Empirical Rule for grouped data
Empirical rule is the general rule of thumb that applies to bell shaped (symmetrical) distribution. The empirical rule can be stated as :
- $68$% of the data will fall within one standard deviation of the mean,
- $95$% of the data will fall within two standard deviations of the mean,
- $99.7$% of the data will fall within three standard deviations of the mean.
Formula
Let $(x_i,f_i)$; $i=1,2,\cdots, n$ be the observed frequency distribution. If the distribution of $x$ is approximately symmetrical, then
$68$% of the data falls in $\overline{x}\pm 1 s_x$
$95$% of the data falls in $\overline{x}\pm 2 s_x$
$99.7$% of the data falls in $\overline{x}\pm 3 s_x$
where,
$\overline{x}=\dfrac{1}{N}\sum_{i=1}^{n}f_ix_i$is the sample mean,$N=\sum_i f_i$, is the total number of observations,$s_x =\sqrt{\dfrac{1}{N-1}\bigg(\sum_{i=1}^{n}f_ix_i^2-\dfrac{\big(\sum_{i=1}^n f_ix_i\big)^2}{N}\bigg)}$is the sample standard deviation.
