Use The Empirical Rule To Determine The Percentage

Using the Empirical Rule to Determine Percentages in Normal Distributions

The empirical rule, also known as the 68-95-99.7 rule, is a handy statistical guideline that helps us understand the distribution of data within a normal distribution. It's a powerful tool for quickly estimating probabilities and percentages associated with different ranges of values. This article delves deep into the empirical rule, explaining its application, limitations, and how to effectively utilize it to determine percentages within a normally distributed dataset.

Understanding the Normal Distribution

Before we dive into the empirical rule, it's crucial to grasp the concept of a normal distribution. A normal distribution, often depicted as a bell curve, is a symmetrical probability distribution where the majority of data points cluster around the mean (average). The curve is characterized by its mean (μ) and standard deviation (σ). The mean represents the center of the distribution, while the standard deviation measures the spread or dispersion of the data. A smaller standard deviation indicates data points are clustered tightly around the mean, while a larger standard deviation implies a wider spread.

The beauty of the normal distribution lies in its predictability. Knowing the mean and standard deviation allows us to estimate the probability of finding a data point within a specific range. This is where the empirical rule comes into play.

The Empirical Rule: A Quick Overview

The empirical rule states that for a normal distribution:

Approximately 68% of the data falls within one standard deviation of the mean. This means that if we consider the interval (μ - σ, μ + σ), we expect to find roughly 68% of all data points within this range.
Approximately 95% of the data falls within two standard deviations of the mean. The interval (μ - 2σ, μ + 2σ) encompasses about 95% of the data points.
Approximately 99.7% of the data falls within three standard deviations of the mean. Almost all the data (99.7%) lies within the interval (μ - 3σ, μ + 3σ).

Applying the Empirical Rule: Step-by-Step Examples

Let's illustrate the application of the empirical rule with some practical examples.

Example 1: Exam Scores

Imagine a class's exam scores are normally distributed with a mean (μ) of 75 and a standard deviation (σ) of 10. Using the empirical rule, we can answer several questions:

Question 1: What percentage of students scored between 65 and 85?

Solution: The range 65-85 represents one standard deviation from the mean (75 ± 10). According to the empirical rule, approximately 68% of students scored within this range.

Question 2: What percentage of students scored between 55 and 95?

Solution: The range 55-95 represents two standard deviations from the mean (75 ± 20). The empirical rule tells us that roughly 95% of students scored within this range.

Question 3: What percentage of students scored below 55 or above 95?

Solution: Since 95% of students scored between 55 and 95, the remaining 5% scored either below 55 or above 95.

Example 2: Manufacturing Process

A manufacturing company produces bolts with a mean length (μ) of 10 cm and a standard deviation (σ) of 0.2 cm. The lengths are normally distributed.

Question 1: What percentage of bolts have lengths between 9.8 cm and 10.2 cm?

Solution: This range represents one standard deviation from the mean (10 ± 0.2). The empirical rule suggests that approximately 68% of the bolts fall within this length range.

Question 2: What percentage of bolts have lengths outside the range 9.6 cm to 10.4 cm?

Solution: This range is two standard deviations from the mean (10 ± 0.4). The empirical rule indicates 95% of bolts fall within this range. Therefore, 5% of bolts have lengths outside this range.

Example 3: Height Distribution

Let's assume the heights of adult women in a particular country are normally distributed with a mean (μ) of 165 cm and a standard deviation (σ) of 5 cm.

Question 1: What percentage of women have heights between 160 cm and 170 cm?

Solution: This range represents one standard deviation from the mean (165 ± 5). According to the empirical rule, approximately 68% of women have heights within this range.

Question 2: What percentage of women are taller than 175 cm?

Solution: A height of 175 cm is two standard deviations above the mean (165 + 10). Since 95% of the data falls within two standard deviations of the mean, (100% - 95%) / 2 = 2.5% of women are taller than 175 cm.

Limitations of the Empirical Rule

While the empirical rule provides a quick and easy estimation, it's important to be aware of its limitations:

Approximation: The percentages given by the empirical rule are approximations. The actual percentages might slightly deviate from the 68%, 95%, and 99.7% values.
Normality Assumption: The empirical rule only applies to data that follows a normal distribution. If the data is significantly skewed or not normally distributed, the empirical rule's predictions will be inaccurate. Statistical tests like the Shapiro-Wilk test can be used to check for normality.
Limited Precision: The empirical rule doesn't provide precise percentages for ranges that don't align exactly with one, two, or three standard deviations from the mean. For more precise calculations, you'll need to utilize Z-scores and a Z-table or statistical software.

Beyond the Empirical Rule: Z-scores and the Standard Normal Distribution

For more precise estimations beyond the boundaries of the empirical rule, we utilize Z-scores. A Z-score standardizes a data point by expressing how many standard deviations it is away from the mean. The formula for calculating a Z-score is:

Z = (x - μ) / σ

Where:

x is the data point
μ is the mean
σ is the standard deviation

Once you have the Z-score, you can consult a Z-table (or use statistical software) to find the corresponding probability or percentage. This allows for precise calculations for any range of values within a normal distribution, providing greater accuracy than the empirical rule's approximations.

Conclusion: Practical Application and Further Exploration

The empirical rule serves as a valuable tool for quickly estimating percentages within a normal distribution. Its simplicity makes it accessible for a broad range of applications, from understanding exam scores to analyzing manufacturing processes. However, remember its limitations and consider employing Z-scores and the standard normal distribution for more precise calculations, particularly when dealing with ranges not directly covered by the 68-95-99.7 rule or when questioning the normality of your data. Mastering both the empirical rule and Z-scores provides a robust toolkit for effectively analyzing and interpreting data within normal distributions. Further exploration of statistical concepts, such as confidence intervals and hypothesis testing, will build upon this foundation and expand your analytical capabilities. Remember to always critically evaluate your data and choose the appropriate statistical method for your specific needs.