Understanding the Area Under the Normal Curve within One Standard Deviation

Have you ever wondered about the area under the normal curve within one standard deviation? Or perhaps, more to the point, why someone would seek such information about the distribution of data points on a normal curve? This article aims to provide clarity on these topics, explaining the key concepts and providing a detailed overview of the empirical rule as it applies to the normal distribution.

What is the Normal Distribution?

The normal distribution, often referred to as the Gaussian distribution, is a continuous probability distribution that is symmetrical around the mean, forming a bell-shaped curve. This distribution is significant in statistics due to the central limit theorem, which states that the sum of many independent and identically distributed random variables will tend to be normally distributed, irrespective of the original distribution of the variables themselves.

Key Terms: Z-Score and Empirical Rule

To understand the area under the normal curve within one standard deviation, it's crucial to familiarize oneself with two key concepts: the z-score and the empirical rule.

Z-Score

The z-score, also known as the standard score, is a measure of how many standard deviations an element is from the mean. It's calculated using the following formula:

[ z frac{(X - mu)}{sigma} ]X: The raw score or the variable value.(mu): The mean of the distribution.(sigma): The standard deviation of the distribution.

A z-score of 0 indicates a data point that is exactly at the mean, while a positive z-score means that the value is above the mean, and a negative z-score means the value is below the mean.

Empirical Rule

The empirical rule, also known as the 68-95-99.7 rule, is a statistical rule that applies to normal distributions. It states that:

About 68% of the values lie within one standard deviation of the mean.About 95% of the values lie within two standard deviations of the mean.About 99.7% of the values lie within three standard deviations of the mean.

Area Under the Normal Curve within One Standard Deviation

To address the original question regarding the area under the normal curve between z-scores of -1 and 1, the empirical rule provides a straightforward answer. As per the rule, approximately 68% of the data points lie within one standard deviation from the mean. In other words, the area under the normal curve between z -1 and z 1 is about 68%, which is illustrated schematically as follows:

Schematic Representation:

This means that if we take a random sample from a normally distributed population, around 68% of the samples will have values between one standard deviation below the mean and one standard deviation above the mean.

Conclusion and Further Reading

This article has aimed to provide a clear understanding of the area under the normal curve within one standard deviation through the concepts of z-scores and the empirical rule. If you're looking for a deeper dive into related topics, the following resources may be helpful:

Textbooks on introductory statistics and resources such as Khan Academy, Coursera, and edX, which offer comprehensive courses on and data science forums such as Stack Overflow and Reddit's r/statistics, where you can ask more detailed questions and discuss ideas.

Remember that places like Quora should be used for crowdsourced knowledge and not as a substitute for conducting your own research. Most answers to statistical questions can be found in textbooks or online resources. Happy exploring!