Standard Deviation & Variance Calculator
Free online tool to calculate population/sample standard deviation and variance with step-by-step results | Learn how to calculate standard deviation and find variance
Input Data
Supports integers and decimals. Separate values with commas, spaces, or newlines
Calculation Results
Definition and Formula of Variance & Standard Deviation
First, understand the definition of variance: Variance is a statistical measure of the spread of data points around the mean. A higher variance indicates greater dispersion of data, while a lower variance means data is more concentrated around the mean.
Variance Formulas
Population Variance (σ²): Calculated for the entire data set, the formula is:
σ² = Σ(xᵢ - μ)² / N
Where: μ = population mean, xᵢ = each data point, N = total number of data points, Σ = sum of values.
Sample Variance (s²): Calculated for a subset (sample) of the population, using n-1 in the denominator for unbiased estimation:
s² = Σ(xᵢ - x̄)² / (n - 1)
Where: x̄ = sample mean, xᵢ = each sample data point, n = sample size.
Standard Deviation Formulas
Standard deviation is the square root of variance, expressed in the original units of the data, making it more interpretable. Here's how to calculate standard deviation:
σ = √σ² (Population Standard Deviation)
s = √s² (Sample Standard Deviation)
Key Differences Between Population and Sample Metrics
- Population Metrics (σ, σ²): Used when you have data for the entire group you want to study (denominator = N)
- Sample Metrics (s, s²): Used when you only have a subset of data and want to estimate population parameters (denominator = n-1, Bessel's correction)
- Sample variance uses n-1 to correct for bias in estimating population variance from a sample
- Standard deviation has the same units as the original data, while variance has squared units
How to Use This Calculator
- Enter your numerical data set (supports commas, spaces, or newlines as separators)
- Select the calculation type: Population or Sample
- Choose your desired precision (decimal places for results)
- Click "Calculate Variance" to get instant results
- View detailed step-by-step calculations to understand how results are derived
- Copy the summary results for your records or further analysis
Calculation Steps
- Calculate the mean (μ for population, x̄ for sample)
- Calculate the difference between each data point and the mean
- Square each of these differences
- Sum all the squared differences (sum of squares)
- Population variance: sum of squares ÷ total number of data points
Sample variance: sum of squares ÷ (number of data points - 1) - Standard deviation: square root of the variance