Our empirical rule calculator can help you to determine if 68%, 95%, and 99.7% of your data fall within three key ranges.
In statistics, data distribution is important as it helps make knowledgeable decisions in many fields, such as healthcare, engineering, social sciences, and business. The Empirical Rule (also called the 68-95-99.7 rule) states that the 68% of the data will range within 1 standard deviation (σ) of the mean (μ), while the 95% will fall within 2 standard deviations (σ) of the mean (μ), and the 99.7% of data will range within 3 standard deviations (σ) of the mean(μ).
The Empirical Rule (also known as the "68-95-99.7 rule", the "3σ rule" or the "three-sigma rule") is a statistical rule that applies to the data sets with a normally (bell-shaped) distribution. It estimates how the data is spread around the mean (μ) in the form of standard deviation (σ).
Specifically, it states that:
Our Empirical Rule Calculator not only provides the result but also a complete step-by-step procedure of calculation and formula, which helps beginners. It simply show a formula and each step involved in the calculation to get the final result, making our calculator straightforward and helps those who were unfamiliar with empirical rules.
Let's Understand the empirical rule more closely.
To understand the Empirical Rule, there is essential two key concepts that need to be understood first:
Now, let's breakdown the 68-95-99.7 rule.
68% of data within One Standard Deviation (μ ± σ):
(μ - σ)(μ + σ)95% Within Two Standard Deviations (μ ± 2σ):
μ - 2σμ + 2σ99.7% Within Three Standard Deviations (μ ± 3σ):
μ - 3σμ + 3σLet's provide an example to help understand the empirical rule and its application. Suppose the average Intelligence Quotient (IQ) in a population is 100, while with a standard deviation of 15, let's look at the calculation of IQ ranges using the 68-95-99.7 rule.
1. 68% of IQ Scores Within One Standard Deviation (μ ± σ)
μ − σ = 100 − 15 = 85μ + σ = 100 + 15 = 11568% of individuals have an IQ score between 85 and 115.
2. 95% of IQ Scores Within Two Standard Deviations (μ ± 2σ)
μ − 2σ = 100 − (2 × 15) = 100 − 30 = 70μ + 2σ = 100 + (2 × 15) = 100 + 30 = 13095% have an IQ score between 70 and 130.
3. 99.7% of IQ Scores Within Three Standard Deviations (μ ± 3σ)
μ − 3σ = 100 − (3 × 15) = 100 − 45 = 55μ + 3σ = 100 + (3 × 15) = 100 + 45 = 14599.7% have an IQ score between 55 and 145.
Table Of the Example Result to better understand:
| Percentage | Range of IQ Scores | Details |
|---|---|---|
| 68% | 85 – 115 | Most individuals fall within this IQ range. |
| 95% | 70 – 130 | A vast majority of individuals fall within this IQ range. |
| 99.7% | 55 – 145 | Almost all individuals fall within this IQ range. |
The application of empirical rules in various fields to make precise data-driven decisions. It is used to calculate the probability of data points or estimate outcomes when some data is missing.
Here are some common fields where the empirical rule is used:
No, the Empirical Rule primarily applies to data sets that follow a normal (bell-shaped) distribution. The rule may not be true for data with skewed or non-normal distributions.
The Empirical Rule may not be accurate for small data sets, but it is more reliable and accurate for large data sets. In large data sets, it increased the reliability because the distribution tends to approximate normality due to the Central Limit Theorem.