Calculating Total Percentage Errors for an Experiment
For an equation that adds or subtracts terms:
y = a(± sa) + b(± sb) + c(± sc)
where a, b, and c are the average values obtained from the measurement and sa, sb, and sc are the
corresponding errors, the propagation of error, s, is given by:
s = (sa^2 + sb^2 + sc^2)^1/2
Example: In an experiment to determine the enthalpy of neutralization of NaOH by HCl, the intial
temperature was (19.2 ± 0.2)˚C, and the final temperature (26.4 ± 0.2)˚C. What is the temperature rise?
Difference in Temperature (DT) = (a – b)
DT = (26.4 – 19.2)˚C
DT = 7.2˚C
The DT error is given by s = (sa2 + sb2)1/2
s = [(0.02 ˚C)2 + (0.02 ˚C)2]1/2
s = [(0.08)(˚C)2]1/2
s = ± 0.28 ˚C
The difference in temperature (a – b) is (7.2 ± 0.28) ˚C
The error in a is (0.2/19.2) x 100 = 1.0 %
The error in b is (0.2/26.4) x 100 = 0.75 %
The error in the difference of a – b is (0.28/7.2) x 100 = 38%
For an equation that multiplies or divides terms:
y = (a*b)/c
the propagation of error will be:
s/y = [(sa/a)^2 + (sb/b)^2 + (sc/c)^2]^1/2
In this case we are dividing the error by its associated average value. Remember that the answer is not
the error propagation itself but s/y. To get the error, multiply this value by y.
Calculating Total Percentage Errors for an Experiment
Example: What is the error in density for an aluminum block that measures 50.50 g ± 0.05g and
displaces 19.0 mL ± 0.5 mL of water? D = m/V; sa = 0.05; sb = 0.5; y = 50.5/19.0 = 2.7
s/y = [sa/a)2 + (sb/b)2]1/2
s/2.7 = [(0.05/50.50)2 + (0.5/19.0)2]1/2
s/2.7 = (0.00000098 + 0.00069252)1/2
s/2.7 = (0.0006935)1/2
s/2.7 = 0.026
s = ± 0.0702 g/mL … Accounting for sig figs this should be rounded to ± 0.1 g/mL
The measured density of the aluminum cube is 2.7 g/mL ± 0.1 g/mL
The error in a is (0.05/50.50) x 100 = 0.09 %
The error in b is (0.5/19.0) x 100 = 2.6 %
The error in density (y) is (0.1/2.7) x 100 = 37%
The actual density of aluminum is 2.70 g/mL
Source(s):
Fundamentals of Analytical Chemistry, 8th ed.
by Skoog, West, Holler, and Crouch