## Notes on Uncertainty AnalysisThe final result from a chemical experiment, such as the value of
ΔH for a particular reaction or the average of several molarities
obtained from an acid-base titration, is often calculated from several
different
measured values. The uncertainty of the result is influenced
by the uncertainty of each of the individual measurements. Suppose,
for example, that one found the density (mass/volume) of a piece of
metal by weighing it on an
analytical
balance
(mass uncertainty ± 0.0001
g) and determined its volume by the water it displaced in a graduated
cylinder (volume uncertainty ± 0.5 mL). The error, or
uncertainty, in the calculated density would be dominated by the uncertainty
in the volume, because there is more uncertainty in the volume measurement. Before
you get started, be sure to read the Significant Figures Summary in
the Lab Manual Appendices.## Uncertainty (a.k.a. Error)Uncertainty is also known as "error." Any measured or calculated value has some uncertainty in the reported value. This does not refer to mistakes, but rather unavoidable error due to the nature of the experiment. For example, if you were measuring the width of a grape using a ruler, you might report a value of 12.3 mm but there would definitely be some error incorporated in that last digit.
It is also vital that you use several significant
figures
throughout
your
uncertainty calculations, so as to get an accurate representation
of your overall uncertainty. There are two ways to represent uncertainty: **Absolute uncertainty (AU)**is a measure of uncertainty with the. For example, the grape's width is 12.3 ± 0.2 mm, where 0.2 mm is the AU.**same units as the reported value****Relative uncertainty (RU)**represents AU as a fraction or percentage.
For example, 0.2mm/12.3mm = 0.02 = 2%. The grape's width is 12.3 mm ± 2%,
where 2% is the RU.
## Absolute uncertainty (AU)A measured quantity is often reported with uncertainty. Absolute uncertainty is the uncertainty given in the same units as the measurement:
where 0.01 g is the absolute uncertainty. There are two primary contributions to absolute uncertainty: .
precision
## Accuracy (systematic error)Systematic error is sometimes reported for specific instruments. For example, Vernier temperature probes claim accuracy to within 0.03 º C. This means that there may be a systematic error of up to 0.03 º C for any specific temperature probe. Similarly, analytical balances are accurate to within 0.0001 g. ## Precision (reproducibility error)
**Ability to read an instrument**. For example, using a ruler that is divided into cm, you may be able to determine that a wire is between 9.2 and 9.6 cm long. This could be written 9.4 ± 0.2 cm. By estimating your ability to read the ruler, you can estimate the absolute uncertainty. In this case, reproducibility error is ± 0.2 cm. Alternatively, if you are using an analytical balance and the ten-thousandths digit is fluctuating between 1 and 5, reproducibility error would be ± 0.0002 g.**Multiple measurements**. When several measurements are averaged, the reproducibility error can be approximated by the standard deviation of the measurements.
Most of the time, we will only be dealing with reproducibility uncertainty. However, if we know both, AU is calculated:
In the case of the analytical balance mentioned above:
Notes: - AUs are positive values with one significant figure.
- AUs have units if the associated value has units.
## Relative uncertainty (RU)Relative uncertainty is a fractional value. If you measure a pencil to be 10cm ± 1cm, then the relative uncertainty is one tenth of its length (RU = 0.1 or 10%). RU is simply absolute uncertainty divided by the measured value. It is reported as a fraction or percent: For the example given under AU:
Notes: - RUs can be represented as either a fractional value or a percent.
- RUs have no units.
## Propagation of UncertaintyWhen you perform calculations on numbers whose uncertainties are known, you can determine the uncertainty in the calculated answer using two simple rules. This is known as propagation of uncertainty. maximum possible uncertainty.**Addition and Subtraction:**AU = ΣAU
When calculating uncertainty for the sum or difference of measured values, AU of the calculated value is the sum of the absolute uncertainties of the individual terms.- RU
_{A+B}≠ RU_{A}+ RU_{B}. - RU can be calculated using the equation RU = AU/|value|.
- Even if you are subtracting
measured values, be sure to
*add*AUs. - Calculate q
_{total}, ignoring uncertainties: - Add absolute uncertainties:
- Calculate relative uncertainty from absolute uncertainty:
- Report your final answer::
**Multiplication and Division**: RU =ΣRU- To determine the correct number of significant figures in your answer, calculate the absolute uncertainty, round it to one significant figure, and then round the calculated value to the same digit.
- AU
_{AxB}≠ AU_{A}+ AU_{B}. - AU can be calculated using the equation AU = RUx|value|.
- Relative uncertainties are always positive. Be sure to calculate RU using absolute values.
- Calculate q
_{cal}, ignoring uncertainties: - Add relative uncertainties:
- Calculate absolute uncertainty from relative uncertainty:
- Report your final answer:
**Combination of addition/subtraction with multiplication/division**- Calculate q
_{total}, ignoring uncertainties: - Determine uncertainties for the multiplication:
- Determine uncertainties for the addition:
AU _{qtotal}= AU_{qsolution}+ AU_{qcal}=__2__0 + 47.4 = 67.4 J=__7__0 JRU _{qtotal}= AU/|(q_{total})| = 67.4J/|-1774J| = 0.04 - Report your final answer:
Notes:
q AU RU q When calculating uncertainty for the product or ratio of measured values, RU of the calculated value is the sum of the relative uncertainties of the individual terms. However, AU cannot be calculated directly from AU or RU values of the measured values.
Notes:
q RU AU q
For a combination of these operations we follow the standard order of operations.
q Note that you did the multiplication first, then the addition. We will approach the uncertainty calculations in the same order. RU AU q
Reference: Taylor, John R., An Introduction to Error Analysis |

Created By: Adilia James '07 and Sarah Coutlee '07 Maintained By: Nick Doe Date Created: July 3, 2006 Last Modified: February 11, 2009 Expiration Date: July 3, 2007 |