## Calculate Standard Deviation

Enter a series of numbers separated by commas:

## What is Standard Deviation?

Standard deviation is a measure of the dispersion of a set of numbers from their mean. It helps in understanding the variability within a dataset.

## Why is Standard Deviation Important?

Understanding standard deviation is crucial in various fields like finance, science, and engineering. Here are some reasons why standard deviation is important:

**Variability:**It helps measure the variability within a dataset.**Normal Distribution:**It is essential for understanding and working with normal distributions.**Risk Assessment:**In finance, it is used to assess the volatility and risk of investments.

## How to Calculate Standard Deviation?

Calculating standard deviation can be broken down into a few steps:

**Calculate the Mean:**The mean (μ) is calculated by summing all the numbers and dividing by the total number of numbers.**Calculate the Deviations from the Mean:**Subtract the mean from each number to find the deviations.**Square the Deviations:**Square each deviation to eliminate negative values.**Calculate the Variance:**For a population: Divide the sum of squared deviations by the total number of data points (n). For a sample: Divide the sum of squared deviations by the number of data points minus one (n-1).**Calculate the Standard Deviation:**Take the square root of the variance to get the standard deviation.

## Formula for Standard Deviation

For a population:

`σ = √(Σ(xi - μ)² / n)`

For a sample:

`s = √(Σ(xi - x̄)² / (n - 1))`

## Example Calculation

Suppose we have the dataset: 2, 4, 4, 4, 5, 5, 7, 9.

**Calculate the mean:****Calculate the deviations from the mean:****Square the deviations:****Calculate the variance:****Calculate the standard deviation:**

μ = (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) / 8 = 5

(2-5), (4-5), (4-5), (4-5), (5-5), (5-5), (7-5), (9-5) = -3, -1, -1, -1, 0, 0, 2, 4

(-3)², (-1)², (-1)², (-1)², 0², 0², 2², 4² = 9, 1, 1, 1, 0, 0, 4, 16

σ² = (9 + 1 + 1 + 1 + 0 + 0 + 4 + 16) / 8 = 4

σ = √4 = 2

The standard deviation of the dataset is 2.