Standard Deviation Calculator

Calculate population and sample standard deviation with step-by-step solutions

Input Values

Separate values with commas or spaces

Results

What is Standard Deviation?

Standard deviation is a measure of the amount of variation or dispersion in a dataset. It indicates how much the individual data points deviate from the mean (average) of the dataset. A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range.

Standard Deviation Formulas

Population Standard Deviation:

σ = √(Σ(x - μ)² / N)

Where σ = population SD, μ = population mean, N = population size

Sample Standard Deviation:

s = √(Σ(x - x̄)² / (n-1))

Where s = sample SD, x̄ = sample mean, n = sample size

Steps to Calculate Standard Deviation

  1. Calculate the mean (average) of the dataset
  2. Subtract the mean from each data point to get deviations
  3. Square each deviation
  4. Sum all squared deviations
  5. Divide by N (population) or n-1 (sample)
  6. Take the square root of the result

How to Use the Calculator

  1. Enter your data points separated by commas or spaces
  2. Choose whether you're calculating population or sample standard deviation
  3. Click "Calculate" to get the result
  4. View the detailed step-by-step solution

Example Calculation

For the dataset: 2, 4, 4, 4, 5, 5, 7, 9

Step 1: Mean = (2+4+4+4+5+5+7+9)/8 = 5

Step 2: Deviations: -3, -1, -1, -1, 0, 0, 2, 4

Step 3: Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16

Step 4: Sum = 32

Step 5: Population variance = 32/8 = 4

Step 6: Population SD = √4 = 2

Population vs Sample Standard Deviation

  • Population SD: Used when you have data for an entire population
  • Sample SD: Used when you have data from a sample of a larger population
  • Sample SD uses n-1 in the denominator (Bessel's correction) to provide an unbiased estimate
  • Population SD uses N in the denominator