In a normal distribution 68 of the data fall within how many standard deviations of the mean

Created by Rita Rain

Reviewed by Bogna Szyk and Jack Bowater

Last updated: Feb 15, 2022

The empirical rule calculator (also a 68 95 99 rule calculator) is a tool for finding the ranges that are 1 standard deviation, 2 standard deviations, and 3 standard deviations from the mean, in which you'll find 68, 95, and 99.7% of the normally distributed data respectively. In the text below, you'll find the definition of the empirical rule, the formula for the empirical rule, and an example of how to use the empirical rule.

If you're into statistics, you may want to read about some related concepts - z-score, confidence interval, and point estimate.

What is the empirical rule?

The empirical rule is a statistical rule (also called the three-sigma rule or the 68-95-99.7 rule) which states that, for normally distributed data, almost all of the data will fall within three standard deviations either side of the mean.

More specifically, you'll find:

• 68% of data within 1 standard deviation
• 95% of data within 2 standard deviations
• 99.7% of data within 3 standard deviations

Let's explain the concepts used in this definition:

Standard deviation is a measure of spread; it tells how much the data varies from the average, i.e., how diverse the dataset is. The smaller value, the more narrow the range of data is.

Normal distribution is a distribution that is symmetric about the mean, with data near the mean are more frequent in occurrence than data far from the mean. In graphical form, normal distributions appear as a bell shaped curve, as you can see below:

The empirical rule - formula

The algorithm below explains how to use the empirical rule:

1. Calculate the mean of your values:

   μ = (Σ xi) / n

• ∑ - sum

• xi - each individual value from your data

• n - the number of samples

1. Calculate the standard deviation:

   σ = √( ∑(xi – µ)² / (n – 1) )

2. Apply the empirical rule formula:

• 68% of data falls within 1 standard deviation from the mean - that means between μ - σ and μ + σ.

• 95% of data falls within 2 standard deviations from the mean - between μ – 2σ and μ + 2σ.

• 99.7% of data falls within 3 standard deviations from the mean - between μ - 3σ and μ + 3σ.

  Enter the mean and standard deviation into the empirical rule calculator, and it will output the intervals for you.

An example of how to use the empirical rule

Intelligence quotient (IQ) scores are normally distributed with the mean of 100 and the standard deviation equal to 15. Let's have a look at the maths behind the 68 95 99 rule calculator:

1. Mean: μ = 100

2. Standard deviation: σ = 15

3. Empirical rule formula:

   μ - σ = 100 – 15 = 85

   μ + σ = 100 + 15 = 115

   68% of people have an IQ between 85 and 115.

   μ – 2σ = 100 – 2*15 = 70

   μ + 2σ = 100 + 2*15 = 130

   95% of people have an IQ between 70 and 130.

   μ - 3σ = 100 – 3*15 = 55

   μ + 3σ = 100 + 3*15 = 145

   99.7% of people have an IQ between 55 and 145.

For quicker and easier calculations, input the mean and standard deviation into this empirical rule calculator, and watch as it does the rest for you.

Where is the empirical rule used?

The rule is widely used in empirical research, such as when calculating the probability of a certain piece of data occurring, or for forecasting outcomes when not all data is available. It gives insight into the characteristics of a population without the need to test everyone and helps to determine whether a given data set is normally distributed. It is also used to find outliers – results that differ significantly from others - which may be the result of experimental errors.

Benford's lawBeta distributionBinomial distribution… 20 more

Normal distribution is commonly associated with the 68-95-99.7 rule, or empirical rule, which you can see in the image below. Sixty-eight percent of the data is within one standard deviation (σ) of the mean (μ), 95 percent of the data is within two standard deviations (σ) of the mean (μ), and 99.7 percent of the data is within three standard deviations (σ) of the mean (μ).

This post explains how those numbers were derived in the hope that they can be more interpretable for your future endeavors. 

What Is the Empirical Rule? 

The empirical rule, also known as the 68-95-99.7 rule, represents the percentages of values within an interval for a normal distribution. That is, 68 percent of data is within one standard deviation of the mean; 95 percent of data is within two standard deviation of the mean and 99.7 percent of data is within three standard deviation of the mean.  

As always, the code used to make everything — including the graphs — is available on my GitHub. With that, let’s get started.

Empirical Rule & the Probability Density Function

To understand where the 68-95-99.7 percentages come from, it’s important to first understand the probability density function, known as the PDF. A PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. The integral of the variable’s PDF over the range gives its probability. 

What Is a Probability Density Function?

A probability density function (PDF) specifies the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. 

That is, it’s given by the area under the density function but above the horizontal axis, and by the area between the lowest and greatest values of the range. This definition might not make much sense, so let’s graph the probability density function for a normal distribution to clear it up. The probability density function for a normal distribution is represented in the equation below:

Let’s simplify it by assuming we have a mean (μ) of zero and a standard deviation (σ) of one.

Now that the function is simpler, let’s graph this function with a range from -3 to 3.

# Import all libraries for the rest of the blog post
from scipy.integrate import quad
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.patches import Polygon
%matplotlib inline

x = np.linspace(-3, 3, num = 100)
constant = 1.0 / np.sqrt(2*np.pi)
pdf_normal_distribution = constant * np.exp((-x**2) / 2.0)
fig, ax = plt.subplots(figsize=(10, 5));
ax.plot(x, pdf_normal_distribution);
ax.set_ylim(0);
ax.set_title('Normal Distribution', size = 20);
ax.set_ylabel('Probability Density', size = 20);

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How to Find the Probability of Events 

The graph above does not show you the probability of events but their probability density. We will need to integrateto get the probability of an event within a given range. Suppose we are interested in finding the probability of a random data point landing within one standard deviation of the mean. We need to integrate from -1 to 1. This can be done with SciPy.

# Make a PDF for the normal distribution a function
def normalProbabilityDensity(x):
   constant = 1.0 / np.sqrt(2*np.pi)
   return(constant * np.exp((-x**2) / 2.0) )

# Integrate PDF from -1 to 1
result, _ = quad(normalProbabilityDensity, -1, 1, limit = 1000)
print(result)

You’ll see that 68 percent of the data is within one standard deviation (σ) of the mean (μ).

If you are interested in finding the probability of a random data point landing within two standard deviations of the mean, you need to integrate from -2 to 2.

Now, 95 percent of the data is within two standard deviations (σ) of the mean (μ).

If you are interested in finding the probability of a random data point landing within three standard deviations of the mean, you need to integrate from -3 to 3.

And now, 99.7 percent of the data is within three standard deviations (σ) of the mean (μ).

It is important to note that for any probability density function, the area under the curve must be one. The probability of drawing any number from the function’s range is always one.

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You will also find that it is also possible for observations to fall four, five or even more standard deviations from the mean, but this is very rare if you have a normal, or nearly normal, distribution.

You can now take this knowledge and apply it to boxplots.

How many standard deviations is 68?

What score falls within 68% of distribution?

How many standard deviations above the mean is the score of 68?

What is the term for a normal distribution in which approximately 68?