How To Use Empirical Rule To Find Percentage

The empirical rule, also known as the 68-95-99.7 rule, is a statistical rule that states that for a normal distribution, almost all observed data will fall within three standard deviations of the mean. Understanding and applying this rule is crucial for quickly estimating probabilities and understanding data spread in various fields, from finance to engineering. This article provides a comprehensive guide on how to use the empirical rule to find percentages in a normally distributed dataset.

Understanding the Empirical Rule

The empirical rule is a convenient way to estimate the proportion of data that falls within certain ranges around the mean in a normal distribution. A normal distribution, often visualized as a bell curve, is symmetrical, with the mean, median, and mode being equal and located at the center. The rule breaks down as follows:

• Approximately 68% of the data falls within one standard deviation of the mean (μ ± 1σ).
• Approximately 95% of the data falls within two standard deviations of the mean (μ ± 2σ).
• Approximately 99.7% of the data falls within three standard deviations of the mean (μ ± 3σ).

Key Components

Before diving into how to use the empirical rule, let's clarify the key components:

• Mean (μ): The average value of the dataset.
• Standard Deviation (σ): A measure of the spread or dispersion of the data around the mean. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range.

Steps to Apply the Empirical Rule

To effectively use the empirical rule, follow these steps:

1. Verify Normality: Ensure that the data is approximately normally distributed.
2. Calculate the Mean (μ) and Standard Deviation (σ): Determine these values from the dataset.
3. Determine the Interval: Identify the interval of interest in terms of standard deviations from the mean.
4. Apply the Empirical Rule: Use the rule to estimate the percentage of data within the specified interval.

Step 1: Verify Normality

The empirical rule applies specifically to data that follows a normal distribution. While real-world data may not be perfectly normal, the empirical rule can still provide a reasonable approximation if the data is approximately normal.

• Visual Inspection: Create a histogram or a normal probability plot (Q-Q plot) to visually assess the distribution. A bell-shaped histogram or a Q-Q plot where the data points fall close to a straight line suggests normality.
• Statistical Tests: Perform statistical tests such as the Shapiro-Wilk test or the Kolmogorov-Smirnov test to formally test for normality. However, these tests can be sensitive to sample size and may not always be necessary for practical applications.

If the data deviates significantly from a normal distribution, the empirical rule may not be applicable.

Step 2: Calculate the Mean (μ) and Standard Deviation (σ)

Calculate the mean and standard deviation of the dataset. These are fundamental to applying the empirical rule.

• Mean (μ): The sum of all values divided by the number of values.
  • μ = (Σxᵢ) / n
• Standard Deviation (σ): A measure of the spread of the data.
  • σ = √[(Σ(xᵢ - μ)²) / (n - 1)] for a sample
  • σ = √[(Σ(xᵢ - μ)²) / n] for a population

Use statistical software (e.g., R, Python, Excel) to calculate these values accurately, especially for large datasets.

Step 3: Determine the Interval

Identify the interval for which you want to find the percentage of data. This interval should be defined in terms of standard deviations from the mean.

• Example: Determine the percentage of data within one standard deviation of the mean (μ ± 1σ), within two standard deviations (μ ± 2σ), or within three standard deviations (μ ± 3σ).

You might also be interested in finding the percentage of data above or below a certain value. In such cases, determine how many standard deviations away from the mean that value is.

Step 4: Apply the Empirical Rule

Apply the empirical rule to estimate the percentage of data within the specified interval.

• Within One Standard Deviation (μ ± 1σ): Approximately 68% of the data falls within this range.
• Within Two Standard Deviations (μ ± 2σ): Approximately 95% of the data falls within this range.
• Within Three Standard Deviations (μ ± 3σ): Approximately 99.7% of the data falls within this range.

If you need to find the percentage of data in one tail (either above or below a certain value), divide the corresponding percentage by 2.

Examples of Applying the Empirical Rule

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

Example 1: Exam Scores

Suppose the scores on an exam are normally distributed with a mean (μ) of 75 and a standard deviation (σ) of 10.

1. Percentage of scores between 65 and 85:
   • This interval is within one standard deviation of the mean (75 ± 10).
   • According to the empirical rule, approximately 68% of the scores fall between 65 and 85.
2. Percentage of scores between 55 and 95:
   • This interval is within two standard deviations of the mean (75 ± 20).
   • Approximately 95% of the scores fall between 55 and 95.
3. Percentage of scores between 45 and 105:
   • This interval is within three standard deviations of the mean (75 ± 30).
   • Approximately 99.7% of the scores fall between 45 and 105.
4. Percentage of scores above 85:
   • 85 is one standard deviation above the mean.
   • Since 68% of scores are within one standard deviation, 32% are outside this range.
   • Half of this 32% is above 85 (the other half is below 65).
   • Therefore, approximately 16% of the scores are above 85.
5. Percentage of scores below 55:
   • 55 is two standard deviations below the mean.
   • Since 95% of scores are within two standard deviations, 5% are outside this range.
   • Half of this 5% is below 55 (the other half is above 95).
   • Therefore, approximately 2.5% of the scores are below 55.

Example 2: Heights of Adults

The heights of adult males in a certain population are normally distributed with a mean (μ) of 70 inches and a standard deviation (σ) of 3 inches.

1. Percentage of males with heights between 67 and 73 inches:
   • This interval is within one standard deviation of the mean (70 ± 3).
   • Approximately 68% of males have heights between 67 and 73 inches.
2. Percentage of males with heights between 64 and 76 inches:
   • This interval is within two standard deviations of the mean (70 ± 6).
   • Approximately 95% of males have heights between 64 and 76 inches.
3. Percentage of males with heights less than 64 inches:
   • 64 is two standard deviations below the mean.
   • Since 95% of heights are within two standard deviations, 5% are outside this range.
   • Half of this 5% is below 64 (the other half is above 76).
   • Therefore, approximately 2.5% of males have heights less than 64 inches.
4. Percentage of males with heights greater than 76 inches:
   • 76 is two standard deviations above the mean.
   • Since 95% of heights are within two standard deviations, 5% are outside this range.
   • Half of this 5% is above 76 (the other half is below 64).
   • Therefore, approximately 2.5% of males have heights greater than 76 inches.

Example 3: Manufacturing Process

A machine manufactures bolts with a mean diameter (μ) of 10 mm and a standard deviation (σ) of 0.1 mm. The diameters are normally distributed.

1. Percentage of bolts with diameters between 9.9 mm and 10.1 mm:
   • This interval is within one standard deviation of the mean (10 ± 0.1).
   • Approximately 68% of bolts have diameters between 9.9 mm and 10.1 mm.
2. Percentage of bolts with diameters between 9.8 mm and 10.2 mm:
   • This interval is within two standard deviations of the mean (10 ± 0.2).
   • Approximately 95% of bolts have diameters between 9.8 mm and 10.2 mm.
3. Percentage of bolts with diameters less than 9.7 mm:
   • 9.7 is three standard deviations below the mean.
   • Since 99.7% of diameters are within three standard deviations, 0.3% are outside this range.
   • Half of this 0.3% is below 9.7 (the other half is above 10.3).
   • Therefore, approximately 0.15% of bolts have diameters less than 9.7 mm.
4. Percentage of bolts with diameters greater than 10.2 mm:
   • 10.3 is three standard deviations above the mean.
   • Since 99.7% of diameters are within three standard deviations, 0.3% are outside this range.
   • Half of this 0.3% is above 10.3 (the other half is below 9.7).
   • Therefore, approximately 0.15% of bolts have diameters greater than 10.3 mm.

Limitations of the Empirical Rule

While the empirical rule is a useful tool, it has limitations:

• Normality Assumption: It assumes that the data is normally distributed. If the data is significantly non-normal, the rule may not provide accurate estimates.
• Approximation: The percentages (68%, 95%, 99.7%) are approximations. For more precise calculations, especially when dealing with values that are not exactly one, two, or three standard deviations from the mean, z-scores and standard normal distribution tables should be used.
• Applicability: The rule is most useful for quick estimations and understanding the spread of data. It is not a substitute for more sophisticated statistical analysis when accuracy is critical.

Advanced Considerations

Using Z-Scores for Precise Calculations

For more precise calculations, particularly when dealing with values that are not exactly one, two, or three standard deviations from the mean, use z-scores. The z-score measures how many standard deviations an element is from the mean.

• Formula: z = (x - μ) / σ
• Where:
  • x is the value of interest.
  • μ is the mean of the dataset.
  • σ is the standard deviation of the dataset.

After calculating the z-score, you can use a standard normal distribution table (also known as a z-table) or statistical software to find the percentage of data below that z-score. To find the percentage above the z-score, subtract the table value from 1.

Example: Using Z-Scores

Suppose the heights of adult women are normally distributed with a mean (μ) of 64 inches and a standard deviation (σ) of 2.5 inches. What percentage of women are taller than 68 inches?

1. Calculate the z-score:
   • z = (68 - 64) / 2.5 = 1.6
2. Use a z-table:
   • Look up the z-score of 1.6 in a standard normal distribution table. The table value is approximately 0.9452. This means that 94.52% of women are shorter than 68 inches.
3. Find the percentage taller than 68 inches:
   • 1 - 0.9452 = 0.0548
   • Therefore, approximately 5.48% of women are taller than 68 inches.

Practical Applications

The empirical rule is widely used in various fields for quick data analysis and decision-making.

• Finance: Estimating the range of stock prices or investment returns. For example, if the average return of a stock is 8% with a standard deviation of 2%, the empirical rule can help estimate the range within which the returns are likely to fall.
• Quality Control: Monitoring manufacturing processes to ensure that products meet specifications. If the diameter of a manufactured part has a mean of 5 cm and a standard deviation of 0.05 cm, the empirical rule can help determine if the process is under control.
• Healthcare: Analyzing patient data, such as blood pressure or cholesterol levels. For instance, if systolic blood pressure has a mean of 120 mmHg and a standard deviation of 10 mmHg, the empirical rule can provide insights into the distribution of blood pressure levels in a population.
• Education: Evaluating test scores and student performance. The examples given earlier about exam scores are a direct application of the empirical rule in education.
• Sports Analytics: Assessing player performance metrics, such as batting averages in baseball or scoring rates in basketball. By understanding the distribution of these metrics, coaches and analysts can make informed decisions about player selection and strategy.

Conclusion

The empirical rule is a powerful yet simple tool for estimating probabilities in a normal distribution. By understanding the mean and standard deviation of a dataset, you can quickly approximate the percentage of data within one, two, or three standard deviations of the mean. While it has limitations, particularly for non-normal data and when precise calculations are needed, the empirical rule provides valuable insights and a quick way to understand the spread and distribution of data in numerous real-world applications. For more accurate calculations, especially when dealing with values that are not exact multiples of the standard deviation from the mean, z-scores and standard normal distribution tables should be employed. By mastering these techniques, you can enhance your ability to analyze and interpret data effectively.