Notes on Error Analysis

Any experimentally derived number is inexact.  Any number that comes from  a measurement has an error associated with it.  This brief note is an introduction to how to record and use quantities and their errors.

Types of Errors:

Limit Errors:

Suppose you measure the distance between two dots on a ticker-tape.  If the ruler that you used is ruled in cm spacings then you will have to estimate the separation to the nearest 0.5 mm.  Thus, you would record the separation, for example as  12.3  cm +/- 0.05 cm.  The +/- 0.05 cm represents the worst that the error is likely to be.  That is, its the limiting bound of the error - hence limit error.

Try  It!  Measure the separation between the dots:

                           .              .

Standard Errors:

Sometimes you can get a much "finer" estimate of a quantity and its error if you can make repeated measurements of the quantity.  Intuition probably tells you that averaging many measurements is a better representation of the actual quantity than is any one measurement.  Mathematics will support this claim - especially if the errors that occur are random.   As an example, Suppose that the following numbers are the timings for a cart to travel 12.3 cm +/- 0.05 cm along an incline:

Dt = 2.34 s, 2,29 s, 2.40 s, 2.36 s, 2.38 s

The average time is found to be 2.35 s.  Notice that 2.35 s is NOT one of our measured times. If 2.35 s is the average time what is the error in the measurement?  Answer: Not obvious but the standard deviation is a good measure of the spread in the values around the average (mean)  The standard deviation for this set of numbers is 0.04 s.   So, the time taken to travel 12.3 +/- 0.05 cm was 2.35 +/- 0.04 s. 

On EXCEL you can easily calculate averages and standard deviations by using the =average(<block of numbers>) and =stdev(<block of numbers>) functions.  Try this!

 

Recapping:

  • Limit Errors:  Use when making a single measurement and estimating

  • Standard Errors:  Use standard deviation when making many measurements of the same quantity.

 

Working With Errors

How fast was the cart used in the previous example moving?

but both Dx and Dt have errors attached to them:

The trouble here is that the errors are of different types (cm and s).  We canfigure out the error by looking at the extreme range of possible speeds:

 

The average velcoity is then 5.23 cm/s +/- 0.11 cm/s.  BUT! there is a simpler way!!

Absolute and Fractional Error Forms

Up to now we have expressed the errors in their absolute forms.  A much easier way to deal with errors is to convert them to a fractional or percentage form.  Then all errors are of the same type (compare apples with apples!).  So:

and

NOW!  To calculate the total error just ADD the percentage errors of each quantity.  This means that the error in average speed will be 0.4% + 1.7% or about 2.1%.  What is 2.1% of 5.23 cm/s? Answer: 0.11 cm/s.

In Abstract Looking Form:

d X= error in quantity, dX/X = fractional error (multiply by 100 to convert to % form)


Summary of Rules for Working With Errors

  1. When adding or substracting quantities take the largest of the two fractional errors as the error for the final result. (There is a more exact way of doing this but this is good enough for now)

  2. When multiply or dividing you must ADD the fractional errors.

  3. When raising to a power you must multiply the fractional error by the power.

  4. When doing complicated calculations you may need to apply all of the above, perhaps several times.

Example:

You make 5 timings of an object rolling from rest down a 1.20 m ramp whose length was measured to the nearest 1 mm. The timings are:

0.45 s, 0.46 s, 0.51 s, 0.47 s, 0.42 s

What acceleration should you record?

  1. To start, it is easy to show that we should use the following expression 
  2. Now, look at the pieces:
  1. Next, note that the value we determine for a will contain error contributions from both d and t.  Also, applying the rules above we see that since t is squared in this expression we must double the error made in t.  Finally, in order to combine these errors we must convert them into fractional form.  This is shown in the following expression:

If we insert numbers we get:

and conclude that a = 11.34 m/s2 +/- 0.13
To convert the fractional error +/1 0.13 into m/s2 we multiply by 'a' and get:   a = 11.34 m/s2 +/- 1.5 m/s2

About Significant Digits?  Notice that I was not particularly careful about sig digs.  Although you may have learned elaborate rules for reporting sig digs the most important thing to note is that you need to be able to justify the number of sig digs that you display.  It is important not to include more digits than you could possibly justify.  If in doubt ask!