Standard Deviation Calculator Formula

Understand the math behind the standard deviation calculator. Each variable explained with a worked example.

Use the Standard Deviation Calculator

Formulas Used

Std Dev

std_dev = sqrt(((a - mean_val)^2 + (b - mean_val)^2 + (c - mean_val)^2) / 3)

Std Dev Sample

std_dev_sample = sqrt(((a - mean_val)^2 + (b - mean_val)^2 + (c - mean_val)^2) / 2)

Variance Pop

variance_pop = ((a - mean_val)^2 + (b - mean_val)^2 + (c - mean_val)^2) / 3

Mean

mean = mean_val

Variables

VariableDescriptionDefault
aValue 110
bValue 220
cValue 330
mean_valDerived value= (a + b + c) / 3calculated

How It Works

How to Calculate Standard Deviation

Formula (Population)

σ = √[ Σ(xᵢ - μ)² / N ]

Steps

1. Find the mean (μ) of the data set 2. Subtract the mean from each value and square the result 3. Find the average of those squared differences 4. Take the square root

For a sample standard deviation, divide by (N-1) instead of N.

Worked Example

Find the standard deviation of 10, 20, and 30.

a = 10b = 20c = 30
  1. 01Mean = (10 + 20 + 30) / 3 = 20
  2. 02Squared differences: (10-20)² = 100, (20-20)² = 0, (30-20)² = 100
  3. 03Variance (population) = (100 + 0 + 100) / 3 = 66.6667
  4. 04Standard Deviation = √66.6667 ≈ 8.1650

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