Uncertainty Calculation

Uncertainty: this is when we measure and we do not know the true value. The error committed cannot be determined. Uncertainty is scientific attempts to estimate error during measurement, where we estimate the error field within which we estimate the true value of this quantity.

Absolute Uncertainty

The absolute uncertainty represents the upper limit of the error committed during the measurement. Thus, if we wish to measure a physical quantity x, we will write that[1]

X = X mes ± ΔX X= X_mes±ΔX

Where

ΔX is the absolute uncertainty

This also means that the real value of the quantity x is in the interval [ X mes − Δ X X_{mes} - %DELTA X , X mes + Δ X X_{mes}+ %DELTA X ], and it is not possible to know it with accuracy.

Example

Length = (6 ± 0.001)m ⇒ 5.999 < length < 6.001 m

Method

  • If the measurement is direct, the absolute uncertainty is the sum of the systematic error[2] ΔXs, the reading error ΔXr and the instrumental error ΔXi

    Δ X = Δ X s + Δ X r + Δ X i %DELTA X= %DELTA X_{s}+ %DELTA X_{r}+ %DELTA X_{i}

  • If the measurement is indirect and the physical quantity is related to other quantities through a mathematical relationship. In that case, we can use the mathematical tool “the total differential or logarithm method ” to determine the uncertainties.

Relative Uncertainty

To determine the accuracy of the measurement, we resort to calculating the relative uncertainty, which is equal to the absolute uncertainty (Δx) over the measured value (x)

ε = ( Δ X ) X × 100 ε = ( %DELTA X ) over X ×100

ε: is the relative uncertainty has no units and is generally expressed in %