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Uncertainty in
Scientific Measurement


Measure the length of a writing utensil provided by your instructor using a ruler marked in decimeters (dm or 10-1 m). Next, measure its length in centimeters (cm or 10-2 m). Finally, measure its length in millimeters (mm or 10-3 m).

For all measurements estimate the next significant figure. For example, say the writing utensil has a length that falls between 1 dm and 2 dm. Estimate the length to the nearest 0.1 dm. When measuring using centimeters estimate to the nearest 0.1 cm. Finally, measure the writing utensil to the nearest 0.1 mm.

Take the black line below as an example. Look at the top ruler and estimate how long it is in decimeters. I get a value of about 0.5 dm. Now look at the middle ruler. It’s clear that the end of the line is between 5 cm and 6 cm so I estimate that its length is about 5.5 cm. Now, looking at the bottom ruler I can see that the line’s length is between 55 and 56 mm. So I estimate that the line has a length of 55.5 mm (5.55 cm or 0.555 dm).

Because I estimated the last digit there is some uncertainty in it: all the digits but the last are precise but the last one was estimated. If I did a good job then that value might range plus 1 or minus 1 (± 1).

Rulers (24K)
Write your results here:
Length in dm   Example: 1.4 ± 0.1 dm
Length in cm   Example: 14.2 ± 0.1 cm
Length in mm   Example: 142.3 ± 0.1 mm
meniscus (28K)

At the front of the room there are three graduated cylinders. These instruments are used to measure out approximate volumes of fluids and can be read to a fairly high level of precision. The smaller the cylinder, the better the precision. Here is how to read the volume in a graduated cylinder to the highest possible precision. First, you read the volume using the meniscus. The meniscus is the curved upper surface of a fluid in a narrow tube. With water the meniscus curves downward. Read the volume using the lowest part of the meniscus as shown in the illustration with the dotted line. In the illustration the volume is between 9.7 mL and 9.8 mL and you can estimate here to 0.1 mL since marks are given every 1 mL. Doing so gives a volume of 9.76 mL. Note: In general you can estimate to a precision of 1/10 (0.1) the size of the smallest division on a measuring instrument. When you measure the volumes in the cylinders always estimate the volume to 1/10 the size of the smallest division.

Vol. in the 1 L cylinder   Example: 451 ± 1 mL
Vol. in the 100 mL cylinder   Example: 82.4 ± 0.1 mL
Vol. in the 10 mL cylinder   Example: 5.68 ± 0.01 mL

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There are three objects at the front of the room that your teacher has set out for you to weigh. Lab balances have limits on their precision just like rulers and graduated cylinders. Just because the balance gives a digital readout doesn’t mean that the last digit is as certain as the others. Think of the last digit as the estimated digit. The last digit is the first uncertain digit. The triple-beam balances achieve a higher precision based on your estimation of the last digit, just as for the linear measurements using the ruler.

  Using 3-Beam Balance Using Digital Balance
Mass of ____________________                ± ______                ± ______
Mass of ____________________                ± ______                ± ______
Mass of ____________________                ± ______                ± ______

Now we will move back to length measurements. Using a short ruler (not a meterstick), measure the length and width of the lab bench. Next, use a meterstick. For both measurements, report your answer in centimeters.

  Using a Short Ruler Using a Meterstick
Length                ± ______                ± ______
Width                ± ______                ± ______


Answer the following questions using complete sentences. Use your best writing and your best hand-writing. Look in your text for help with these questions. Write answers on a separate piece of paper.
  1. Define the terms precision and accuracy. Explain the difference between them by using an example.
  2. Why are measurements always at least a little uncertain?
  3. Why is estimation a necessary part of making measurements?
  4. Explain the phrase: “an instrument is only as good as its user”.
  5. What factors determine the precision of an instrument?
  6. What variable determines the accuracy of an instrument?
  7. How do you get the maximum precision out of an instrument?
  8. What is a meniscus and why is it important for measuring volumes?
  9. What is the first uncertain digit in any measurement? Does it make sense to report digits after that digit?
  10. What is the maximum precision of a 500 mL graduated cylinder that has marks every 10 mL? So the volume reading from such a cylinder would be plus-or-minus how much?
  11. A balance shows a mass of 21.134 g. What is the first uncertain digit? What is the range of the true mass?
  12. Convert the length measurements from the first exercise to cm. Which one is the most precise? Why?
  13. You measured the length and width of the lab bench in two different ways. Which way was more precise? Why? You probably reported a value ± 0.1 cm. Is this a reasonable expression of the precision of your measurement? Why or why not?
  14. Use one pair of length/width measurements to find the area of the top of the lab bench (assuming no holes). What units are correct for reporting this result? What is the proper level of precision for this calculated result?
  15. What is the rule governing significant figures in calculations involving multiplication and division? Give an example.
  16. What is the rule governing significant figures in calculations involving addition and subtration? Give an example.
Last updated: Aug 29, 2006       Keller Home  |