**Combination of Errors in a series of Measurements**

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

Arithmetic mean, *a*mean = (*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 **|Δ a|** (or Mod of Delta a). The mod value is always positive even if Δ

*a*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 Δ*a*mean.

Δ*a*mean = (|Δ*a1*| + |Δ*a2*| +|Δ*a3*| + …. +|Δ*an*|) / n

For single measurement, the value of ‘a’ is always in the range *a*mean± Δ*a*mean

So, a = *a*mean ± Δ*a*mean Or *a*mean – Δ*a*mean*< a <a*mean + Δ*a*mean

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

Relative Error = Δ*a*mean / *a*mean

o **Percentage Error: **It is the relative error expressed in percentage. It is denoted by δ*a*.

δa = (Δ*a*mean / *a*mean) 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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