Scientific Measurement
Introduction
One of the most important elements of a physics lab is to make accurate, precise, and
independent measurements. Equally important is understanding how accurate, precise, and
independent your measurements actually are! In this lab, you will learn how to use some
common equipment found in physics labs, and will also gain a greater understanding of how to
make measurements using this equipment and how to characterize the accuracy and precision of
these measurements. You will also learn why independent measurements are important when
characterizing these values for a data set.
Another very important skill is to understand the uncertainty or error on a set of data. To have
“error” doesn’t mean you have necessarily done anything wrong — although you do want to
minimize it to the extent possible. However, it’s very important to quantify and state the
estimated errors on an empirically-derived value. If you have a number (generally > 5) of
independent measurements, you can actually use your data to estimate the error! This is a very
powerful technique, and involves the calculation of the standard deviation of the mean, which
you will learn about in this lab as well.
Objectives
After completing this lab, you should:
● Understand how to make measurements using common equipment;
● Understand the meaning and importance of the estimated digit in a measurement;
● Know how to determine the correct number of significant figures to use for analog
measurements;
● Understand the importance of independent measurements;
● Know how to calculate the mean and standard deviation for a set of measurements;
● Know how to calculate the standard deviation of the mean for a set of measurements;
● Understand the usefulness of the standard deviation of the mean in error estimation;
● Be able to state a measurement, with associated error range, using proper formatting;
● Know how to properly set up and format a scientific data table.
Accuracy vs. Precision
Perhaps the best way to illustrate these concepts is with a graphic:
In physics, high accuracy and high precision is ideal, but often difficult to achieve in a
college-level lab. Low accuracy and high precision often results when you are making the best
measurements possible with the equipment but have a systematic error (such as friction, more on
that later) or an error in a calculation. High accuracy and low precision may be acceptable, as
long as you can make many measurements and take an average value. We strive to avoid low
accuracy and low precision — but more than this, we strive to understand our accuracy and
precision!
Equipment
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Meter stick
Tape measure
Vernier calipers
Metal cylinder & plastic “wire”
Digital micrometer
Digital scales (2)
Balance scales (2)
Photogate & timer
Experiments
This lab is divided up into four separate experiments. Be sure to answer all the questions and
present your data as requested. Answers should be provided in a legible, distinct color text.
All data should be entered into Google Sheets, and calculations should be done in this
spreadsheet when possible (feel free to check them with a calculator if desired). Be sure to ask
for help if you are unfamiliar with Google Sheets. Do not “divide and conquer” the components
of the lab—all lab partners must participate as a group for each experiment in order.
Experiment 1, “Large” Length Measurement
In this experiment, you will learn how to make and estimate the error for measurements using a
meter stick and a tape measure. When making measurements with an analog device (such as a
ruler), you must always provide an estimated digit. For example, if the ruler is marked in
centimeters (cm), you estimate where the object falls between the markings—in this case,
measuring to a millimeter (mm).
1. Examine one of the lab room’s meter sticks. What is the smallest marked unit of
measurement on the meter stick? (Be sure to use the side with centimeters!)
1 cm
2. Considering your answer to question 1, what should be the unit of your estimated digit
when you make measurements with this meter stick? Give decimal place (e.g. “tenth” or
“hundredth”) as well as unit (e.g., “cm” or “mm).
Hundredth mm
Now, consider using the meter stick to measure the length of a book. (Any book is fine!) Before
making measurements, think about the best way to make measurements which will have high
accuracy.
3. For the best accuracy, should you start at the end of the meter stick? What about starting
at the one-cm mark? Argue why one of these will likely be more accurate, and be sure to
look carefully at your meter stick when considering your answer.
At the end of the stick
All measurements should be independent as well. Before making measurements, think about the
best ways to make them independently—that is, each measurement is completely separate from a
previous one.
4. How are you going to ensure that each measurement is independent? Give an example of
two measurements that are not independent, and what can be done to make them
independent: Measure twice from the same spot on the book. Measure from different spots on the book.
⇒ Create a Google Sheets spreadsheet, and share it with all lab members as well as your
instructor. Add a link to the URL of the spreadsheet here: ______________________________
https://docs.google.com/spreadsheets/d/1s4mLjFauPt3TKbPUEY
yleGyVIt9CEu4GlLnnkSKiCrM/edit?usp=sharing
⇒ Each lab member should now make three independent measurements of the book’s length,
and enter the data in a single column in the spreadsheet (there should be 12 measurements total).
Using these data, make the following calculations and answer the associated questions:
5. What is the average, or mean, value of the 12 measurements? In the spreadsheet, use the
“=AVERAGE(##)” function to calculate this, where “##” is the range of data cells, e.g.
A1:A12. Be sure to provide units as well as a value.
28.125 cm
The standard deviation for a set of measurements tells you how spread out the measurements
are. Here is a good resource for understanding standard deviation, as well as the equation that is
used to calculate it. Fortunately, you don’t have to do this by hand, the spreadsheet will do the
calculation for you!
6. What is the standard deviation of the measurements? Again, do this in the spreadsheet:
put “=STDEV(##)”. Be sure to provide correct units for your value!
0.1138180366 cm
7. Do you think that the standard deviation would be higher or lower for measurements that
are not independent?
It would be lower because there is less spread
Standard deviation is useful in characterizing the overall spread of data, but if you want to
estimate the uncertainty on your mean value (which is almost always what you will want to do!)
you need to calculate the standard deviation of the mean. The standard deviation of the mean is
simply the standard deviation divided by the square root of the number of measurements you
have made (σm = σ / √N). Think about this: the more measurements you make, the better you
understand the mean value, even if the spread in measurements is relatively large. The more
measurements the better!
8. What is the standard deviation of the mean (σm) of the measurements? Again, you can do
this in the spreadsheet: put “=STDEV(##)/SQRT(12)”.
0.03 cm
Since the standard deviation of the mean is a good way to estimate the precision of your
measurements, it will tell you how many significant figures you should keep on the mean value.
A good rule of thumb is to keep one digit in σm, and round your mean value to have the same
precision. Then, you can state your result as “µ ± σ𝑚”. For example, let’s say your original
mean value for the length of the book was µ = 0.25458934 m, and σm = 0.000243543 m. You
would state your result as l = 0.2546 ± 0.0002 m.
9. What is the appropriately-formatted mean length of the book, with error range, using the
format µ ± σ𝑚?
28.12 cm + 0.03 cm
10. If the measurements were made in such a way that they were not independent, would the
error range be estimated as too large or too small? State why this might give a researcher
a false sense of confidence about the precision of their results.
If they are not independent the error range would be too small. This could give a higher confidence about the
precision
⇒ Now, use the tape measure to make measurements of book’s length. Follow the same
procedure as for the meter stick.
⇒ Finally, repeat the experiment with both the meter stick and the tape measure for the length of
a lab bench. Include your final values in the table below, using the format µ ± σ𝑚.
Double-check with your instructor that this table is properly formatted before continuing:
Measuring Device
Meter Stick
Length of Book (m)
0.2812 m
Length of Lab Bench (m)
0.765 m
Tape Measure
11. How do the values you found for σm compare with the magnitude of the estimated digit
for the meter stick (0.0001 m)? What about the tape measure?
12. Was there any difference in σm between the measurement of the book vs. the lab bench?
Which measuring device gave a more precise measurement of the book? Which gave a
more precise measurement of the lab bench?
Experiment 2, “Small” Length Measurement
There are some situations where measuring length with a meter stick is impractical (or even
impossible), such as measuring the diameter of a wire, or length of a small object. Here, you will
learn to use Vernier calipers and a digital micrometer.
⇒ First, record the serial numbers of the Vernier calipers and digital micrometer:
Vernier calipers: _______________
00868
Digital micrometer: _______________
000886
This should become a habit for all labeled equipment (in case you need to check it out again or
track down any issues with your data after you turn it in). Always be sure to include the serial
numbers in your equipment list.
Both the Vernier calipers and digital micrometer report a digital value for length. Therefore,
there is no estimated digit. The best way to estimate their precision is to make multiple
measurements. For each of these devices, be sure to tighten the jaws but don’t clamp down too
hard on your sample! Also, be sure to zero out the devices between measurements. Ask for help
if you are uncertain how to do these things.
Start by measuring the diameter of the wire. Using the Vernier calipers, each lab member should
make three independent measurements, so you should again have a total of 12 measurements.
Making a new column in the spreadsheet, repeat the measurements using the micrometer. Using
the same procedure as for Experiment 1, determine the mean for each set of 12 measurements, as
well as σm.
Next, repeat the same procedure to measure the diameter of the small cylinder.
⇒ Create a table of results below, using the same format as the table for Experiment 1:
⇒ Using the results from this experiment, answer the following questions:
1. For the wire, how does σm compare with the magnitude of the final reported digit for the
Vernier caliper? What about for the digital micrometer?
Vernier - 0.02
Micrometer - 0.002
2. Answer question (1) for the small cylinder.
Vernier - 0.03
Micrometer - 0.0008
3. For what kinds of measurements does it make the most sense to use the Vernier caliper,
and why?
For larger objects
4. When does it make the most sense to use the digital micrometer, and why?
For smaller objects and more accurate measurements
Experiment 3, Mass Measurement
Determining the mass of objects will be a frequent measurement in physics labs. Achieving an
accurate and precise mass measurement, as well as understanding the associated uncertainty, will
be critical for getting good results. Often, the weights in the lab room have masses stamped on
their sides. One thing we will check is whether these values are reasonable given the precision
of our scales. The second thing we will check is the readings from several different scales, to try
to get a measure of accuracy.
To do this, you will be measuring the mass of one of the “100-g” classroom weights using digital
scales as well as balance scales.
⇒ Look at one of the balance scales to answer the following questions.
1. What is the smallest marked unit on the balance scales?
Grams
2. To what precision must you report a measurement using one of these balance scales?
Milligrams
Each lab partner should weigh the same mass eight times: twice on two different balance scales
(be sure to zero out the balance between each measurement) and twice on two different digital
scales (one of these scales is located in each lab room). So, for each of the four scales, you
should have eight measurements of the mass (two from each lab partner).
⇒ Make a neat data table for these measurements. Keep in mind that since the units (grams) are
indicated in the column headers, you don’t need to put units in the cells, just values:
m1A (g)
m2A (g)
m1D (g)
m2D (g)
100.15
100.14
100.16
100.1
100.13
100.09
100
100
100
100
100
100
100
100
100
100
100.14
100.09
100.12
100.1
100.13
100.08
100
100
100.13
100.1
100
100
100.13
100.09
100
100
m1A: the mass in grams as reported by the first balance scale;
m2A: the mass in grams as reported by the second balance scale;
m1D: the mass in grams as reported by the first digital scale;
m2D: the mass in grams as reported by the second digital scale.
⇒ Answer the following questions:
3. What is directly below the table, and what is the purpose of this information?
Explains what the labels mean
4. What is the difference between the values in the columns vs. what is in the rows, and why
is the table organized this way (rather than the other way around)?
Columns is different scales while rows is different attempts
⇒ For each of the four measuring devices, calculate the mean mass and σm of the mass and put
these derived values in a properly-formatted table:
⇒ Answer the following questions:
5. How many σm is the mean away from the stamped value on the mass for each of the
scales? To find this value, subtract the mean value from the stamped value and divide by
the calculated σm. You should have four different numbers; be sure to label them
appropriately:
M1A - 27
M2A - 15.45
M1D - 0
M2D - 1
6. Is the difference between the stamped value and the mean value significant (i.e., is it
greater than 3σm)? Considering these results, when should you trust the value that is
stamped on the mass, and when is it best to measure yourself?
It is significant for the two manual scales and not significant for the digital scales
7. Consider the values you found for σm from the balance scales. What is a reasonable
estimate of the precision of a balance scale?
.01 Grams
8. Consider the values you found for the mean mass and σm from the digital scales. How
many σm apart were the mean values for the two different scales? What does this tell you
about the accuracy of the digital scales?
Digital Scales are more accurate because there isn't room for human error
Experiment 4, Time Measurement
As you might expect, time measurements are also very common in Physics 1A labs—along with
length and mass, these comprise the trio of fundamental physical quantities. We will make most
time measurements using a photogate detector, which is a device that emits a beam of infrared
light from one side and detects it on the other. When the beam is broken, the photogate timer is
either started or stopped, depending on the mode of operation.
First of all, please be gentle with your photogate detector. There are a lot of screws for loosening
and tightening various parts, and they’re mostly made of plastic. Never tighten anything past
“finger tight.”
Before starting your experiment, be sure to plug in the photogate. Set MEMORY to OFF. Note
that the value it displays is in seconds. Play around a bit by moving your hand or an object
between the detector to learn about its behavior, and to be able to answer the following
questions.
1. When the photogate is set to GATE and right toggle is at “1 mS”, what causes the timer
to start running?
When the laser is crossed
2. What causes the timer to stop running?
The timer stops running when the laser stops being crossed
3. What is the maximum length of time it can record?
19.999
4. What is the precision of the measurement (instrumentally reported, not experimentally
determined)?
milliseconds
5. Now, set the right toggle to “0.1 mS.” Does this change affect what starts and stops the
timer?
No
6. Now, what is the maximum length of time it can record?
1.999
7. What happens when it reaches the maximum time?
It resets 1.000
8. What is the precision of this measurement?
milliseconds
9. Next, set the photogate to PULSE mode and the right toggle back to “1 mS.” What starts
the timer going in PULSE mode?
The timer starts when something crosses the laser
10. What stops the timer?
The timer stops when the laser is crossed twice