The Empirical Rule, often called the 68-95-99.7 rule, is a statistical guideline used to understand the spread of data. It is commonly associated with a specific type of data distribution. This article explores whether this rule can be reliably used for data that does not follow a normal distribution.
Understanding the Empirical Rule
The Empirical Rule describes how data points are distributed around the mean in a specific type of dataset. It states that approximately 68% of data falls within one standard deviation of the mean. About 95% of data points are expected to lie within two standard deviations. Nearly all data, about 99.7%, will be found within three standard deviations from the mean.
These percentages provide a way to estimate the proportion of observations within certain ranges. For example, if heights of a large population are normally distributed with a known mean and standard deviation, this rule allows an estimation of the percentage of people whose heights fall within specified intervals around the average height. The rule serves as a practical tool for understanding data spread when certain conditions are met.
The Significance of Normal Distribution
The Empirical Rule is intrinsically linked to the normal distribution, also known as the Gaussian distribution or bell curve. A normal distribution is characterized by its symmetrical, bell-shaped curve. In this distribution, the mean, median, and mode are all equal and located at the center.
This symmetry ensures that data points taper off equally in both directions from the central peak. The predictable shape of the normal curve allows the precise percentages of the Empirical Rule to hold true. Without these characteristics, the fixed proportions would not apply.
Limitations with Non-Normal Data
The Empirical Rule cannot be reliably used for distributions that deviate from normal. Many real-world datasets are non-normal and exhibit different characteristics that invalidate the rule’s fixed percentages. Common types of non-normal distributions include skewed distributions, which are asymmetrical.
In a positively (right) skewed distribution, the tail extends longer on the right side, meaning there are more extreme high values, such as income distribution. Conversely, a negatively (left) skewed distribution has a longer tail on the left, indicating more extreme low values. These asymmetries mean that the data is not evenly spread around the mean, making the 68-95-99.7 percentages inaccurate.
Distributions can also differ in kurtosis, which describes the shape of their tails and peak, further affecting data spread.
Broader Applications and Alternative Approaches
While the Empirical Rule is specific to normal distributions, it can offer a rough approximation for data that is nearly normal, unimodal, and symmetric. However, for data that significantly deviates from normality, more general statistical tools are necessary.
Chebyshev’s Inequality is a notable alternative, providing a minimum percentage of data that falls within a certain number of standard deviations from the mean for any distribution, regardless of its shape. For example, Chebyshev’s Inequality guarantees that at least 75% of data will fall within two standard deviations of the mean, and at least 89% within three standard deviations. These bounds are less precise than the Empirical Rule’s percentages but are universally applicable.
When dealing with non-normal data, analysts also employ other methods:
- Using descriptive statistics like the median and interquartile range.
- Visualizing data with histograms and box plots.
- Applying data transformations to make the data more normal.
- Using non-parametric statistical tests, which do not assume a specific distribution for the data.