How to combine errors
Robin Hogan
June 2006
1 What is an “error”?
All measurements have uncertainties that need to
be communicated along with the measurement itself.
Suppose we make a measurement of temperature, ˆ
T,
but the “true” temperature is T. In this case our instan-
taneous error is εT=ˆ
T−T. Obviously we don’t
know the value of εTfor any specific measurement
(otherwise we could simply subtract it and report the
true value) but we should be able to estimate the root-
mean-squared error, given by
∆T=qε2
T,(1)
and then report our measurement in the form ˆ
T±∆T
(for example T=284.6±0.2 K). In Eq. 1, the overbar
denotes the mean taken over a large number of mea-
surements by an identical instrument.
Usually the quantity ∆Tis referred to simply as
the “error” in the measurement. This is a bit mislead-
ing and is easy to confuse with the instantaneous error;
a better term would have been “uncertainty”, but “er-
ror” is in such common use that we had better stick
with it. Just keep in mind that what we usually mean
is the root-mean-squared error.
If the instantaneous errors have aGaussian distri-
bution (also known as a Normal or Bell-shaped distri-
bution) then approximately 68% of the individual mea-
surements will lie between T−∆Tand T+∆T, and
95% of them between T−2∆Tand T+2∆T. Be aware
that sometimes errors are stated to indicate the “95%
confidence interval”, in which case they are equal to
2∆T. It should be noted that the “measurement” may
well be the mean of a number of samples, in which
case we might take the standard error of the mean as
an estimate of the error ∆T.
2 Errors for functions of one variable
Usually we will have a formula we want to use to
derive a new variable from one or more of our mea-
sured variables. Section 3 describes the general case
0 100 200 300
0
100
200
300
400
εT
εF
T
F
T + εT
F + εF
Temperature (K)
Blackbody irradiance (W m−2)
Figure 1: Illustration of the estimation of the error in F
from the error in Tusing the gradient of the relationship
between them (using Eq. 2).
in which we use the definition of an error in Eq. 1 to
estimate the error in the new variable given the errors
in the measured variables. However, if only one mea-
surement is involved then we can use a simpler method
based on differentiation.
Suppose we wish to derive the irradiance Femit-
ted by a blackbody with a temperature Tusing
F=σT4,(2)
where σis a constant that is known very accurately. It
can be seen from Fig. 1 that, provided the error in T
is relatively small, the ratio of instantaneous errors in
Fand Tis approximately equal to the gradient of the
relationship between them:
εF
εT≃dF
dT.(3)
From now on we will replace “≃” with “=”, but al-
ways be aware that error estimation is an approximate
business so it is not worth quoting errors to high preci-
sion (certainly no more than two significant figures).
With the help of Eq. 1, it can be shown that the
ratio of root-mean-squared errors is
∆F
∆T=
εF
εT
=
dF
dT
,(4)
1
