Why all this bother about measurements?

You may wonder why we spend so much time talking about measurement standards and handling data properly. Two stories....

1. Suppose you are buying fabric, and you and the seller agree to a price over the phone for 10 yards of upholstery fabric. When she makes delivery, you discover that her tape measure went through the washer and dryer and is only 3/4 as long as yours. She insists she is selling you 10 yards (and "proves" it by measuring with her tape measure.)

2. You need ten half inch holes drilled in a piece of aluminum. The shop gives you an acceptable price, so you write up the order for drilling ten holes each 0.500 inches in diameter. The shop owner raises his eyebrows and charges you far more than the original estimate.

It is important that everyone in a transaction agree on units of measurement, and that close attention be paid to the stated accuracy of units!

Precision

To understand the physical world, physicists attempt to find relationships between quantities. For example, how does the stopping distance of a car depend upon its initial speed? Or, how does the length of time it takes a kettle of water to boil depend on the amount of water in the kettle? Usually these relationships are expressed as mathematical equations. To verify a relationship, measurements must be made of the relevant quantities.

Of course, it is important to make measurements as precise as possible. However, no measurment is ever absolutely precise. Every measurement has some uncertainty associated with it. (We are not talking here about outright mistakes in measuring, for example, misreading the scale of a ruler.) For example, the precision of a measurement is limited by the ability (or inability) to read between the smallest divisions.

Let's say you are measuring the distance between two lines on a piece of paper. (Figure 1) The ruler is accurate to around 0.1 cm ( or, 1 mm), which is the distance between the smallest divisions. The measured distance has an uncertainty of about 0.1 cm, since it's difficult to interpolate between the lines on the ruler. The distance between the lines could be stated as 7.7 ± 0.1 cm (or, 77±1 mm).

 Figure 1 measurement of a piece of paper

The distance between the smallest divisions, in this case 0.1 cm, is sometimes called the precision of the measurement. When using this particular ruler, it doesn't matter how wide the piece of paper is, the precision is still 0.1 cm because that is the smallest unit on the ruler. The precision of many bathroom scales is 1 pound, although digital scales may be precise to 0.1 pound. If you have a thermometer (outdoor, oven, candy, etc) can you determine its precision?

Often, the uncertainty in a measurement isn't stated but is implied by the way the number is written. The measurement 7.7 cm implies that the ruler had a smallest division of 0.1 cm so the uncertainty is about 0.1cm. The odometer of my car reads 23568 miles, so the uncertainty is about one mile (the smallest unit in the measurement).

ARITHMETIC WITH MEASUREMENTS: ADDING AND SUBTRACTING

Suppose I need to know the width of the paper in Figure 1 with more precision. I could use a vernier caliper and find the width to the nearest 0.01 cm ( or tenth of a cm). I do that and find the width is 7.72 cm. Now, I think I could get a better idea of the true width if I averaged the two measurements together. The first step in finding the average is to add the two measurements: 7.7 cm + 7.72 cm. Is the answer 15.42 cm? NO!!

The problem is that the first measurement wasn't precise enough to provide a digit in the 0.01 cm place. That is, the number is 7.7? where the ? indicates I have no idea what the digit should be (my ruler wasn't good enough). What do you get when you add ? to 2? I don't know!

7.7 ? + 7.72 = 15.4 ?

THE RULE: When adding measurements of different precisions, the sum cannot be any more precise than the least precise measurement.

EXAMPLE: My odometer reads 23123 miles and I set the trip meter to 0. After I drive to school the trip meter reads 4.2 miles. What does the odometer read?
23123 miles + 4.2 miles = 23127 miles (the answer is rounded to the nearest mile)

While this may not look like "correct arithmetic", it is correct if the numbers being added (or subtracted) are measurements. To state an answer to excess precision is to pretend to know the sum of numbers you never measured!

Accuracy

Precision was defined as the smallest unit measured with a particular instrument, such as, 0.1 cm for a plastic centimeter ruler. Now we define accuracy, or significant digits, using the idea of precision to help us.

Suppose you measure the width of a piece of paper to be 2.0 cm (precision 0.1 cm). The uncertainty of the measurement is 0.1 cm (in terms of centimeters). We can express the uncertainty in terms of a percent as well. That is, the measurement can be stated as 2.0 cm ±5%

Now suppose you measure a larger piece of paper with the same ruler. The new measurement is 25.4 cm. The precision of the measurement is the same (0.1 cm) because it is the same ruler. The uncertainty in the measurement is only 0.4%, however, because the measurement is larger. Another way of saying this is that in the second case, more "units" or digits are measured.

The first measurement included only 2 digits- whole cm and tenths of cm. The second measurement included 3 digits. We call these significant digits (measured units) and say that the number with more significant digits is more accurate, because of the smaller percent uncertainty.

ARITHMETIC WITH MEASUREMENTS: MULTIPLYING AND DIVIDING

When using your calculator, you'll notice that answers to multiplication and division problems often give you LOTS of digits- the whole display is lit up with digits! You may have learned in a math class to write all these down as the "answer". But, just as in the case of precision, if you include all these digits you are implying that you know the answer with more accuracy than you do.

THE RULE: When multiplying/dividing measurements of different numbers of significant digits, the answer cannot have more significant digits than any of the measurements

EXAMPLE: A piece of paper is 2.0 cm long and 125.4 cm wide. What is its area?
The area of a rectangle is length times width, so

A = (123.4 cm ) (2.0 cm) = 246.8 cm2 = 250 cm2 (two significant digits- just the 2 and 5)

Sometimes, it is difficult to determine how many significant digits a measurement has. For example, does 1200 meters have two signficant digits (between 1100 and 1300 meters) or three significant digits (between 1190 and 1210 meters), or four significant digits (between 1199 and 1201 meters)? In such cases the ambiguity can be removed by using scientific notation. (See the tutorial on scientific notation if you need a refresher.)