Combination of Errors in a series of Measurements
Suppose the values obtained in several measurement are a1, a2, a3, …, an.
Arithmetic mean, amean = (a1+ a2 + a3+ … + an)/n
o Absolute Error: The magnitude of the difference between the true value of the quantity and the individual measurement value is called absolute error of the measurement. It is denoted by |Δa| (or Mod of Delta a). The mod value is always positive even if Δa is negative. The individual errors are:
Δa1 = amean – a1, Δa2 = amean – a2, ……. ,Δan = amean – an
o Mean absolute error is the arithmetic mean of all absolute errors. It is represented by Δamean.
Δamean = (|Δa1| + |Δa2| +|Δa3| + …. +|Δan|) / n
For single measurement, the value of ‘a’ is always in the range amean± Δamean
So, a = amean ± Δamean Or amean – Δamean< a <amean + Δamean
o Relative Error: It is the ratio of mean absolute error to the mean value of the quantity measured.
Relative Error = Δamean / amean
o Percentage Error: It is the relative error expressed in percentage. It is denoted by δa.
δa = (Δamean / amean) x 100%
Combination of Errors
If a quantity depends on two or more other quantities, the combination of errors in the two quantities helps to determine and predict the errors in the resultant quantity. There are several procedures for this.
Suppose two quantities A and B have values as A ± ΔA and B ± ΔB. Z is the result and ΔZ is the error due to combination of A and B.
|Criteria||Sum or Difference||Product||Raised to Power|
|Resultant value Z||Z = A ± B||Z = AB||Z = Ak|
|Result with error||Z ± ΔZ = (A ± ΔA) + (B ± ΔB)||Z ± ΔZ = (A ± ΔA) (B ± ΔB)||Z ± ΔZ = (A ± ΔA)k|
|Resultant error range||± ΔZ = ± ΔA ± ΔB||ΔZ/Z = ΔA/A ± ΔB/B|
|Maximum error||ΔZ = ΔA + ΔB||ΔZ/Z = ΔA/A + ΔB/B||ΔZ/Z = k(ΔA/A)|
|Error||Sum of absolute errors||Sum of relative errors||k times relative error|
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