Measure your progress

Perhaps you are thinking that you are having a great day. Well, let’s hope that you still feel that way after this “pop quiz” on measurement uncertainty terminology and symbols. Dr. Gooddata wants to see how much you’ve learned so far from this series. Hopefully, this quiz will be fun and educational rather than a downer for you. The answers to the questions appear later in this article, but no cheating.

1. In the equation X = (measured) - ( true). X is:
    a. the unknown quantity
    b. the uncertainty
    c. the error
    d. the horizontal axis
   Hint: There’s no uncertainty here.
2. The limit to which an error can go with some confidence is:
    a. unbounded
    b. the uncertainty
    c. the degrees of freedom
    d. inconsequential
   Hint: Forget about freedom of choice in this case.
3. A systematic uncertainty is one that:
    a. doesn’t change for an experiment or test
    b. arises from a random error source
    c. varies from one test to another
    d. relates only to hardware not procedures
   Hint: Pick the best answer to avoid random variability.
4. The degrees of freedom frequently associated with systematic uncertainties are:
    a. uncertain
    b. poorly defined
    c. infinity
    d. irrelevant
   Hint: It’s neither uncertain nor poor.
5. You need to calculate degrees of freedom for:
    a. random uncertainties taken alone
    b. the root-sum-square of systematic uncertainties when infinity cannot be assumed
    c. the result uncertainty when infinity cannot be assumed
    d. all the above
   Hint: Think inclusively.
6. The proper descriptor for the ith random uncertainty is:
    a. 
    b. 
    c. neither
    d. either
   Hint: Think average.
7. In ISO notation, Type A uncertainties:
    a. have no data to calculate standard deviation
    b. have data to calculate standard deviation
    c. don’t involve the standard deviation
    d. are higher than Type B uncertainties
   Hint: The data are the thing.
8. In ISO notation, Type B uncertainties:
    a. have no data to calculate standard deviation
    b. have data to calculate standard deviation
    c. don’t involve the standard deviation
    d. are higher than Type A uncertainties
   Hint: The data aren’t the thing.
9. When evaluating the combined effects of errors, you should:
    a. root-sum-square their estimates
    b. algebraically add their estimates
    c. use only the larger of the errors
    d. use only the smaller of the errors
   Hint: Sum’s such as they are.
10. When evaluating the combined effects of uncertainties, you should:
     a. root-sum-square their estimates
     b. algebraically add their estimates
     c. always give more weight to the random uncertainty
     d. always give more weight to the systematic uncertainty
    Hint: What does RSS stand for?
11. A “measurement uncertainty” estimate is:
     a. the error of an experiment
     b. an estimate of an experiment’s or test’s error
     c. a reason for legal action in border disputes
     d. an estimate of an experiment’s or test’s error limit with some confidence
    Hint: Long answers are only sometimes tricks.
12. When a temperature measurement is reported as 75°F  5°F, the 5°F is: 
     a. the experimental error
     b. an estimate of the experimental error
     c. twice the measurement uncertainty
     d. none of these
    Hint: Have no degree of uncertainty about this.
13. Random uncertainties are reported as:
     a. two standard deviations
     b. the random error
     c. one standard deviation
     d. one standard deviation of the average
    Hint: Remember that averages count.
14. Systematic uncertainties are reported as:
     a. two standard deviations
     b. the random error
     c. one standard deviation
     d. one standard deviation of the average
    Hint: Consider the number you are dealing with.
15. For ISO classifications of uncertainties, both Type A and Type B are:
     a. two standard deviations
     b. the random error
     c. one standard deviation
     d. one standard deviation of the average
    Hint: Averages are all that count here.
    The answers are (no cheating now, take the test then look at these answers): 1. c; 2. b; 3. a; 4. c; 5. d; 6. a; 7. b; 8. a; 9. b; 10. a; 11. d; 12. d; 13. d; 14. d; and 15. d.

How did you do?

Dr. Gooddata hopes that you aced this quiz (without checking out the answers first, of course). However, just in case you don’t agree with the answers, let’s go over the tough questions.
Consider Question No. 5:

5.  You need to calculate degrees of freedom for:
 a. random uncertainties taken alone
 b. the root-sum-square of systematic uncertainties when infinity   cannot be assumed
 c. the result uncertainty when infinity cannot be assumed
 d. all the above

Well, you need degrees of freedom for all the above: for the random uncertainties in the absence of systematic ones, the RSS (root-sum-square) of systematic uncertainties if degrees of freedom can’t be assumed to be infinity, and for the uncertainty of the result. The last is because we must have the proper degrees of freedom to look up Student’s t95 in the equation for uncertainty even if there are no systematic uncertainties.

How about Question No. 9:

9. When evaluating the combined effects of errors, you should:
 a. root-sum-square their estimates
 b. algebraically add their estimates
 c. use only the larger of the errors
 d. use only the smaller of the errors

Here, remember that errors (actual signed displacements from the true value) add algebraically but uncertainties (the estimates of the limits of errors with some confidence) add in root-sum-square. [This also should clear up Question No. 10, too.]

Now for Question No. 11:
11. A “measurement uncertainty” estimate is:
 a. the error of an experiment
 b. an estimate of an experiment’s or test’s error
 c. a reason for legal action in border disputes
 d. an estimate of an experiment’s or test’s error limit with some confidence

Remember that the measurement uncertainty is an estimate of the limits to which an error can go with some confidence. The uncertainty is not the error. It is also an estimate of those limits. Some errors will actually exceed the uncertainty limits. In fact, because we calculate uncertainty at 95% confidence, 5% of the time the actual errors will surpass the uncertainty limits.

Don’t forget Question No. 12:

12. When a temperature measurement is reported as 75°F  5°F, the 5°F is: 
 a. the experimental error
 b. an estimate of the experimental error
 c. twice the measurement uncertainty
 d. none of these

Note that here no answer is correct but “d.” That  5°F is not the error, not an estimate of the error, and not twice the uncertainty. It is an estimate of the limits to which the experimental error can go with some confidence. It is the uncertainty.

Finally, let’s look at Question No. 14:

14. Systematic uncertainties are reported as:
 a. two standard deviations
 b. the random error
 c. one standard deviation
 d. one standard deviation of the average

This one is really tricky. Remember that all root-sum-square operations in measurement uncertainty calculations utilize   and not  . That is, we always use the standard deviation of the average,   . When we are dealing with systematic uncertainties, most people estimate them as two standard deviations of the average and then root-sum-square them. How can we do that? Well, we actually have only one shot at each systematic uncertainty in each experiment or test. Remember it is the constant error in our experiment we are estimating with the systematic uncertainty. Well... that means we also have the standard deviation of the average, right? Think about it.

The equation for calculating the standard deviation of the average is:  . Because we only have one shot at each systematic error, we’ve only got N =1, so, like magic,  =  for systematic uncertainties.

So, we’re okay with this. Does that help?

Well, that’s all for now. We’ll try this all on some actual data next time.  

Until then, remember, “use numbers, not adjectives.”

Ronald H. Dieck is principal of Ron Dieck Associates, Inc., Palm Beach Gardens, Fla. E-mail him at rondieck@aol.com.