Getting the Most Precision from a Metric Scale

We will use the metric ruler in lab.

NO measurement is exact; some error (uncertainty) is always involved. This means that every result in science has some uncertainty associated with it. We might be fairly confident we have the correct answer, but we can never be 100% certain we have the EXACT correct answer.

Measurements always have two parts, a numerical part (sometimes called a factor) and a dimension (sometimes called a label or a unit). The reason for this designation is that we are measuring quantities such as length, elapsed time, temperature, mass, etc. Not only do we have to tell how much there is, but we have to tell how much of what.

In a mathematics class units are inconsistently used. This situation is because much of mathematics discusses the relationships between pure numbers, not the use of a number to describe the amount of something. Many astronomy students have the unfortunate tendency to see units as unnecessary. THEY ARE NOT.

Measuring gives significance (or meaning) to each digit in the number produced. This concept of significance, of what is and what is not significant is VERY IMPORTANT. Especially the "what is not" portion. The concept of significant figures (or significant digits) is important and will play a role in almost every exercise undertaken in the astronomy lab.

The dimensions of an object refer to some property the object possesses. Examples include mass, length, area, density, and electrical charge. Dimensions are usually called units in this course.

The metric scale is a simple but important measuring device. The whole numbers in the images below represent centimeters. The divisions are tenths of a centimeter, or millimeters. An arrow will represent the end of an object being measured. Zero is always to the left.

Example #1: photo 1

1) We know for sure the object is more than 2 cm, but less than 3 cm.

2) We know for sure the object is more than 0.8 cm, but less than 0.9 cm.

How do we know these two things?

Look where the arrow is, it is to the right of 2, but short of 3. It is to the right of 0.8, but short of 0.9. So, I can say the object is more than 2.8 cm, but less than 2.9 cm. We can say this with complete confidence because of the markings on the ruler.

Can I say anything more about the length?

1) Look at the gap between the 0.8 and 0.9 cm, where the arrow is and, mentally, divide that gap into 10 equal divisions.

2) Estimate how many tenths to the right the arrow is from the 0.8 cm.

Let us say your answer is two-tenths. We then say the object's length as 2.82 cm. The first two digits are 100% certain, but the last, since it was estimated, has some error in it. But all three digits are significant.

This issue of estimation is important. Experience tells us that the human mind is capable of dividing a short distance into tenths with acceptable reliability. However, there is error built in and it cannot be escaped. Since the reliability is acceptable, we say the digit is significant, even with the built-in error.

However, the process stops there. ONLY ONE estimated digit is allowed to be significant.

Example #2: photo 2

What length is indicated by the arrow?

1) More than 4 cm, but less than 5 cm.

2) More than 0.5 cm, but less than 0.6 cm.

Correct answer = 4.50 cm. The arrow is pointing directly at the mark and is neither to the left nor to the right of it.

Notice that whatever the smallest division in your scale is, you can always estimate to the next decimal place after. In this case, the smallest division is in the tenth place, so we can estimate to the 0.01 place.

Be aware that there is some error, some uncertainity in the last digit of 4.50. While one should make an effort to estimate as carefully as possible, there is still some room for error.

The rule about uncertain digits is that there can be one and only one estimated or uncertain digit in a measurement. It is always the last digit in the measurement.

Here are two more examples of centimeter rulers. Decide what length is being shown, then click the link for the answer.

Example #3: photo 3 Check Answer to Example #3

Example #4: photo 4 Check Answer to Example #4