## Empirical Rule for ungrouped 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 $x_1,x_2,\cdots, x_n$ be $n$ sample observations. 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,

`$\overline{x}=\dfrac{1}{n}\sum_{i=1}^{n}x_i$`

is the sample mean,`$s_x =\sqrt{\dfrac{1}{n-1}\bigg(\sum_{i=1}^{n}x_i^2-\dfrac{\big(\sum_{i=1}^n x_i\big)^2}{n}\bigg)}$`

is the sample standard deviation.