三西格玛原理 (68-95-99.7)

3 Sigma Rule

Also known as the "empirical rule," the 3 sigma rule provides a quick way to analyze data that follows a normal distribution.

What it entails:

• 68% of the observations fall within σ of the mean μ
• 95% of the observations fall within 2σ of the mean μ
• 99.7% of the observations fall within 3σ of the mean μ

(Video) The Normal Distribution and the 68-95-99.7 Rule

by patrickJMT

In this video by patrickJMT, you can learn about the normal distribution and the $68-95-99.7$ rule.

Summary

This video walks you through the normal distribution and the $68-95-99.7$ rule, which helps us to understand how data is spread around the mean.

In a normal distribution, $68$ of the data falls within one standard deviation of the mean, $95$ falls within two standard deviations, and $99.7$ falls within three standard deviations. For example, when looking at IQs, there is a mean of $100$ and a standard deviation of $15$. You can use the $68-95-99.7$ rule to find the values that are one, two, or three standard deviations from the mean.

We can extend this to finding the percentage of data above or below a certain value. For example, about $50$ of IQ scores are above the mean, about $16$ are below one standard deviation from the mean, and about $0.15$ are above three standard deviations from the mean.

The $68-95-99.7$ rule works with a bell-shaped, symmetric distribution and helps you estimate how much data falls within different numbers of standard deviations from the mean.

The Three Sigma Rule

We know that a normal distribution is a data pattern that looks like a bell-shaped curve. The mean, $\mu$, is at the center of the curve. The standard deviation tells us how far data values tend to be from the mean.

The $3 Sigma$ rule, also known as the $68-95-99.7$ rule, helps us to understand where most data values fall.

About $68$ of the data is within one standard deviation of the mean. About $95$ of the data is within two standard deviations of the mean. And About $99.7$ of the data is within three standard deviations of the mean.

This means that very few data values are more than three standard deviations from the mean. The rule is useful for understanding the spread of data in a normal distribution.

The Three Sigma Rule

The 3 sigma rule is a rule that helps us understand how data is spread out in a data set. It is also called the 68-95-99.7 rule or the empirical rule. The empirical rule states that data will fall within certain distances from the mean when the distribution is bell shaped. A bell-shaped distribution is also called a normal distribution. We've learned that before!

The rule is based on the number of standard deviations from the mean. About $68$ of the data will fall within one standard deviation of the mean. Approximately $95$ of the data will fall within two standard deviations. About $99.7$ of the data will fall within three standard deviations.

This pattern is why the rule is called the $68-95-99.7$ rule. The distribution is symmetric, so the data is spread out in a similar way on both sides of the mean. We can also use the area under the curve to show these percentages. For example, the area within three standard deviations represents about $99.7$ of the data.

The rule can be useful for data such as IQ scores, which are often modeled with a bell-shaped distribution. It gives us a quick way to estimate how much of the data should be near the mean. However, the rule does not work well for every data set. It is mainly used when the distribution is approximately bell shaped and symmetric - so, the data must follow a normal distribution.

The $68-95-99.7$ rule comes from empirical research about normal distributions. It is often used as a rule of thumb when we want a quick estimate. For data that is not bell shaped, Chebyshev's theorem can give us a more general way to describe how much data will fall near the mean. Understanding the $3$ Sigma Rule will help you work with normal distributions and understand how standard deviation describes the spread of data.