SME
Measurements & Instrumentation · Problem type
Uncertainty of a sum or difference
Use this for “Two thermocouples read 85.2 ± 0.4 °C and 60.7 ± 0.3 °C. Find the temperature difference and its uncertainty.”
Diagram
ud
2
=
(u₁)
2
+
(u₂)
2
:
ud
=
√
0.4
2
+
0.3
2
=
0.5
°C.
Drawn to scale: u₂ up from the end of u₁, ud closing the triangle.
First reading x₁ = 85.2 °C
Second reading x₂ = 60.7 °C
Difference d = +24.5 °C
Relative uncertainty rd = 2.0408%
This is an example. Type your own values to start a new problem.
d (°C)
=
x₁ (°C)
−
x₂ (°C)
ud (°C)
=
(
u₁ (°C)
)
2
+
(
u₂ (°C)
)
2
rd
Relative uncertainty
calculated · %
Formulas
d
=
x₁
−
x₂
+24.5 = 85.2 − 60.7
ud
=
√
u₁
2
+
u₂
2
0.5
=
√
0.4
2
+
0.3
2
rd
=
100
×
ud
÷
|
d
|
2.0408 = 100 × 0.5 ÷ |24.5|
Assumptions
•
The two errors are random and independent, so they add in quadrature, not straight.
•
The same rule holds for a sum: absolute uncertainties combine, whatever the sign.
•
Both readings are in the same unit.
Step-by-step
WE KNOW
x₁ = 85.2 °C, u₁ = 0.4 °C, x₂ = 60.7 °C, u₂ = 0.3 °C
FIND
difference (d), difference uncertainty (ud), relative uncertainty (rd)
Step 1 · Find difference (d)
Use d = x₁ − x₂
Take the second reading from the first.
d
=
85.2
−
60.7
d
=
+24.5
°C
Step 2 · Find difference uncertainty (ud)
Use ud = √(u₁² + u₂²)
Independent errors add in quadrature, like the legs of a right triangle and its long side.
ud
=
√
0.4
2
+
0.3
2
ud
=
√
0.16 + 0.09
ud
=
√
0.25
ud = 0.5 °C
Step 3 · Find relative uncertainty (rd)
Use rd = 100 × ud ÷ |d|
The
uncertainty
as
a
share
of
the
difference,
in
percent:
a
small
d
makes
it
large.
rd = 100 × 0.5 ÷ |24.5|
rd
=
100
×
0.5
24.5
rd
=
50
24.5
rd = 2.0408%
Check
Put the numbers back in. Both sides should match.
+24.5 = 85.2 − 60.7 ✓
0.5
=
√
0.4
2
+
0.3
2
✓
2.0408 = 100 × 0.5 ÷ |24.5| ✓
← Previous lesson
Mean, standard deviation and a confidence interval
Next lesson →
Sensors and transducers
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.