SOME THOUGHTS ON MEASUREMENT
Harold Brochmann

Assignment: Use a ruler to find the dimensions of a rectangular piece of paper, and then use a calculator to find it's area. Round your answer to the nearest cm2

 

These three students did what they were asked, performed the calculation correctly and rounded off the answer as per instructions. Yet the three answers are not the same. Which is correct? All three? Is #3 'better' because the student was 'more accurate before' rounding? Should #1 lose marks because he was 'less accurate'?

Most people are vaguely aware that all measurements inherently have limited precision. For example, it just isn't sensible to give credence to the measurements given by Student 3 if you are dealing with a piece of ordinary paper and an ordinary wooden or plastic ruler, no matter how careful the student has been. This has something to do with the number and thickness of the markings on standard rulers.

It should also be realized that there is likely to be variations in the order of 0.1 cm in the positions of the markings on the various brands and types of rulers that students in the typical class have. Even if this were not so, one might not unreasonably expect variation on this order of magnitude in the dimensions of even standard sizes of paper from one batch to another. This means that it is unfair to expect a conscientious student to produce the same measurement to the nearest 0.1 cm that you do when you determine the 'right' measure.

What is also true is that in the real world, with measurements done by real people, accuracy is invariably sacrificed as magnitude increases. Give a student a metre stick and ask him to measure the width of his desk; you'd be confident his stated measurement and the actual one would be within a few millimetres of each other... provided the desk did not have bevelled edges, of course, like most of them have. If the same metre stick were used to measure the length of, say, the school parking lot, is the result likely to be within a few millimetres of the actual length of the parking lot? In a word, "No!".

Yet another factor is what might be called 'human variation'. Try asking five of your colleagues (who are presumably very accurate in what they do) to measure the dimensions of the same rectangular paper to the nearest 0.1 cm. I'm willing to bet a nickel that you would not get five identical responses. It's not that anybody is wrong so much as that measurements have limited reproducibility.

All of this means that when students are asked to measure things, expectations of ranges of measures are far more valid than specific measures. And it follows that calculations based on measurements are valid only to the extent that there is an assumed or specified range.

We have used the terms 'accuracy' and 'precision'. Are these the same thing, and if not, does the difference matter?

'Accuracy' refers to the validity of your measuring procedure, the quality of your 'ruler' and how careful you are. 'Precision' refers to 'the number of significant digits' or the 'number of decimal places' you use when recording the result. Even if the person making a measurement is as accurate as humanly possible, limitations imposed by the measuring instrument, variations in dimensions of the object being measured, and the conditions under which the measurement is being made, dictate that the result should not imply greater precision than circumstances warrant. Here is what I mean by 'implied precision':

If you state that a certain measure is 23.4 cm, you are implying that the actual measure is more than 23.35000... cm and less than 23.45000.... cm. If this needs to be stated explicitly you should say 23.4 +/- 0.05 cm.

For reasons we have already discussed it is probably not sensible to ask students to supply measurements with this degree of implied precision. If variations of 0.2 cm in measurements are acceptable (as I suggest they should be for many things), then this is not conveniently expressible in implied precision format. It must be written with stated precision, for example 23.4 +/- 0.2 cm.

What happens when you multiply things? There is a tendency for us to assume, without giving the matter much thought, that the fact that there are more digits and more decimal places in a product than there are in the numbers being multiplied, suggests that a calculated area is more precise than the measures that were used to produce it. To explore this notion, let's determine the range of possible areas for a rectangle which measures 123.3 +/- 0.2 cm by 46.2 +/- 0.2 cm:

The product of the minima, 123.1 and 46.0, is 5662.6 The product of the maxima, 123.5 and 46.4, is 5730.4. The mean of these is 5696.5. The difference between the mean and the product of the maxima is 33.9. Therefore the area should be given as 5696.5 +/- 33.9 cm2.

Another acceptable answer would be 0.57 m2 because this implied precision is less than the previously stated one.

There are two acceptable ways of dealing with precision in measurements and calculations based on them. We have just illustrated the more cumbersome method, involving implied and stated of ranges. The other is involves the notion of 'significant digits'.

The measures "5 cm", "5.0 cm"and "5.00 cm" have 1, 2, and 3 implied significant digits, respectively. You get the idea.

But how do you know how many significant digits there are in for example, "5000 cm"... 1, 2, 3, or 4?. You don't, and that's a problem. The problem is effectively eliminated if we use scientific notation; but we aren't going into that at this time. If scientific notation is not to be used, and you have a measure like "5000 cm" where it is impossible to determine an implied number of significant digits, the precision must be stated in words. For example "5000 cm to to three significant digits", or whatever.

Suppose now we find the area of a rectangle which measures 123.3 cm by 46.2 cm. Here the implied precision of the measures are four and three significant digits, respectively. The product, 5696.46, should now be rounded off to the number of significant digits in the measurement with the lowest precision... ie three digits. The result in this case is 5700 cm2 to three significant digits. This had to be explicitly stated in this case because there is no way of telling, just from looking at it whether 5700 has 2, 3, or 4 significant digits. If, on the other hand, we write it as 0.570 m2, then there is no problem because this implies a precision of three significant digits.

In this article I have attempted to draw attention to a topic which I perceive to be generally ignored in the mathematics curriculum. It is probably an awkward topic to teach, and nobody seems to know much about what is the best way of going about it. On the other hand, when you come to think about it, ignoring the notion of precision in measurements and calculations based on them at least partly invalidates what we do when we teach applied geometry.

Although I get correspondence from people in Cyberland on vaccum cleaners and Fractal Geometry, I haven't had any comments on this particular article, until this morning when I received the following. The names of the parties involved have been removed for reasons of privacy.

(From a friend of mine)

Harold, just dropped by your web site and perused your treatise on
accuracy. Nice, and clearly stated! Thought you might be interested in this
exchange I had with our local astronomical guru a few years ago.

This topic is still being ignored in our new curriculum; however, we still
include a supplementary unit on it in Math 9!

BTW, what is your snailmail address. Sent you a christmas letter to
Scotton Place and it came back. Do I recall you mentioning a move a few
years ago?

Regards to you and Candace.

jon

 

To: (x@space-centre)
From: JC (My Friend)
Subject: Night Sky Bulletins
Date: 1996/10/07
Cc:
Bcc:
X-Attachments:

I accessed your "night sky" bulletin in September to get info about the
eclipse, for discussion with my (grade 9 Math) students. I found it very
helpful and informative -- thank you.

But, as a math teacher who tries to make my students aware of the
importance of "significant digits" in measurements and calculations, I'm a
little concerned with the implied accuracy in your reporting of distances,
e.g. " As the month opens Venus is 151,798,894.039 kilometers from us and
as the month closes it will be 182,575,688.2162 kilometers away." I can't
believe you know those figures within an accuracy of one metre or
decimetre; I WOULD be interested to know how accurately you know them?

Thanks.

JC

---------------

To: JC
From: (x@pacific-space-centre
Subject: Accuracy of Planet Positions

There is no real trick to the accuracy, I simply look up the data in the
(current year) version of THE ASTRONOMICAL ALMANAC. There all the planets'
principal data are listed including the geocentric distance (in Astronomical
Units) to the 7th signifcant decimal. I then multiply that by the how many
kilometers in an astronomical unit (149,597,870) and, voila, there is the
answere.

I trust this is of assistnace.

--------------

To: x@pacific-space
From: JC
Subject: Re: Accuracy of Planet Positions
Cc:
Bcc:
X-Attachments:

x

I asked you a question about the accuracy of distances in your Night Sky
bulletin, such as " As the month opens Venus is 151,798,894.039 kilometers
from us . . ."

You replied:
>There is no real trick to the accuracy, I simply look up the data in the
>(current year) version of THE ASTRONOMICAL ALMANAC. There all the planets'
>principal data are listed including the geocentric distance (in Astronomical
>Units) to the 7th signifcant decimal. I then multiply that by the how many
>kilometers in an astronomical unit (149,597,870) and, voila, there is the
>answere.
>
>I trust this is of assistnace.

Perhaps I didn't make the point of my question clear. I wasn't asking how
you obtained the distance, but how much accuracy it implied.

If, as you say, you have the geocentric distance given to 7 significant
figures, and the astronomical unit to 8 significant figures, then surely
the product can have no more than 7 significant figures? (It can be no more
accurate than the least accurate of the two measurements you're
multiplying!) That means you should round off the product to 151,798,900
kilometers. The other 5 digits you gave are quite meaningless, and give a
false impression of extreme accuracy.

You have a calculated distance that is accurate to within about a hundred
kilometers, not to within 1 metre as your figure of 151,798,894.039
kilometers implies.

Thanks.

JC

---------------

No reply received!!!

 

Frightening, isn't it?