Why are independent uncertainties added in quadrature instead of simply added?

Conditions

• The two uncertainty effects are random.
• The two effects are independent from one another.
• The combination is modeled by a perpendicular geometric construction.

Reasoning, step by step

1. Consider two error contributions, $e_1$ and $e_2$.
2. Observe that if they always oppose, they partially cancel (best case).
3. Observe that if they always align, they give the worst-case total $e_1 + e_2$.
4. Model the average relationship for random independent effects by placing them at a 90-degree angle.
5. Use the Pythagorean theorem to find the hypotenuse: $\sqrt{e_1^2 + e_2^2}$.
6. Conclude that this quadrature sum is less than $e_1 + e_2$ but greater than either component alone.

Example

The video animates arrows $e_1$ and $e_2$ rotating from opposing to aligned to perpendicular. The screen explicitly shows the inequality: hypotenuse < $e_1 + e_2$, illustrating that the quadrature combination is smaller than the fully additive worst case.

Common misconceptions

• Assuming that all uncertainties simply add together to give the total error, which ignores the random nature of independent measurements.
• Believing that independent errors always cancel each other out completely.
• Thinking that the quadrature rule applies to systematic errors that are perfectly correlated.

Watch the explanation

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