Error Propagation
In the following I present a simple method for calculating how errors in experimental
values affect the precision of results calculated from these values.
(See also E. Steiner, The Chemistry Maths Book, pp. 443-446)
The errors (for example x) are to be understood as ”maximum errors” and not as
standard deviations.
1. Additions and Subtractions
Given two measurements with the average values x and y and their errors ±x and
±y, the error ±z in a result calculated from
z=x+y
or
z=x–y
is given by:
z = x + y
Proof:
For additions the largest possible value of z is:
z + z = x + x + y + y
and therefore (since z = x + y): z = x + y,
whereas the smallest possible value of z is:
z – z = x – x + y – y
and therefore (since z = x + y): –z = –x – y.
For subtractions the largest possible value of z is:
z + z = x + x – (y – y)
and therefore (since z = x – y): z = x + y,
whereas the smallest possible value of z is:
z - z = x – x – (y – y)
and therefore (since z = x – y): –z = –x – y.
In all four cases z = x + y.
2. Multiplications and Divisions
Given two measurements with the average values x and y and their errors ±x and
±y, the error ±z in a result calculated from
z=xy
or
z=x/y
is given by:
z / z = x / x + y / y
Proof:
For multiplications the largest possible value of z is:
z + z = (x + x) (y + y) = x y + x y + y x + x y
Replacing x y by z and approximating x y as zero we get
z = x y + y x
and therefore (dividing by z = x y): z / z = x / x + y / y.
The smallest possible value for z is:
z – z = (x – x) (y – y) = x y – x y – y x + x y
Replacing x y by z and approximating x y as zero we get
–z = –x y – y x
and therefore (dividing by z = x y): –z / z = –x / x – y / y.
For divisions the largest possible value of z is:
z + z = (x + x) / (y – y)
and therefore:
x + x = (z + z) (y – y) = z y – z y + y z – z y
Replacing z y by x and approximating z y as zero we get
x = –z y + y z
or
y z = x + z y
and therefore (dividing by x = z y): z / z = x / x + y / y.
The smallest possible value for z is:
z – z = (x – x) / (y + y)
and therefore:
x – x = (z – z) (y + y) = z y + z y – y z – z y
Replacing z y by x and approximating z y as zero we get
–x = z y – y z
or
–y z = –x – z y
and therefore (dividing by x = z y): –z / z = –x / x – y / y.
In all four cases z / z = x / x + y / y.
3.Constants
In connection with multiplications and divisions we can also consider the special case
that y is free (or almost free) of experimental error, for example when y is a constant
(such as R).
For a multiplication z = a x, where x has an experimental error of ±x, but a is “errorfree” we get:
z / z = x / x
or
z = a x
For a division z = x / a, where x has an experimental error of ±x, but a is “error-free”
we get:
z / z = x / x
or
z = x / a
Adding or subtracting an “error-free” constant a leaves the error unchanged:
For z = x + a or
z = x
z=x–a