[Tutorial] Empirical rule (68–95–99.7 rule)

If your data is with normal or nearly normal distribution, unimodal distribution which is symmetric. Empirical rule should be applied on your data.

photo from statisticshowto.com


We call location of distribution which is located at this line as μ+σ.

 

and scope in red rectangle is from μ to μ+σ

On the other hand, we call location of distribution which's located at this line as μ-σ.

and scope in red rectangle is from μ to μ-σ.

we call the location of distribution here as μ+2σ.

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Suppose :

• X is an observation from a normally distributed random variable
• μ is the mean of the distribution
• σ is its standard deviation:

Empirical rule means

1) Around 68% of values fall within one standard deviation of the mean.

In math we use this to represent the probability a random distributed value X located between μ-σ to μ+σ :

In figure,the scope [μ-σ, μ+σ] is shown at area in blue.

2) Around 95% of the values fall within two standard deviations from the mean.

In math we use this to represent the probability a random distributed value X located between μ-2σ to μ+2σ :

 

In figure,the scope [μ-σ, μ+σ] is shown at area in blue and brown.

3) Almost all of the values — about 99.7% — fall within three standard deviations from the mean.

In math we use this to represent the probability a random distributed value X located between μ-3σ to μ+3σ :

In figure,the scope [μ-σ, μ+σ] is shown at area in blue ,brown and green.

Example

http://www.oswego.edu/~srp/stats/wts_males.htm 

Reference

http://www.statisticshowto.com/68-95-99-7-rule/
http://www.oswego.edu/~srp/stats/6895997.htm