Empirical Rule Calculator/ 68 95 and 99.7 Rule Calculator Online

The empirical rule calculator/ 68 95 99 rule calculator is a free tool created by us by which you can easily find the ranges that fall between 1 standard deviation, 2 standard deviations, and 3 standard deviations from the mean, in which you will find to have 68, 95, and 99.7% of the normally distributed data respectively. In this article, you will find all the information related to the empirical rule calculator/ 68 95 99 rule calculator that includes a definition of the empirical rule, the formula related to the empirical rule, and also an easy self-illustrative example on how to use the empirical rule calculator/ 68 95 99 rule calculator.

Empirical Rule Calculator/ 68 95 and 99.7 Rule Calculator Online

What empirical rule is?

The empirical rule is a statistical rule (also known as 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 essentially under three standard deviations on either side of the mean.

More deeply, you will find:

68% of data under 1 standard deviation

95% of data under 2 standard deviations

99.7% of data under 3 standard deviations

Now, you will be explained all the concepts used in the above definition:

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

Normal distribution: It is a distribution that is symmetric about the mean, with data near the mean to be more frequent in occurrence than that data that is far from the mean. 

Calculate the mean of your values:

μ = (Σ xi) / n

∑ - sum

xi - individual value from your data

n - the number of samples

Calculate the standard deviation:

σ = √( ∑(xi – µ)² / (n – 1) )


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 fall within 3 standard deviations from the mean - between μ - 3σ and μ + 3σ.

Now, what you need to do is to just enter the standard and mean deviation into the empirical rule calculator, and it will give you the output as the intervals for you.

An example of how the empirical rule calculator/ 68 95 99 rule calculator works

Suppose IQ scores are being normally distributed with the mean equal to 100 and the standard deviation equal to 10. Let's have a look at the maths behind the empirical rule calculator/ 68 95 99 rule calculator:

Mean: μ = 100

Standard deviation: σ = 10

Empirical rule formula:

μ - σ = 100 – 10 = 90

μ + σ = 100 + 10 = 110

68% of people have an IQ between 90 and 110.

μ – 2σ = 100 – 2*10 = 80

μ + 2σ = 100 + 2*10 = 120

95% of people have an IQ between 80 and 120.

μ - 3σ = 100 – 3*10 = 70

μ + 3σ = 100 + 3*10 = 130

99.7% of people have an IQ between 70 and 130.

For quicker, easier, and more efficient calculations, enter the mean and standard deviation into this empirical rule calculator/ 68 95 99 rule calculator, and then fold your arms as it will do the rest for you.

Where is the empirical rule used?

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

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