Propagation of Error: An Error Analysis Activity

 

Errors of a Calculated Result

More often than not, we use measured quantities to determine derived quantities. For instance, we would take mass and volume measurements to make density determinations.

 

If we use a balance with a ± 0.001g error and a pipet with a ± .01ml error, how should we combine these errors to determine the error in density?  The use of significant figure is an approximation of error, but for a more exact representation we use the following.Text Box:   The first thing we need to do is define two values:

 

Absolute error is the approximate error of a single measurement.

The absolute error is best estimated as the standard deviation for a measurement.  Absent this data an estimation should be made based upon the confidence the experimenter has with their ability to read the measuring device.

For example – when reading the volume of liquid in the graduated cylinder to the right, you would estimate 64 mL but it might be 63 or 65 so the value would be reported as 64 ± 1 mL and the absolute error (DV) = 1 mL.

 

Relative error is the ratio of the size of the absolute error to the size of the measurement being made.

 

Relative error =

 

The relative error for the measurement above is 1/65 = 0.015 which is sometime reported as a 1.5 % relative error.

 

Using these values we can approximate the error in calculated values.  For addition and subtraction the absolute value of the sum or difference can be roughly approximated as the sum of the absolute values.  If measure the volume of a rock by displacement of water we might take two volume measurements:

 

Volume water = 2.5 ± 0.3 mL                                                  (range = 2.2-2.8 mL)

Volume water + rock = 6.8 ± 0.4 mL                                       (range = 6.4-7.2 mL)

 

The volume of the rock is then 6.8mL – 2.5 mL = 4.3 mL.  But the two volume measurements may range over a variety of values.  Subtracting the high value for water volume from the low value for water + rock will give a low value for the rock volume (3.6 mL) and subtracting the low value for water volume from the high value for water + rock will give a high volume for the rock volume (5.0 mL).  The range of values for the rock volume is 3.6-5.0 mL or 4.3± 0.7mL.  The absolute error in this measurement, 0.7 mL is equal to the sum of the absolute errors for the two original volumes.  This idea is illustrated in the picture that follows.  This method of error propagation overestimates the combined error because of the possibility that errors can cancel when more than one measurement is made.  In fact, if we look at this process statistically, we find that a better approximation of error in a sum or a difference is given by the formula:

 

                                     

 

 

When performing multiplication and division, the propagation of error must use relative rather than absolute errors.  This is illustrated when one calculates density.  The error in mass is in grams, while the error in volume is in ml.  We cannot add grams to milliliter to calculate the total error.  However, the relative error is unitless.  The sum of the relative errors is an approximation of the total relative error (although it is an overestimation as before).  Again, if we look at this simple idea using statistics we find that a better approximation of error results from the following

 

                

 

And the final absolute error in the result is given by:

 

 DR = relative error·R   

 

 Where R is the calculated value.

 

Table 5 below summarizes the equations required for propagation of error.

 

 

 

Table 5 – Error Approximation

Operation

Example

Error

Addition

S = A+B

Subtraction

D = A-B

Multiplication

P = A x B

Division

Q = A / B

 

Let’s apply these rules in the following example:

 

A chemist analyzes a compound containing only H, C and O. He does a C-H analysis with the following results:

            % C = 54.80 ± 0.05

            % H = 6.92 ± 0.05

 

What is the uncertainty in the % O?

 

% O     = 100.00% – (54.80% C + 6.92%H)

 = 32.28%O ± ?

 

Applying the rule for subtraction we get

Text Box:

D%O

 

 

 

 

 

 

So we can report % O as 38.38 ± 0.07

 

Now let’s apply these rules to a density calculation:

 

A student finds the density of a liquid by allowing a 10.00mL volumetric pipet filled with the liquid to drain into a previously weighed Erlenmeyer flask. The following data was recorded:

 

            Volume liquid = 10.04 ± 0.01mL

            Mass empty flask = 22.452 ± 0.002g

            Mass flask + liquid = 33.629 ± 0.002g

 

 

For error in mass we use the rule for addition

 

 

For error in density we use the rule for division

 

The student can now report the error in density as

            D = 1.11 ± 0.04 g/mL

Note that the final density and the error in the density end at the same decimal place.  Normally, we only report the error to one significant digit.  Generally speaking, as demonstrated in the above example, error in the least accurate quantity determines the magnitude of overall error.

 

See the following web sites for good discussions of error propagation

http://www.rit.edu/~uphysics/uncertainties/Uncertaintiespart2.html

http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc55.htm

http://teacher.nsrl.rochester.edu/phy_labs/AppendixB/AppendixB.html

Experimental Design

 

As you know, density is a physical property of matter.  It is defined as the ratio of the mass of an object divided by its volume (D = m/V).  For solids and liquids, density is usually expressed in g/cm3.  It is evident that you must know the mass and volume of an object to determine its density.  In this part of the procedure, a cylindrical shaped metal slug will be used.  Its mass will be determined by a simple weighing.  The volume, however, will be determined by two different methods.  An estimate of error will then be made for each method, and then the method or methods with the highest degree of precision will be decided.

 

a.       Using dimensions of the cylinder to determine volume - Prepare your data table in your notebook so there is space to record the quantities indicated in each part.  Be sure to estimate and record the uncertainty of each measurement as you are making the measurements.  Obtain the metal cylinder assigned to you.  Use Vernier calipers to measure the diameter and length of your metal cylinder. Your instructor will demonstrate how to use these calipers.   Make your measurements in centimeters and don’t forget to include the uncertainty of your measurements.  This will give a volume measurement in cm3.   

 

b.      Determining volumes by water displacement – partially fill a 50 mL graduated cylinder with water and read the volume.  You must initially have enough water in the cylinder to completely cover the metal slug, but be careful that the displaced water does not exceed the 50 mL mark.  Tip the graduated cylinder at a sharp angle to the vertical and carefully slide the metal slug into the graduated cylinder.  Do not drop the metal slug into the graduated cylinder.  Take care to avoid splashing water up the sides of the graduated cylinder, or breaking the graduated cylinder.  Place the graduated cylinder back on your bench and record the new volume.  Don’t forget to include the uncertainty of your measurement and to take the temperature of your water.