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:

  1. Calculate the Mean: The mean (μ) is calculated by summing all the numbers and dividing by the total number of numbers.
  2. Calculate the Deviations from the Mean: Subtract the mean from each number to find the deviations.
  3. Square the Deviations: Square each deviation to eliminate negative values.
  4. 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).
  5. 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.

  1. Calculate the mean:

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

  2. Calculate the deviations from the mean:

    3. (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. Square the deviations:

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

  4. Calculate the variance:

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

  5. Calculate the standard deviation:

    6. σ = √4 = 2

The standard deviation of the dataset is 2.