Understanding the spread of your data is crucial in statistics, finance, and research. This tool calculates population and sample standard deviation, variance, mean, sum, and a 95% margin of error in real time.
Standard deviation quantifies how spread out numbers are from the mean. Use this calculator to:
Average of all values:
μ = (Σx) / NWhere Σx is the sum of values, N is population size, n is sample size.
x̄ = (Σx) / n
Mean squared deviation:
σ² = Σ(xᵢ - μ)² / N (Population)
s² = Σ(xᵢ - x̄)² / (n - 1) (Sample)
Square root of variance:
σ = √(σ²) (Population)
s = √(s²) (Sample)
Example: Data = [5, 7, 9]
μ = (5 + 7 + 9) / 3 = 7
(5 − 7)² = 4, (7 − 7)² = 0, (9 − 7)² = 4
Population: (4 + 0 + 4) / 3 = 2.6667
Sample: (4 + 0 + 4) / (3 − 1) = 4
σ = √2.6667 ≈ 1.633
s = √4 = 2
Bessel’s correction (n−1) compensates for bias when estimating a population from a sample, improving accuracy.
Data tightly clustered around mean (σ < ⅓ mean)
Typical spread (σ ≈ ⅓–⅔ mean)
Wide dispersion (σ > ⅔ mean)
