This empirical rule calculator can help you determine if a given data set follows a normal distribution by checking if 68% of data falls within first standard deviation (σ), 95% within first 2 σ and 99.7% within first 3 σ.

## How does this empirical rule calculator work?

The empirical rule states that 68% of the observed values are expected to fall within first standard deviation of the mean, 95% of the given figures are expected to fall within two standard deviations and 99.73% of the measured values can be expected to fall within three standard deviation interval. It is often referred to as **68-95-99.7 rule** or **three sigma rule**.

The steps to calculate it which are also applied by this empirical rule calculator are explained here:

a) Determine the count of the numbers within the given data set (n).

b) Find the mean/average of the set (m) as the sum of the specified values (from x_{1} to x_{n}) divided by (n).

c) If the statistical data set refers to a population find the population standard deviation (p):

d) In case the statistical data set represents a sample calculate the sample standard deviation (s):

e) If the data set is a population then:

- ER at 68% falls between [A] and [B]

Where:

[A] = (m - p)

[B] = (m + p)

-ER at 95% falls between [C] and [D]

[C] = (m - 2p)

[D] = (m + 2p)

-ER at 99.7% falls between [E] and [F]

[E] = (m – 3p)

[F] = (m + 3p)

f) In case the data set is a sample then:

-ER at 68% falls between [G] and [H]

[G] = (m - s)

[H] = (m + s)

-ER at 95% falls between [I] and [J]

[I] = (m - 2s)

[J] = (m + 2s)

-ER at 99.7% falls between [K] and [L]

[K] = (m - 3s)

[L] = (m + 3s)

Where: ER = Empirical Rule

Please note that this statistical tool allows using both positive and negative numbers, integers or decimals while the single rule to take account of is that the entries should be separated by semicolon.

## Example of a calculation

Let’s consider the simplest data set from zero to nine and input it within the form: 0;1;2;3;4;5;6;7;8;9.

It will return these results:

■ Mean (Average) = 4.50

■ Population standard deviation = 2.8723

■ Sample standard deviation = 3.0277

■ If data set is a population:

- ER at 68% falls between 1.63 and 7.37

- ER at 95% falls between -1.24 and 10.24

- Empirical Rule at 99.7% falls between -4.12 and 13.12

■ If data set is a sample:

- ER at 68% falls between 1.47 and 7.53

- ER at 95% falls between -1.56 and 10.56

- ER at 99.7% falls between -4.58 and 13.58

07 Jul, 2015