BL2.ru
Boosty
ENRU

Measurement Uncertainty Calculator: Direct and Indirect

Paste a column of measurements — the calculator finds the mean, standard deviation, random error via Student t, adds the instrument error, rejects outliers and writes the rounded result. For indirect measurements enter a formula (e.g. g = 4π²l/T²) and get the uncertainty via partial derivatives.

One per line or separated by spaces, “;”, tabs — paste a column from Excel. Decimal point or comma.
θ is the instrument (systematic) error: half the smallest division for rulers and scales, one division for calipers and digital stopwatches, or from the accuracy class. Leave empty to ignore.
Calculate

Processing repeated direct measurements

  1. Mean: x̄ = Σxᵢ / n.
  2. Standard deviation: S = √(Σ(xᵢ − x̄)² / (n − 1)).
  3. Standard error of the mean: S_x̄ = S / √n.
  4. Outliers: the value farthest from the mean is checked with the Grubbs statistic G = |x − x̄| / S; if G exceeds the critical value for n, it is removed and everything is recalculated.
  5. Random error: ε = t(P, n) · S_x̄ with Student t (P = 0.95: n = 3 → 4.30; n = 5 → 2.78; n = 10 → 2.26; large n → 1.96).
  6. Instrument error θ — half a division, one division or from the accuracy class.
  7. Total: in lab courses usually Δ = √(ε² + θ²).

GOST R 8.736-2011 rule

If θ / S_x̄ < 0.8 the systematic part is neglected (Δ = ε), if > 8 the random part is neglected (Δ = θ); otherwise Δ = K · S_Σ with S_θ = θ / √3, S_Σ = √(S_θ² + S_x̄²), K = (ε + θ) / (S_x̄ + S_θ).

Rounding

Round the error to one significant digit (two if it starts with 1 or 2) and the result to the same place: not “2.00213 ± 0.0259 s” but (2.002 ± 0.026) s, P = 0.95. Relative error δ = Δ / x̄ · 100 %.

Propagation of uncertainty

For f(x₁, …, xₙ): Δf = √((∂f/∂x₁·Δx₁)² + … + (∂f/∂xₙ·Δxₙ)²). For a pendulum g = 4π²l/T²: δg = √(δl² + (2δT)²) — the period error counts twice. The calculator differentiates numerically and shows each contribution.

FAQ

How do I calculate the measurement error for a lab report?
Paste your measurements, set the instrument error and confidence level (usually 0.95). You get the mean, standard deviation, Student-t random error, the total error and the rounded result x = (x̄ ± Δ).
What is the Student t coefficient?
A factor that turns the standard error of the mean into confidence limits for a small number of measurements. For P = 0.95: 2.78 for 5 measurements, 2.26 for 10.
Absolute vs relative error?
Absolute error Δ is in the units of the quantity (±0.03 s); relative error δ = Δ / x̄ is in percent (1.5 %).
How is the error of an indirect measurement found?
Via partial derivatives: Δf = √Σ(∂f/∂xᵢ · Δxᵢ)². Enter the formula and values with errors in the “Indirect” tab.
When should an outlier be rejected?
When the Grubbs statistic G = |x − x̄| / S exceeds the critical value for your n. Uncheck the box if all values must be kept.