The Empirical Rule and Other Rules in Statistics

Empirical Rule For the Normal Distribution

\[\left\{ \mu -\sigma \le X\le \mu +\sigma \right\}=\left\{ -\sigma \le X-\mu \le \sigma \right\}=\left\{ -1\le \frac{X-\mu }{\sigma }\le 1 \right\}\]

\[\left\{ \mu -\sigma \le X\le \mu +\sigma \right\}=\left\{ -\sigma \le X-\mu \le \sigma \right\}=\left\{ -1\le \frac{X-\mu }{\sigma }\le 1 \right\}=\left\{ -1\le Z\le 1 \right\}\] \[Pr \left( \mu -\sigma \le X\le \mu +\sigma \right)=\Pr \left( -1\le \frac{X-\mu }{\sigma }\le 1 \right)=\Pr \left( -1\le Z\le 1 \right)\] \[=\Pr \left( Z\le 1 \right)-\Pr \left( Z\le -1 \right)\approx 0.\text{841345}-0.\text{158655}\approx 0.\text{682689}\] \[\Pr \left( \mu -2\sigma \le X\le \mu +2\sigma \right)=\Pr \left( -2\le \frac{X-\mu }{\sigma }\le 2 \right)=\Pr \left( -2\le Z\le 2 \right)\] \[=\Pr \left( Z\le 2 \right)-\Pr \left( Z\le -2 \right)\approx 0.\text{977249868}-0.0\text{2275}0\text{132}\approx 0.\text{9544997}\] \[\Pr \left( \mu -3\sigma \le X\le \mu +3\sigma \right)=\Pr \left( -3\le \frac{X-\mu }{\sigma }\le 3 \right)=\Pr \left( -3\le Z\le 3 \right)\] \[=\Pr \left( Z\le 3 \right)-\Pr \left( Z\le -3 \right)\approx 0.\text{99865}0\text{1}0\text{2}-0.00\text{1349898}\approx 0.\text{9973}00\text{2}\]

The Rule of Thumb for the Standard Deviation

\[s\approx \frac{Range}{4}\]

Chebyshev’s Rule

\[\Pr \left( \mu -k\sigma \le X\le \mu +k\sigma \right)\ge 1-\frac{1}{{{k}^{2}}}\] \[\Pr \left( \mu -2\sigma \le X\le \mu +2\sigma \right)\ge 1-\frac{1}{{{2}^{2}}}=0.75\]
This tutorial is brought to you courtesy of

In case you have any suggestion, please do not hesitate to contact us.

log in

reset password

Back to
log in
Copy Protected by Chetan's WP-Copyprotect.