Significant Figures and Uncertainty

Lab Objectives

• Formatting using Significant Figures

• Reporting a measurement

• Propagation of Uncertainty

• Accuracy versus Precision

Lab Equipment

• Short Ruler

Making a measurement

1. These numbered gray boxes on this Lab Instruction sheet denote that there is a question that you need

to answer. There is a companion worksheet in Microsoft Word format. You should answer the questions

in the text box corresponding with the question number. Once you have completed the lab and answer

worksheet, you should save the file as a PDF and upload the PDF to the Canvas assignment associated

with this lab. If you have any trouble converting to a PDF, please ask your Lab Instructor or Lab Assistant.

You should have a small ruler at your station. Use this ruler to measure the width (the shorter side) of your lab

computer keyboard, not your laptop keyboard. Once you and your partner have agreed on a number, estimate

the uncertainty of your measurement. There are many ways to determine the uncertainty of a measurement in

physics, and we will look at a few of them this semester.

For this estimate, we want you to pay attention to the precision of your ruler. The ruler has division markings

on it that define the distance between known units of measurements. Your estimate should be based on how

accurately you are able to discern where the object you are measuring falls on the ruler. For example, on the scale

in Figure 1, the gradations are in tenths of a centimeter (not to scale!). With these gradations, you can certainly

measure to within a tenth of a centimeter, but what about objects that fall between the gradations? Is it reasonable

that you would be able to tell whether the object was closer to one gradation than another? It is this uncertainty

that you are to estimate.

Your estimate should be in the form of a range of possible values expressed as a plus-or-minus value around

your measurement. For instance, if you measure an object that appears to be 2.5 cm, but the ruler makes it hard

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Figure 1: An example of a ruler (not to scale).

to tell if the object is longer or shorter than that up to a tenth of a centimeter, you would report a measurement

of 2.5 cm with an uncertainty of ± 0.1 cm. This indicates that you are reasonably certain that the actual length of

the object falls between 2.4 and 2.6 cm.

The format for your measurement should be “measured value” ± “uncertainty” (x ± δx). Once you have

these values, enter them below, then go to the board and write down your measurement and uncertainty. Once

everyone has reported a measurement, the Lab Instructor will talk about the results, so please wait before you

continue with this sheet.

1. Width of your keyboard = ±

Reporting a measurement

A properly formatted measurement should have the following attributes:

• The correct number of significant figures for the uncertainty: One significant figure unless the leading digit

is a 1 or 2, in which case you should use two significant figures.

• The correct number of significant figures for the measurement: At the same order of magnitude (the same

decimal position) as the uncertainty. (Numbers with the same order of magnitude, when written in scientific notation, have the same factor of ten. 10 and 20 are the same order of magnitude, 100 is an order of

magnitude higher than 10).

• All measurements must have units attached to them.

Update your entry on the board as well as here on the worksheet with all the proper attributes of a measurement.

If your response in question 1 follows all of the rules above, your answer for this question will be the same.

2. Width of your keyboard = ±

From now on, all measurements that you report in physics labs will be required to have these attributes, or

an explanation for why it does not. This is part of the scientific language and is common across many disciplines.

Familiarity with this format is a useful skill wherever you need to make and report accurate measurements.

Types of errors

• Systematic errors are due to identifiable causes and can, in principle, be eliminated. They are due to biases in the equipment or experimental procedure that cause measurements to be consistently higher (or

consistently lower) than the true value.

• Random errors are the positive and negative fluctuations that cause about half the measurements to be too

high and half to be too low. Sources of random errors cannot always be identified, and are usually inherent

in the limitations of the equipment.

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• Mistakes are human errors. A person may record a wrong value, misread a scale, forget a digit when

recording a measurement, or make another of the many possible mistakes. These mistakes should stick out

like a sore thumb if you make multiple measurements or if one person checks the work of another. Mistakes

should not be included in the analysis of data.

Propagation of uncertainties

Using your ruler, measure the long side of your keyboard. Because your ruler is not long enough to measure the

length in one measurement, you will need to take multiple measurements, each with their own uncertainties.

3. Length of keyboard = ( ± ) + ( ± ) + . . .

Adding the measurements is trivial, but what about adding the uncertainties? In the case of addition and

subtraction, the equation for combining uncertainties is

δx = δx1 + δx2 + δx3 , (1)

where δx is the total uncertainty in your length calculation and δx1, δx2, and δx3 are the uncertainties of your

individual measurements.

This is called error propagation. Notice that the error in your final measurement is much larger than the error

of any of your individual measurements. Report your completed measurement below and on the board.

4. Length of keyboard = ±

5. If you could change anything besides the length of the keyboard, what might you be able to change to

reduce the error in this measurement?

6. Are there any additional sources of error that you did not take into account when reporting this measurement?

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Figure 2: Navigate to jupyterhub.wpi.edu and sign in with your WPI email address and password, choose

an instance to spawn (either is fine) and create a new Python 3 file as shown here

Python

Python is a high level programming language that is regularly used by the scientific community for general and

research computation. We will be introducing it and using it in a very limited way in this lab course although we

encourage you to explore it more on your own. We will primarily make use of the Python to calculate statistical

values and propagate uncertainties. We will be using an interface for Python called Jupyter and WPI has set up

a JupyterHub server that you can access from any browser (Figure 2). The files you create on the server will stay

for at least the term, probably much longer.

Jupyter uses a cell based system and evaluated variables carry over to the next cell. There are a few different

types of cells, Figure 4 shows 2 kinds, the code cell, which we will be using most of the time, and the markdown

cell, which you can use to add nicely formatted notes to you file.

Statistical uncertainty

Instead of taking a single measurement and estimating the uncertainty, it is often desirable and easier to take

multiple measurements and calculate the mean value (often denoted by x¯) and the uncertainty (often denoted by

δx. The formulas for calculating these two values are

x¯ =

1

N

X

N

i=1

xi (2)

and

δx =

vuut

1

N − 1

X

N

i=1

(xi − x¯)

2

(3)

where N is equal to the number of measurements that you have taken and xi are the individual measurements. 1

While the formula to calculate the mean is fairly simple, calculating the standard deviation becomes more

complicated as you add more measurements. This is a perfect opportunity to use a computational tool in our

case Python. Using the measurements that your peers recorded on the board for the length and width of the

keyboards, calculate the standard deviation of the length and widths of the keyboards using Python. Don’t

forget to use the proper significant figures and units.

7. Length = ± , Width = ±

1You may be wondering why the formula for standard deviation contains 1

N−1

instead of 1

N

. This is more properly called the sample

standard deviation, and this correction, called Bessel’s correction, fixes a systematic error that occurs when taking a finite number of

measurements. If you try to put 1

N

in the above formula, you will consistently underestimate the true standard deviation.

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Figure 3: Above is the code that you could use use to propagate uncertainty for values that are added or subtracted. Always remember to comment every line of your code in this class.

Figure 4: Above is the code that you could use use to calculate the mean and standard deviation. Always remember to comment every line of your code in this class.

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Now is also a good time to do a sanity check on your calculations.

8. Is your mean value on the same order of magnitude as the individual measurements reported by the

class?

9. How does the standard deviation compare to the estimated uncertainties?

Propagation of uncertainties

Using the calculated measurements you just found, calculate the area of the keyboard. Again, the initial calculation is trivial, but what about the uncertainties? This time we will use a slightly different method of error

propagation because we are multiplying them instead of adding them. This method is valid for both multiplication and division of measurements with uncertainties. The formula is

δA

|A|

=

δx

|x|

+

δy

|y|

, (4)

where A is the area, x is the length, y is the width, δx and δy are the uncertainties associated with these

measurements, and δA is the propagated uncertainty of the area.

Using the equation above, report the area of the keyboards in the room with the propagated uncertainty and

correct units. Also paste your code that you used to calculate these quantities including comments about what

each line is doing.

10. Area of the keyboard = ±

11. Python code with comments used to calculate the above values.

Note that this measurement is not of your individual keyboard, but represents the average of all the keyboards

in the room. This may seem like a minor distinction, and in this case it probably is; however, if we wanted to

know how tall Olin Hall is, we wouldn’t measure all the buildings on campus and then take the average—we

would make multiple measurements of Olin Hall alone.

Accuracy versus precision

Figure 5 is a cartoon illustrating the difference between accuracy and precision. There are five data points shown

for each scenario, with the “actual” point represented by the bullseye. While an accurate and precise measurement is the best, an accurate measurement is most important. You can make an accurate measurement with most

measurement devices by minimizing systematic errors. Precision is almost completely dependent on the type of

instrument and the level of precision it is capable of.

12. The calibration of a measurement device would help ensure what about our measurements?

13. What is one way that we could make more precise length measurements if we were to repeat today’s

measurements?

You should consider saving this file as a reference sheet; many of the topics presented here will be useful

throughout your physics coursework and your scientific career. For further information on uncertainties and

error propagation, a good reference is John R. Taylor’s An Introduction to Error Analysis, which is available in the

physics library.

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Figure 5: A cartoon explaining the difference between accuracy and precision of collections of measurements.

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