Standard Deviation Calculator

Instructions: Enter your numbers separated by commas, spaces, or new lines. Examples:

  • 1, 2, 3, 4, 5
  • 1 2 3 4 5
  • 1
    2
    3
    4
    5

Results:

Count (n):
Mean (μ):
Sum of Numbers:
Sum of Squares:
Population Variance (σ²):
Population Standard Deviation (σ):
Sample Variance (s²):
Sample Standard Deviation (s):

Formulas Used:

Mean (Average)
μ = (Σx) / n
The sum of all numbers divided by the count of numbers.
Sum of Squares
SS = Σ(x - μ)²
The sum of the squared differences between each data point and the mean.
Population Variance
σ² = SS / n
The average of the squared differences from the Mean (using population size n).
Population Standard Deviation
σ = √(σ²)
The square root of the population variance, showing how much variation exists from the average.
Sample Variance
s² = SS / (n - 1)
When working with a sample (not entire population), we divide by n-1 (Bessel's correction) for an unbiased estimate.
Sample Standard Deviation
s = √(s²)
The square root of the sample variance, showing how spread out the sample data is.
Key
μ = population mean
σ = population standard deviation
σ² = population variance
s = sample standard deviation
s² = sample variance
Σ = sum of
x = each value in the dataset
n = number of values in the dataset



Features of Standard Deviation Calculator

Standard deviation is a fundamental statistical measure that quantifies the amount of variation or dispersion in a dataset. Our calculator helps you compute both population standard deviation (σ) and sample standard deviation (s) with step-by-step formulas.

How to Use the Standard Deviation Calculator

Population Standard Deviation

Enter your complete dataset and click "Calculate" to find the population standard deviation (σ). Use this when you have data for the entire population. Example: For data 5,7,3,7, σ = 1.71

Sample Standard Deviation

Enter your sample data to calculate the sample standard deviation (s). Use this when working with a subset of a larger population. Example: For sample 12,15,17, s = 2.52

Data Input Options

Enter numbers separated by commas, spaces, or new lines. The calculator automatically parses your input. Example formats: "1,2,3" or "1 2 3" or one number per line.

Variance Calculations

View both population variance (σ²) and sample variance (s²) in the results, along with the sum of squares used in the calculations.

Formula Display

Click "Show Formulas" to reveal all mathematical formulas used in the calculations, including step-by-step explanations of each statistical measure.

Standard Deviation FAQs

What's the difference between population and sample standard deviation?

Population standard deviation (σ) uses the entire dataset (divide by N) while sample standard deviation (s) uses a sample (divide by N-1 for unbiased estimation). Use population when you have all data, sample when working with a subset.

When should I use standard deviation?

Standard deviation is used to measure dispersion in data. It's essential in statistics, quality control, finance (risk measurement), research (data analysis), and any field dealing with variability in measurements.

What does a high standard deviation mean?

A high standard deviation indicates data points are spread out over a wider range of values, suggesting greater variability. A low standard deviation means data points tend to be close to the mean.

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