A3: It applies only to normally distributed (bell-shaped curve) data sets.

Q4: How does the Empirical Rule relate to standard deviation? A4: It quantifies how data is spread around the mean in terms of standard deviations.

Q5: What's the link between Empirical Rule and z-scores? A5: Z-scores measure how many standard deviations a value is from the mean, closely related to the Empirical Rule.

Q6: Is the Empirical Rule accurate for real-world data?

A6: While it provides an excellent framework for understanding data distribution, it may not be precisely accurate due to possible outliers or non-normal distribution in real-world data.

Q7: Can we use the Empirical Rule for skewed distributions?

A7: No, the Empirical Rule is best applied to symmetric, bell-shaped distributions (normal distributions). For skewed or heavily-tailed distributions, it may not provide accurate estimations.

Q8: What is the connection between the Empirical Rule and the Central Limit Theorem?

A8: Both relate to normal distributions. The Central Limit Theorem states that, with a large enough sample size, the sampling distribution of the mean will approximate a normal distribution, regardless of the shape of the population distribution.

Q9: How does the Empirical Rule relate to confidence intervals?

A9: The Empirical Rule helps to estimate confidence intervals in a normal distribution, with 68%, 95%, and 99.7% confidence intervals corresponding to one, two, and three standard deviations from the mean, respectively.

Q10: How is the Empirical Rule used in quality control?

A10: In quality control, the Empirical Rule can be used to identify whether a process is in control. If data points fall outside the expected range (more than three standard deviations from the mean), it suggests a process anomaly.