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PHY 241

Lab 1: Uncertainty


A central idea for all the laboratory experiments is UNCERTAINTY.

Rule 1: Every measurement/calculation MUST also report the uncertainty in that measurement/ calculation in the form

is the measured value. is the uncertainty in that measurement. When we state that we are promising the reader of the lab that we are certain that is between the values

and .


Example 1

Imagine that we are given a nice straight piece of wire and told to measure the length. Using some measurement device like a ruler, we find that the length of the wire is . We must report exactly how close to it is.

  1. If we are absolutely certain that the length is between and , we would report: In this example and This is a very precise measurement because is so much smaller than ; it requires a significant investment of time and resources to measure a wire this carefully.
  2. On the other hand, if our measurement tool is not very good or the wire is full of kinks, maybe we are only certain that the length of the wire is between and , we would report: . This is not a precise measurement; however, we should be able to make this measurement in less than a minute with a basic ruler. In this example and .

Notice that both a) and b) follow the following rules for all uncertainties.

Rule 2: The uncertainty in any measurement, can never be smaller than the half the smallest division on the measurement device.

                Most rulers have tick marks that are apart, therefore any length that we get directly from a ruler must have an uncertainty that is . or larger.

If a sensor is reporting that the temperature is and the number is not changing at all then the uncertainty will be . On the other hand, if the sensor is fluctuating between and we will use Rule 3 below because we are not certain what the value is.




Rule 3: When writing the lab report, the uncertainty in any value, must have exactly ONE “significant figure.”

                To find from a minimum and maximum we can use:

Where we ALWAYS ROUND UP so there is one non-zero digit.



Rule 4: The measured value, must be rounded to the precision of the uncertainty.  

                To find from a minimum and maximum we can use:

Where we must round to exactly the same digit as the uncertainty.


Now unfortunately, we will very rarely be able to measure a desired quantity directly. Our tools measure distance, time, and weight. And from these direct measurements we must use equations to calculate things like Force, Energy, or Density.   The rules for uncertainty equations ARE DIFFERENT and will require practice.



Rule 5: If two measured values are ADDED or SUBTRACTED, you must ADD the uncertainties to get the uncertainty in the result.

Example: Imagine you measure two different times, and , and you are interested in the amount of time between these two measurements, . In this case you would find that

But more importantly the uncertainty in the time between measurements would be

And now we can report that for this example the time between our two measurements is


Rule 6: If two measured values are MULTIPLIED or DIVIDED, you must ADD the FRACTIONAL UNCERTAINTIES.

Example: Velocity comes from the formula . If we measure and     then the calculated velocity must be

But more importantly, the uncertainty follows the equation above,

But of course must obey Rule 3, so the uncertainty is rounded up to

And therefore must obey Rule 4 and be rounded to the same precision



To explore the rules above we will perform a series of difficult but straightforward experiments. Our goal for today is to answer the following question.

“How dense is an average sheet of copier paper?”


Density, is a property of a material that is measured in units of and defined by the ratio of the mass, , of the object to its volume, ,

Because the density of our paper is constant, we can instead use the formula

Experiment 1.1- The Silly Experiment.


1 sheet of paper

1 ruler or meter stick

1 triple beam balance

Excel sheet- “Lab 1 Template PHY 241”


  • From Blackboard, Course Content, Lab Files open “Lab 1 Template PHY 241.”
  • Before you make any modifications, save your Excel Template as “Lab 1-Names of group” in the folder “Section XXX” that can be found on the desktop.
  • Measure the Length (L), the Width (W) and the Thickness (T) of a single sheet of paper using a ruler, notice that the units must be recorded for the column in the header for that column. Record the minimum and maximum possible values for each of these measurement in the yellow cells between B6 and C10.
  • Notice that as you enter data in the yellow cells, the dark blue cells automatically update their values according to the equations that have been already been saved in those cells. On a scrap of paper you should check that these values obey Rules 3 and 4 except that calculated values have NOT been rounded. We always round our values by hand when we quote data in the lab report, not while calculations are being done in the excel file.
  • Make sure that cells F6, F8, and F10 obey Rule 2. If not you will need to adjust your values in B6 throu gh C10.
  • Notice that the Excel is also calculating the total Volume of the sheet in cell H7. However because there is a change of units, cell H4 is used repeatedly.
  • Open the program “Fundamental Science Skills” on the computer and ensure that all group members know how to calibrate and read a triple beam balance to measure the mass of an object.
  • Measure the mass of one sheet of paper, and record the results in B12 and C12.

Important information about Excel Formula:

To let Excel know you are typing a formula, you must begin the entry with an equals sign “=”

Excel uses cell references to do calculations. Because we do not want to spend all of our time typing equations, again and again, we will take advantage of Absolute References and Relative References.

                Absolute References always have “Dollar Signs” in them, like $B$2. When you copy and paste a formula into a new location, the Absolute references remain exactly the same and the new formula continues to reference the same cell.

                Relative References do NOT have “Dollar Signs” in them, like B2. While in a formula, B2 and $B$2 behave exactly the same. BUT when you copy and paste the formula into a new cell, all the references without dollar signs will reference a new cell based on where you do the copying.

  • Light blue cells are waiting for you to enter an equation. Using nearby equation cells as guides, enter equations in cells E12, F12, H12, and I12 to calculate the Density and uncertainty in the density . Be careful with the fact that the units are changing from g to kg.


  • Notice that the graph, has been pre-saved to plot the density of the paper where the length of the bar represents the experimental uncertainty that the designer of the lab built into this experiment. The uncertainty for this measurement is awkwardly large, not because of mistakes on the part of the students, but because the experiment was completed with inappropriate equipment and/or a naïve procedure.
  • In fact the uncertainty is SO LARGE that our error propagation formulas have failed us! If you look at the values for and , you should notice that So when the density is calculated you should have .   Any time the Percent Uncertainty of any measurement is greater than our error analysis will become less reliable and in this case the percent uncertainty is greater than 100%.


  • To get a feel for how uncertainty is propagating through this simple measurement we want to adjust the values in the yellow cells and see how the density of paper plot changes.
    1. Go to cell C6 and adjust so that is twice as large. Is there a noticeable change in the plot of the density? (After finding your answer return to its proper value.)
    2. Go to cell C8 and adjust so that is twice as large. Is there a noticeable change in the plot of the density? (After finding your answer return to its proper value.)
    3. Go to cell C10 and adjust so that is twice as large. Is there a noticeable change in the plot of the density? (After finding your answer return to its proper value.)
    4. Go to cell C12 and adjust so that is twice as large. Is there a noticeable change in the plot of the density? (After finding your answer return to its proper value.)
    5. Which of the above adjustments (a…d) made the greatest change to the plot?



Experiment 1.2 –Fancier Equipment.


1 sheet of paper (the same sheet from Experiment 1.1)

1 caliper

1 micrometer


  • Because thickness was the main source of uncertainty in Experiment 1.1, we will remeasure the thickness using better equipment.
  • If you are uncertain how to use the Caliper or the Micrometer please watch a 5 or 10 minute video on Youtube. Just google: “How to use caliper” or “How to use micrometer.” The lab computers also have the program “Fundamental Science Skills” which will help you be certain you know how to use the equipment.
  • Measure the thickness of the sheet of paper using the caliper and ruler. Record the appropriate values in Cells E19, E20, F19, and F20.
  • Make sure that cells F19 and F20 obey Rule 2.
  • On a scrap of paper check to make certain that cells B19, B20, C19 and C20 are calculated correctly following Rule 1. Notice, that one of the values is not correct. Using the equations in the cells nearby, correct the equation in the erroneous cell.
  • Finally, fill in the equations in cells H19 through K20 to calculate the density for the sheet of paper using these tools.


  • All three bars on the graph represent the density of the paper as you have measured it so far. Since you are still measuring the same sheet of paper, you must get the same answer. However, because the different equipment have different levels of precision, your three bars will not be identical but they should overlap. We know from the analysis section above, that the “Ruler” bar should extend much farther, but our Error Propagation does not work very well when .
  • If all of the bars on the graph overlap in the same region, then all three of your measurements agree. If the bars of the graph for Caliper and Micrometer do not all overlap in the same region, most likely you have made a mistake in your lab and you need to review the lab and look for mistakes or physical reasons that your different measurements disagree.
  • Longer bars mean less precision, shorter bars indicate greater precision. In the Conclusion section of the report, make certain you answer the question: Based on the observed precision, how do we evaluate which of the methods so far is the best?


Experiment 1.3 –Lots of paper.


Lots of copier paper

1 caliper or micrometer


  • For this experiment, we will not record the uncertainty in each individual measurement and instead calculate the uncertainty that arises from the collection of measurements using the idea of “Standard Deviation.”
  • Get a stack of at least ten sheets of paper. Measure and record the mass of your stack in cell B27.
  • Using your chosen measurement tool measure the thickness of the stack. Record your measurement in cell C27.
  • Using at least 10 additional sheets each time, Repeat steps 2) and 3) until you have a total of ten measurements, filling the cells from B27 through C36 with your data. Notice trial ten should involve more than 99 sheets of paper.
  • For cells D28 through E36, we don’t want to type in the formula each time. Instead, we will highlight cell D27 AND E27, copy, highlight cells D28 through E36, and paste. Check and make sure that the absolute and relative references were are calculated appropriately.


  • Since we have the Mass vs. Volume data points conveniently plotted in the excel file evaluate using the concept of derivative you learned in your calculus class. In the conclusion section of your report, make certain you answer the question: How does the Mass vs Volume graph reflect the fact that we are told on page 3 that density is constant for this experiment?
  • Because we are taking the time to measure the same thing multiple times, we will not need to get the uncertainty of each individual measurement. Instead, we will use the excel function “=Average(XX:YY)” to average all of our measurements and determine . The excel function “=stdev.s(XX:YY)” finds the uncertainty of our measurements, using the formula from statistics called the “Standard Deviation.”
  • Notice that the Standard Deviation method is not limited by the smallest division of our measurement device. Therefore, we should have much more precise measurement of the paper thickness from this method.
  • Notice that there is a tradeoff between taking time to find the uncertainty of measurements directly, or taking lots of measurements to take advantage of the Standard Deviation formula, both methods require a significant investment of time. Sometimes you will be required to do it one way, and sometimes you will be required to do the other.



At this point you should have four measurements of the density of paper plotted on the graph in excel. Because all three are direct measurements of the paper’s density, all three should give overlapping results. If this is not the case, you have done something significantly incorrect. (Maybe you were careless with your units, maybe you need to use the computer program “Fundamental Science Skills” to see if you are using the caliper or micrometer correctly, maybe you accidently altered the excel template formulas and need to find the mistake, or reopen the template and reenter your raw data.)

You are now ready to open the Lab Report Template file. Each group member will write one or two sections of the report based on your role, and then you will all combine your work to form the full report.