Understanding the 68-95-99.7 Rule and Contradiction with Data Beyond 2 Standard Deviations

In the realm of statistics and data analysis, understanding the behavior of normally distributed data is fundamental. A common point of confusion arises when considering how data beyond two standard deviations from the mean doesn't seem to align with the 68-95-99.7 rule. Let's delve into this detail to clarify the misunderstanding.

The Empirical Rule in Context

The 68-95-99.7 rule, also known as the empirical rule, is a useful heuristic for understanding the distribution of data in a normal distribution. This rule states the following:

About 68 percent of the data falls within 1 standard deviation of the mean. About 95 percent of the data falls within 2 standard deviations of the mean. About 99.7 percent of the data falls within 3 standard deviations of the mean.

These percentages are based on the cumulative distribution function (CDF) of the normal distribution. It's important to note that these percentages relate to the regions sandwiched between standard deviations, not the regions exceeding them.

Addressing the Confusion: Data More than 2 Standard Deviations from the Mean

The confusion often stems from a misinterpretation of the rule. Specifically, when someone says that data more than 2 standard deviations from the mean includes about 5 percent of the data, they are likely referring to the tails of the distribution. Here's how to understand this:

The 95 percent of the data that falls within 2 standard deviations of the mean actually means that 2.5 percent of the data is in the lower tail (below -2 standard deviations), and 2.5 percent is in the upper tail (above 2 standard deviations). This is because the normal distribution is symmetrical around the mean.

Thus, when considering the data that is more than 2 standard deviations away from the mean (both in the lower and upper tails), we are looking at the total area of these combined tails:

Total area beyond 2 standard deviations 100% - 95% 5%.

This explains why data more than 2 standard deviations from the mean is 5 percent, which is half of the 10 percent (2.5% 2.5%) that is in the tails of the distribution.

Further Clarification with Probabilities

To provide a more precise understanding, consider the following calculation:

The probability of data being less than -2 standard deviations or greater than 2 standard deviations can be calculated using the z-score. The z-score for -2 standard deviations is -2, and for 2 standard deviations is 2.

The cumulative probability for z -2 is approximately 0.0228 (or 2.28%), and the cumulative probability for z 2 is also approximately 0.0228 (or 2.28%). Therefore, the sum of these probabilities is:

0.0228 0.0228 0.0456 (or 4.56%).

This confirms that data more than 2 standard deviations from the mean is approximately 4.6%, which aligns with the 95.44% of data within 2 standard deviations.

The Central Limit Theorem and Its Implications

The Central Limit Theorem (CLT) further explains the behavior of sample means. It states that for a sufficiently large sample size, the distribution of the sample means will approximate a normal distribution, regardless of the population distribution.

When applying the CLT, the probability of a data point being more than 2 standard deviations from the mean is approximately 4.6%, as mentioned earlier. This 4.6% is the combined probability for both tails. So, when we say "MORE THAN 2 std dev," we are considering both the left and right tails, which yields a probability of 2.3% for each tail, totaling 4.6%.

This provides a more accurate understanding of the distribution of data points in a normal distribution beyond two standard deviations from the mean.

Conclusion

In conclusion, the 68-95-99.7 rule and the probabilities discussed here do not contradict each other. Understanding the distribution of data in the tails of a normal distribution, the combined probabilities of the tails, and the implications of the Central Limit Theorem all contribute to a clearer picture of the data's behavior.

By grasping these concepts, you can better analyze and interpret data in a variety of statistical and analytical contexts. Whether you're working with normally distributed data or dealing with probabilities in more complex scenarios, a solid understanding of the empirical rule and the Central Limit Theorem is invaluable.