Standard Deviation

Definition: 

    Standard deviation (SD) is a measure of the amount of variation or dispersion in a set of values. A low SD means the data points tend to be close to the mean, while a high SD indicates that the data points are spread out over a wider range.

Formula:

$$\text{SD} = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n}}$$
where $x_i$ represents each value, $\bar{x}$ is the mean, and $n$ is the number of values.

Example: Given the dataset: $[1, 3, 5, 7, 9]$

1. Calculate the mean $\bar{x}$: $$\bar{x}=\frac{1+3+5+7+9}{5}=\frac{25}{5}=5$$$$\bar{x} = \frac{1 + 3 + 5 + 7 + 9}{5} = \frac{25}{5} = 5$$
2. Calculate each squared deviation from the mean: $$(1-5{)}^2=16,(3-5{)}^2=4,(5-5{)}^2=0,(7-5{)}^2=4,(9-5{)}^2=16$$$$(1-5)^2 = 16,\quad (3-5)^2 = 4,\quad (5-5)^2 = 0,\quad (7-5)^2 = 4,\quad (9-5)^2 = 16$$
3. Sum the squared deviations: $$\sum (x_i-\bar{x}{)}^2=16+4+0+4+16=40$$$$\sum (x_i - \bar{x})^2 = 16 + 4 + 0 + 4 + 16 = 40$$
4. Divide by the number of values and take the square root: $$\text{SD} = \sqrt{\frac{40}{5}} = \sqrt{8} \approx 2.83$$