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 %
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 .