Triangulation combines several independent measurements of the same unknown quantity into one estimate that is more precise than any single source. Each method here draws N noisy readings around its own true center (its bias plus Gaussian noise of size σ):
reading = bias + σ · Z, Z ~ N(0,1)
sample mean x̄ᵢ, standard error SEᵢ = σᵢ / √N
The standard way to fuse independent estimates is inverse-variance weighting: a source is trusted in proportion to how precise it is (the inverse of its variance). This minimizes the variance of the combined estimate:
weight wᵢ = 1 / SEᵢ²
x̂ = Σ(wᵢ · x̄ᵢ) / Σwᵢ
σ̂² = 1 / Σwᵢ
- Noise sliders — shrink a method's σ and its beeswarm of samples tightens around its mean; its weight in the combination grows automatically.
- Observation bias slider — this is the key lesson of triangulation: a systematic bias is not noise. Even a very precise (tight, low-noise) biased source pulls the combined estimate away from the truth — more precision on a biased method makes the pull worse, not better.
- Sample size N — larger N shrinks every method's standard error (SE = σ/√N), tightening all three clusters and the combined uncertainty band together, exactly as "larger samples increase statistical power" in the article above.
- Resample — redraws all readings from the same distributions, showing that x̂ jitters run to run purely from sampling noise, while a systematic bias stays put.
- Drag the plot / zoom buttons — pan and zoom the shared value axis to inspect any region closely; all four lanes stay locked to the same scale so distances stay comparable.
This is the mathematical form of what research-methods texts call triangulation: cross-checking quantitative surveys, controlled experiments and field observation against each other. It is also exactly the inverse-variance meta-analysis used to pool independent studies, and the same weighting a GPS receiver uses to fuse noisy satellite ranges into one position fix.