Uncertainty analysis Introduction

The concentration of a sodium hydroxide solution is determined by titration against a standard solution of oxalic acid.  The absolute error in the calculated concentration of sodium hydroxide can be determined from the error in each of the measurements made during the determination.  
 
These are mass of the oxalic acid dihydrate (1.5432), the volume of the volumetric flask used to make the standard solution (250
), and the volumes delivered by the pipette (20.00
) and the burette (21.76
).

The absolute error in each measurement depends on the manufacturer's specification for the piece of equipment used and has the unit of the measurement.  To find the absolute error in the calculated concentration (0.04896 
), the percentage errors in four measurements must be combined.  To do this, follow the steps below.  An example is given underneath.

Convert each absolute error to a percentage error using error in the measurement specified by the manufacturer.
The total %error is the square root of the sum of the squares of the percentage errors.

error total =
√(%error 1)2 + (%error 2)2 + (%error 3)2 + (...)2


Multiply the decimal equivalent of the percentage error by the concentration.

Round the absolute error to one significant figure.

Round the calculated concentration so that the last digit is in the same decimal place as the absolute error.
 
Quantity measured Magnitude of 
measurement		 Absolute error % error
mass
oxalic acid		 1.5432 g balance
±0.0002 g
0.0002 g  × 100 = 0.013 %
1.5432 g
volume
flask		 250.0 mL volumetric flask
±0.2 mL
0.2 mL  × 100 = 0.080 %
250 mL
volume
pipette		 20.00 mL pipette
±0.04 mL
0.04 mL  × 100 = 0.200 %
20 mL
volume
titre		 21.76 mL burette
±0.2 mL
0.2 mL  × 100 = 0.919 %
21.76 mL
Total percentage error in concentration

√(0.013)2 + (0.080)2 + (0.200)2 + (0.919)2 %

Absolute error in concentration
 
0.944  × 0.04896 mol L–1 =
0.0005 mol L–1
100

Report oxalic acid concentration as
0.0490 ± 0.0005 mol L–1
Dr Sheila Woodgate
© The University of Auckland 2002