Errors in a series of Measurements

Suppose the values obtained in several measurement are a₁, a₂, a₃, …, a_n.

Arithmetic mean, a_mean = (a₁+ a₂ + a₃+ … + a_n)/n=

• 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 = a_mean – a₁, Δa₂ = a_mean – a₂, ……. ,Δa_n = a_mean – a_n

• Mean absolute error is the arithmetic mean of all absolute errors. It is represented by Δa_mean.

Δa_mean = (|Δa₁| + |Δa₂| +|Δa₃| + …. +|Δa_n|) / n =

      For single measurement, the value of ‘a’ is always in the range a_mean± Δa_mean

                So, a =  a_mean ± Δa_mean Or a_mean – Δa_mean< a <a_mean + Δa_mean

• Relative Error: It is the ratio of mean absolute error to the mean value of the quantity measured.

Relative Error = Δa_mean / a_mean

• Percentage Error: It is the relative error expressed in percentage. It is denoted by δa.

δa = (Δa_mean / a_mean) x 100%

Combinations 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.