How do you calculate standard deviation?

Definition

The standard deviation is a measure of how spread out numbers are in a data set. It quantifies the amount of variation or dispersion from the mean (average). For a set of $n$ numbers $x_1, x_2, \ldots, x_n$, the standard deviation ($\sigma$ for population, $s$ for sample) is calculated as follows:

• Population standard deviation:

$$ \sigma = \sqrt{ \frac{1}{n} sum_{i=1}^{n} (x_i - \mu)^2 } $$
where $\mu$ is the population mean.

• Sample standard deviation:

$$ s = \sqrt{ \frac{1}{n-1} sum_{i=1}^{n} (x_i - \bar{x})^2 } $$
where $\bar{x}$ is the sample mean.

Worked Example

Suppose we have the sample data: $4, 8, 6, 5, 3$ **Step 1: Calculate the mean** %%MATH_DISPLAY_2%% **Step 2: Find the squared differences from the mean** - $(4 - 5.2)^2 = 1.44$ - $(8 - 5.2)^2 = 7.84$ - $(6 - 5.2)^2 = 0.64$ - $(5 - 5.2)^2 = 0.04$ - $(3 - 5.2)^2 = 4.84$ **Step 3: \sum the squared differences** %%MATH_DISPLAY_3%% **Step 4: Divide by $n-1$ (since this is a sample)** %%MATH_DISPLAY_4%% **Step 5: Take the square root** %%MATH_DISPLAY_5%% ## Takeaways - Standard deviation measures how much data values deviate from the mean. - Use $n-1$ in the denominator for a sample, $n$ for a population.

• A higher standard deviation means more spread in the data.