Standard Deviation - Statistics

Standard Deviation - Statistics

Here's the formula for calculating the standard deviation of a population :

\begin{equation} \sigma = \sqrt{\frac{1}{N} \sum_{i=1}^{N}(x_i - \mu)^2} \end{equation}

where:

• \( \sigma \) is the population standard deviation
• \( N \) is the total number of observations in the population
• \( x_i \) is the value of the \( ith\) observation
• \( \mu\) is the population mean

This formula calculates the square root of the average of the squared differences between each observation and the population mean.

Example 1:

Suppose we have a population of test scores with the following values:

\(x_1 = 75, x_2 = 80, x_3 = 85, x_4 = 90, x_5 = 95\)

To find the population standard deviation, we need to first find the population mean:

\(\mu = \frac{1}{N}\sum_{i=1}^{N} x_i = \frac{75 + 80 + 85 + 90 + 95}{5} = 85\)

Next, we can use the formula for the population standard deviation:

\begin{align} \sigma = \\&= \sqrt{\frac{1}{N} \sum_{i=1}^{N}(x_i - \mu)^2} \\ &= \sqrt{\frac{1}{5}[(75-85)^2 + (80-85)^2 + (85-85)^2 + (90-85)^2 + (95-85)^2]} \\ &= \sqrt{\frac{1}{5}[(-10)^2 + (-5)^2 + 0^2 + 5^2 + 10^2]} \\ &= \sqrt{\frac{1}{5}(100 + 25 + 25 + 100)} \\ &= \sqrt{50} \approx 7.07 \end{align}

Therefore, the population standard deviation of the test scores is approximately 7.07.