# Python – 68-95-99.7 rule in Statistics

The **Empirical Rule**(also called the **68-95-99.7 Rule** or the **Three Sigma Rule**) states that for any normal distribution, we have the following observations :

- 68% of the observed values lie between 1 standard deviation around the mean :
- 95% of the observed values lie between 2 standard deviations around the mean :
- 99.7% of the observed values lie between 3 standard deviation around the mean :

Below is a standard normal distribution graph with (mean = **0** and standard deviation = **1**), illustrating the Empirical Rule.

We, can verify this using functions provided by Python’s **SciPy** module.

We can use the **cdf()** function of the **scipy.stats.norm** module to calculate the cumulative probability(area under a distribution curve).

Syntax :cdf(x, mean, SD)Parameters :

x :value up to which cumulative probability is to be calculatedmean :mean of the distributionSD :standard deviation of the distribution

Below is the implementation :

`import` `matplotlib.pyplot as plt` `import` `numpy as np` `from` `scipy.stats ` `import` `norm` ` ` `# setting the values of` `# mean and S.D.` `mean ` `=` `0` `SD ` `=` `1` ` ` `# value of cdf between one, two` `# and three S.D. around the mean` `one_sd ` `=` `norm.cdf(SD, mean, SD) ` `-` `norm.cdf(` `-` `SD, mean, SD)` `two_sd ` `=` `norm.cdf(` `2` `*` `SD, mean, SD) ` `-` `norm.cdf(` `-` `2` `*` `SD, mean, SD)` `three_sd ` `=` `norm.cdf(` `3` `*` `SD, mean, SD) ` `-` `norm.cdf(` `-` `3` `*` `SD, mean, SD)` ` ` `# printing the value of fractions` `# within each band` `print` `(` `"Fracton of values within one SD ="` `, one_sd)` `print` `(` `"Fracton of values within two SD ="` `, two_sd)` `print` `(` `"Fracton of values within three SD ="` `, three_sd)` |

**Output :**

Fracton of values within one SD = 0.6826894921370859 Fracton of values within two SD = 0.9544997361036416 Fracton of values within three SD = 0.9973002039367398

Hence, we see that the fraction of values are almost equal to **0.65**, **0.95** and **0.997**. Thus, the empirical Rule is verified.

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