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Measurements & Uncertainty in Science

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  • 0:03 Making Measurements
  • 4:13 Example Measurements
  • 5:50 Calculations With Measurements
  • 10:47 Lesson Summary
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Lesson Transcript
Instructor: Elizabeth (Nikki) Wyman

Nikki has a master's degree in teaching chemistry and has taught high school chemistry, biology and astronomy.

In this lesson, you will discover the importance of precision and accuracy in science while learning to make measurements. Also, you will understand how to perform calculations with measurements that conserve precision and limit uncertainty.

Making Measurements

Our vast collection of scientific knowledge explains the universe as thoroughly as possible, from the motions of the planets to the melting points of chemicals. None of this knowledge would be possible, however, without the ability to make measurements and perform calculations with them. Making measurements is the root of science. It's pretty important that you, as a scientist, can make the best, most accurate and precise measurements possible.

Let's say that you need to determine the average length of pencil in a classroom. Your goals are to be as accurate and as precise as possible.

Accuracy is the agreement of a measured value with the actual value. When you make accurate readings, you get as close as possible to the true value. The ability to be accurate depends on the tool used to make a measurement. If you are using a tool that is damaged or not calibrated properly, you will not get an accurate reading!

Precision is the degree of agreement between several measurements of the same quantity or item. When you are precise, you are getting nearly the same values every time you measure the same object. The ability to be precise also depends on the tool used to make a measurement. More sensitive instruments will allow for more detailed measurements.

A great way to imagine the concepts of accuracy and precision in action is to think of throwing a set of three darts at a target. If you hit the target in three different places you are neither precise nor accurate.

  • If one of your darts hits the bull's-eye and the other two are close, you are accurate.
  • If all of your darts miss the bull's-eye but are grouped in the same area, you are precise.
  • If all of your darts hit the bull's-eye, you are precise and accurate.

In the case of measuring a pencil, we'll want to measure with units made up of the smallest increments. To understand why, we'll measure a pencil using two different rulers: a ruler on which the smallest unit is the centimeter and a ruler on which the smallest unit is a millimeter. In case you don't remember, a millimeter is ten times smaller than a centimeter; ten of them can squeeze evenly into a centimeter.

Pencil and ruler

First, let's line the bottom of the pencil up with the 0 cm mark on the centimeters only ruler. Mine goes somewhere between 3 and 4 centimeters. Because of the markings on the ruler, I can be certain that this pencil is at least three centimeters, but I am uncertain just how much longer. I must estimate the length of the pencil to the millimeter. To make life easier, I'm going to refer to millimeters as tenths of a centimeter. To me, it looks like the top of the pencil is about 8-tenths of a centimeter. Thus, my measurement is 3.8 cm.

Notice that I was certain that the pencil was at least three centimeters. Anyone measuring this pencil would also be certain that the pencil was three centimeters. However, the digit I estimated (0.8) might not be the same for someone else. This digit is therefore called the uncertain digit because its value varies with the person making the measurement. Generally, only one uncertain digit can be given.

Now I measure the same pencil with a ruler that contains millimeters. I will treat the millimeters as tenths of centimeters. Again, the pencil is at least three centimeters. But now that I have the millimeter marks I can tell it is at least 3.8 cm. These are my certain digits. The top of the pencil seems to be half way between the 0.8 cm mark and 0.9 cm mark, so I estimate the length of my pencil to the hundredths of a centimeter. To me, this appears to be 5-hundredths of a centimeter or 0.05 cm. This is my uncertain digit. Collectively, my measurement is 3.85 cm.

My second measurement was much more accurate because I could measure all the way to the hundredths place. The first measurement had a higher uncertainty because my uncertain digit was in the tenths place, instead of the hundredths.

pencil and ruler

Example Measurements

Below are two diagrams of liquid in a cylinder. Which one will give you the most accurate reading? If you picked A, nice work! A is capable of reading to the tenth of a milliliter, while B is only capable of reading to the milliliter.

Which is more accurate?
two beakers

Now pause the video (4:38) and measure the volume in each cylinder. Your answer should contain certain digits and one uncertain digit.

The volume in B is certainly at least 2 mL, but not quite 3 mL. Since we don't have increments smaller than the mL, we must estimate to the tenth of the mL. I estimated this to be about 0.6 mL. Thus, the volume of the liquid is 2.6 mL. 2 is my certain digit, and .6 is my uncertain digit.

The volume in A is certainly 2.6 mL, but not quite 2.7 mL. We then estimate the volume to the hundredth of a milliliter, which to me looks like 0.05 mL. My measured volume of liquid is 2.65 mL. 2.6 are my certain digits, 0.05 is my uncertain digit. If your certain digits are different than mine, rewind the video, and check your work. If your uncertain digit is different than mine, that's fine! That's why we call it uncertain.

Try again with these two volumes. Pause the video (5:41) and make some measurements.

two beakers for taking measurement

The volume of A is 1.2 mL. The volume of B is 1.20 mL.

Calculations with Measurements

Let's imagine we've collected data about all the lengths of pencils in the classroom: 3.85 cm, 19.0 cm, 13 cm, and 12.055 cm.

Initially, we want to determine the total length of all the pencils combined. So, we add them together: 3.85 + 19.0 + 13 + 12.055 = 47.905 cm.

47.905 cm. . . that's our answer, right? Kind of; we have to be mindful of the fact that some of these measurements are more accurate than others and some have a higher degree of uncertainty than others. Since we are making calculations with a series of measurements, we have to be mindful of precision. Our final answer can only be as precise as our least precise measurement. So that begs the question, what was our least precise measurement?

The number of digits in a measurement - including all of the certain digits and the lone uncertain digit - is known as significant figures. Often, significant figures are nicknamed 'sig figs.' By paying attention to how many significant figures are in each measurement, we can determine how many should be in our answer.

There are five rules to determining what makes a number significant or not.

  1. Any non-zero integer is significant
    The number 3.85 has three significant figures.
  2. Any zero between two integers is significant.
    The number 12.055 has five significant figures.
  3. Zeros after an integer without a decimal are not significant, they are place holders.
    The number 100 has one significant figure.
  4. Zeros after an integer with a decimal point are significant.
    The number 19.0 has three significant figures.
  5. Leading zeros are not significant.
    The number 0.001 has one significant figure.

A note about number three and five: just because a number isn't significant doesn't mean it can be eliminated. Zeros that are not significant are still important as placeholders.

So back to our adding problem. What is our least precise measurement? Pause the video (8:09) and count significant figures as well as the place of the most precise value to determine which one it is.

  • 3.85 has three sig figs and values in the hundredths.
  • 19.0 has three sig figs and values in the tenths.
  • 13 has two sig figs and values in the ones.
  • 12.055 has five sig figs and values in the thousandths.

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