How to Use the Empirical Rule Calculator
This empirical rule calculator allows you to quickly find the 68-95-99.7 rule based on the mean and standard deviation of a normally distributed variable. To use the tool:
- Enter the mean, μ.
- Enter the standard deviation, σ.
- Click Calculate.
The calculator will instantly return the 68-95-99.7 rule ranges and show you exactly how to find these ranges manually, step-by-step. The tool also displays a graph to help you see how the intervals expand as you move farther from the mean.
Example 1
Suppose test scores are normally distributed with a mean of 70 and a standard deviation of 10. Find the intervals that contain about 68%, 95%, and 99.7% of the scores.
Solution
Here, the mean is 70 and the standard deviation is 10. To find the required intervals using the calculator:
- Enter mean, μ = 70
- Enter standard deviation, σ = 10
- Click Calculate
The calculator returns the correct intervals as follows:
- 68% of the data falls between 60 and 80.
- 95% of the data falls between 50 and 90.
- 99.7% of the data falls between 40 and 100.
This means about 68% of students scored between 60 and 80. About 95% scored between 50 and 90. Almost all students, about 99.7%, scored between 40 and 100.
Want to find a probability from a z-score? Use the Z-score probability calculator.
Example 2
Suppose a company fills bags of flour with an average weight of 500 grams. The standard deviation is 4 grams, and the weights are approximately normally distributed. Use the empirical rule to find the main weight intervals.
Solution
In this example:
- Mean, μ = 500
- Standard deviation, σ = 4
To find the correct intervals using the calculator:
- Enter mean, μ = 500
- Enter standard deviation, σ = 4
- Click Calculate
The calculator returns the 68%, 95%, and 99.7% intervals as follows:
- 68% of the data falls between 496 and 504 grams.
- 95% of the data falls between 492 and 508 grams.
- 99.7% of data falls between 488 and 512 grams.
This means about 95% of bags are expected to weigh between 492 and 508 grams.
What Is the Empirical Rule?
The empirical rule is a shortcut used to describe how data is spread in a normal distribution. It is also called the 68-95-99.7 rule because it gives three common percentages for data that falls near the mean.
For a normal distribution:
- 68% of the data falls within 1 standard deviation of the mean.
- 95% of the data falls within 2 standard deviations of the mean.
- 99.7% of data falls within 3 standard deviations of the mean.
This rule is useful because it gives you a quick way to understand the spread of a bell-shaped distribution. For example, if the mean is 10 and the standard deviation is 2, the empirical rule tells you that 68% of the data falls between 8 and 12, 95% falls between 6 and 14, and 99.7% falls between 4 and 16.
Empirical Rule Formula
The empirical rule uses the mean and standard deviation to create three intervals. Thus, the empirical rule formula varies depending on the intervals you want to calculate. Thus, the formulas are:
- 68% range: μ − σ to μ + σ
- 95% range: μ − 2σ to μ + 2σ
- 99.7% range: μ − 3σ to μ + 3σ
Sometimes, the formulas are written as:
- 68% range: μ ± σ
- 95% range: μ ± 2σ
- 99.7% range: μ ± 3σ
In the formulas, μ is the mean, and σ is the standard deviation.
How to Interpret the 68-95-99.7 Rule
The empirical rule is centered on the mean. Since a normal distribution is symmetric, the left and right sides of the curve are mirror images.
That means the 68% range is split evenly around the mean. About 34% of values fall between the mean and 1 standard deviation above it, and about 34% fall between the mean and 1 standard deviation below it.
The same idea applies to the 95% and 99.7% ranges.
This is helpful when you want to estimate values in the tails. For example, if 95% of values fall within 2 standard deviations, then about 5% fall outside that range. Since the distribution is symmetric, about 2.5% fall below the lower limit and about 2.5% fall above the upper limit.
When Should You Use the Empirical Rule?
Use the empirical rule when the data is approximately normal. In simple terms, the distribution should look like a bell curve.
The empirical rule is commonly used for:
- Test scores
- Heights
- Measurement errors
- Product weights
- IQ scores
- Standardized exam results
- Quality control data
Note. The empirical rule is not the best choice when the data is strongly skewed, has many outliers, or does not follow a bell-shaped pattern.
Frequently Asked Questions
An empirical rule calculator is a tool that uses the mean and standard deviation to find the intervals that contain about 68%, 95%, and 99.7% of data in a normal distribution.
The 68-95-99.7 rule says that about 68% of data falls within 1 standard deviation of the mean, 95% falls within 2 standard deviations, and 99.7% falls within 3 standard deviations.
No. The empirical rule should only be used when the data is approximately normally distributed. If the distribution is skewed, the results may be misleading.
Within 1 standard deviation means the interval from mean − 1 standard deviation to mean + 1 standard deviation. In a normal distribution, about 68% of values fall in this range.
About 5% of values fall outside 2 standard deviations in a normal distribution. Since the normal curve is symmetric, about 2.5% fall below the lower limit and about 2.5% fall above the upper limit.
The empirical rule applies to normal distributions and gives the 68%, 95%, and 99.7% estimates. However, Chebyshev’s theorem can be used with many types of distributions, but it gives more conservative minimum percentages.
About 0.3% of values fall outside 3 standard deviations in a normal distribution. This is why values beyond 3 standard deviations are often considered unusual.
