Explain the 68-95-99.7 rule and its significance in understanding the spread of data in a normal distribution.

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27-04-2025
Mathematics

High School

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Explain the 68-95-99.7 rule and its significance in understanding the spread of data in a normal distribution.

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samayetaghosal5555 samayetaghosal5555 · Answer 1 · 2024-11-25T04:26:51-05:00

The 68-95-99.7 rule, also known as the Empirical Rule, describes how data is distributed in a normal distribution, indicating that about 68%, 95%, and 99.7% of data fall within one, two, and three standard deviations of the mean, respectively.

The 68-95-99.7 rule, often referred to as the Empirical Rule, is significant for understanding the spread of data in a normal distribution. It states that for a dataset with a bell-shaped and symmetric distribution:

Approximately 68% of the data fall within one standard deviation of the mean.

About 95% of the data fall within two standard deviations of the mean.

Over 99.7% of the data fall within three standard deviations of the mean.

This rule allows us to quickly estimate the proportion of data values that are close to the mean and those that are more extreme. For example, if we have a normal distribution with a mean (µ) of 50 and a standard deviation (σ) of 10, the rule tells us that:

68% of the distribution is between 40 and 60 (µ ± σ).

Approximately 95% is between 30 and 70 (µ ± 2σ).

About 99.7% is between 20 and 80 (µ ± 3σ).

The rule highlights the predictability and consistency of the normal distribution, making it a powerful tool for statistical analysis and probability inference.