uncertainty and error in measurement ©2010 travis multhaupt, m.s.,

Uncertainty and Error in Measurement

©2010 Travis Multhaupt, M.S., www.travismulthaupt.com

http://www.atr.com.my/store/index.php?main_page=product_info&cPath=1_186_261&products_id=429

http://wattsupwiththat.com/2009/06/10/quote-of-the-week-9-negative-thermometers/

Measurements Science often involves quantifying measurements to

a standard.

A problem that often arises is that when two different instruments are used to take the same measurement, rarely do they give the exact same reading.

So how so we deal with this?


Useful Definitions

First we need to identify a few things to assist us with our understanding of this problem. Precision is the closeness of the experimental results to each

other. Accuracy represents closeness to the actual value. Repeatable refers to close measurements which have been taken

by one person. Reproducible refers to the case where several similar readings

were taken by different people. Systematic Error arises when we use faulty equipment or poor

technique when taking a measurement. This type of error tends to accumulate.


Significant Figures

To determine the correct number of sig figs when doing calculations, follow these rules: 1. Multiplication & Division: give as many sig figs in the answer

as there are in the measurement with the least number of sig figs. 2. Addition & Subtraction: give the same number decimal places

in the answer as there are in the measurement with the least number of decimal places.


Uncertainty

When making a single measurement, absolute uncertainty and percentage uncertainty can be easily calculated.

For instance, if a 25.0 cm3 measuring pipette measures to ±0.1 cm3, then: Absolute uncertainty is simply 0.1 cm3, And the percentage uncertainty is 0.1/25.0 x 100% =

0.4% A 50.0 cm3 measuring device with the same

tolerance would have a percentage uncertainty of 0.1/50.0 = 0.2%


http://www.cardinal.com/us/en/distributedproducts/images/P/P4675-125.jpg

Uncertainty

Now, if we use this 50.0 cm3 measuring device to measure a smaller quantity, say 20.0 cm3, the tolerance doesn’t change, but the percentage uncertainty will. We still have an absolute uncertainty of ±0.1 cm3, But our percentage uncertainty will be 0.1/20.0 =

0.5%



Uncertainty

When we add and subtract two measurements, then we add the uncertainties. Again, using a 25.0 cm3 ±0.1 cm3 measuring device,

we might actually measure 24.9 cm3 and 24.9 cm3 to get a total volume of 49.8 cm3. Alterntatively, we might measure them to be 25.1 cm3 and 25.1 cm3 whereby we’ll get a volume of 50.2 cm3. So, our measurement would be somewhere between

49.8 cm3 and 50.2 cm3. In other words, 50.0 cm3 ±0.2 cm3.



Uncertainty

When multiplying, dividing, or using powers, then percentage uncertainties should be used in the calculations and then converted back into absolute uncertainty when the final result is reported.

For example, let’s say we are performing a titration of an unknown acid to determine its molar mass.


http://www.dartmouth.edu/~chemlab/chem3-5/ionx2/overview/procedure.html

Uncertainty

We begin by dissolving 2.500 g of an unknown acid in dH2O in a volumetric flask with a final volume of 250 cm3.

Next we use 25.0 cm3 of a standardized base for each titration. We pipette this volume into a conical flask.

Into the burette, we place 50 cm3 of the acid and perform our titrations.


http://www.dartmouth.edu/~chemlab/chem3-5/ionx2/overview/procedure.html

Uncertainty

Now consider we’ve used a digital scale, volumetric flask, a pipette, and a burette for the titration—each one of which contributes to the overall uncertainty of our experiment. The conical flask doesn’t contribute to the error,

because it is just a collecting jar.


Accounting for Uncertainty

The balance weighs to ±0.001 g, so the uncertainty is 0.001/2.500 x 100% = 0.04%

The pipette measures 25.00 cm3 ±0.10cm3, so the uncertainty is 0.10/25.00 x 100% = 0.40%

The volumetric flask measures 250.00 cm3 ±0.15 cm3, so the uncertainty is 0.15/250.00 x 100% = 0.060%

The burette measures 50.00 cm3 ±0.10 cm3, so the uncertainty is 0.10/50.00 x 100% = 0.20%

Thus the overall uncertainty is:0.04% + 0.40% + 0.060% + 0.20% ≈ 0.70%


Accounting for Uncertainty

So, if the answer for the molar mass is determined to be 129 g/mol, the uncertainty is 0.70%. Now we must convert it back to absolute uncertainty: 0.007 x 129 = 0.903 g/mol.

Thus, the answer should be reported as 129 ± 1 g/mol


Percent Error

Let’s say that the actual value of the molar mass of the acid is 126 g/mol. To find our percentage error, we use the following equation:

% Error = |(Observed – Expected)| x 100% Expected

= |(129 – 126)| x 100% = 2.4% 126


Other Uncertainties

There are other sources of uncertainty such as the tolerance of the equipment used to prepare the standard solution as well as the end-point reading.

These should be mentioned in your evaluation of the data, but do not have to be included in your calculations.


uncertainty and error in measurement ©2010 travis multhaupt, m.s.,

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