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Scientific Measurement & Data
Understanding measurements through making measurements
Author: Sean S. Lindsay
Version 1. 1 created August 2018
Version 3. 1 by S. Lindsay in September 2019
Learning Goals
In this lab, students will engage with taking direct and indirect measurements, examining
collected data to understand accuracy and precision, and performing remote distance
calculations similar to stellar parallax distance determinations. In this lab, students will
- Perform simple scientific measurements
- Carefully record data
- Examine the distribution of data points over several different variables
- Explore the concepts of accuracy, precision, and uncertainty
- Perform remote distance calculations
- Play a game
Materials
- Game board, bean bags, direct distance measurement tools, and devices to measure
angles, including a protractor.
- Prepared Google response form:
1. Background
1.1 Introduction to Scientific Measurement
How do we go about determining things like the number of galaxies in the universe, how
much “stuff” the universe contains, the value of the gravitational constant of the universe,
and the distance to the stars or how bright they are? Where do we get the evidence to
evaluate some of our biggest astronomical questions, such as how did the universe, the
solar system, and Earth get their start? To answer all of these questions, scientist must
collect observations and empirical evidence to test the predictions of their hypotheses and
theories against. The collection of these observation and evidence almost certainly involves
measuring some quantity of something. Whether that be the amount of light being received
by a distant supernova, the total amount of mass (both luminous and dark) in a galaxy
cluster, or the distance to the stars of the Milky Way, observations must be made, and data
must be collected. How well we can measure the related quantities is intimately tied to how
well we can evaluate our deepest scientific understandings of the universe.
Answering some of the above questions is a daunting task, and often times we only have a
ballpark idea of what the answer might be. In more intriguing cases, we find ourselves in
completely new scientific territory where we have little theory to rely upon. Here, science
must tackle challenging questions in face of the unknown. There may be no known methods
from which we can start trying to solve the problem. We are left to rely on our own creativity
and ingenuity to develop new problem-solving techniques. If we are clever, we often can
devise multiple ways to address a question. Some of the methods will provide more reliable
answers than others. With hypothesis, theory, and ingenuity as our resources, we then have
to determine, using the principles of science, which methods are best. This frequently comes
down to how those methods make their measurements, what they are measuring, and which
one gives the explanations we have the most confidence in. This means we need to
understand the empirical evidence, frequently just called data in science, itself. How reliably
does it give an answer close to the correct one, and how much does the data vary from
measurement to measurement?
In this lab, we will explore the some of the techniques scientists and astronomers use to
begin addressing how to approach scientific problems and gather the required evidence. We
will explore the difficulty in making measurements, the uncertainty and errors associated
with measurement, and how we can compare measurements of the same thing made with
different tools or techniques.
1.2 Scientific Measurement
Measurement often is thought of as a simple task. For most people, daily life measurements
are simply measuring how long something is with a measuring stick, how much volume you
need with a measuring cup, the mass of something with a scale or a mass balance, or how
long something takes with a stopwatch. If you are off by a few centimeters, milliliters, grams,
or seconds, then so be it, there was no real harm done. In science, however, we strive to
arrive at the real value as close as possible, or rather, to have measurements with high
levels of accuracy. To have an accurate measurement, or set of measurements, is to be
close to the real value with the measurement, or the average of the set of measurements.
To accomplish accurate measurements, much more care and sophisticated instruments are
needed than you experience in everyday life.
In science, we also never rely upon a single measurement. Science demands that we make
measurements repeatedly to check and re-check our evidence. In these repeated trials,
scientist aim for measurement methods that consistently give values that are nearly the
same, or rather, have high levels of precision. How close they are together, or how precise
your measurements are, determines how certain we can be that our evidence provides a
good test for the predictions of our hypotheses and theories. In other words, the precision of
our measurements quantifies the uncertainty in our measurements, or rather, what is the
probability that our measurement is accurate within a certain range of values. A graphical
representation of these concepts is provided in Figure 1.
Figure 1. A graphical representation of
accuracy and precision. For repeated
measurements, the accuracy is how
far the average of the measurements
is from a reference value, represented
by the vertical line through the
measurement distribution. The
precision is the spread of all
measurements. A quantification of the
width of the distribution of
measurements would be the
uncertainty.
1.2.3 Astronomy Indirect Measurement Example 2: Distance from stellar parallax
In the other example, astronomers use the technique of stellar parallax to measure the
distances to stars. Stellar parallax is a method to determine distance via triangulation.
Triangulation is a method to determine the distances between two points through drawing
series of (right) triangles, where the distance of one side of the triangle can be directly
measured, and at least one of the two non-right angles can be directly measured. With a
known distance for one of the sides of the triangle, called the baseline distance, and a
measured angle of a right triangle, it is possible to calculate the distance of another side of
the triangle, which in this case, is the distance of a far-off object.
Astronomers use a similar technique, but the angle measured is determined by the
phenomenon known as parallax, which is the angular displacement an object appears to
have against a fixed background when viewed from two different positions. Think about
holding your thumb out at arm’s length and looking at it by blinking between using only our
right and left eye. You thumb appears to shift with respect to the background. When applied
to the stars, this apparent angular shift is referred to as
stellar parallax. The general method
of stellar parallax is shown in Fig. 2. With parallax, the baseline is the distance between the
two viewing vantage points. Since, the stars are so very far away, astronomers need to use
the largest baseline possible in order to actually measure an angle. For this, astronomers
use the diameter of Earth’s orbit as the baseline distance because it is the largest baseline,
a full 2 AU, accessible to humans! The angular displacement is the apparent angular shift a
star appears to have when viewed from opposite sides of Earth’s orbit. The displacement
angle is measured directly, and used to indirectly determine the distance. Half of that
measured displacement is called the parallax angle, p, which often is just referred as the
stellar parallax [See Fig. 2 ]. It can be translated into distance using a relatively simple
equation:
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 [𝑝𝑐] =
7
8 [9:;<=;]
Since the stars are very far way, the stellar parallax angle is very small and hard to measure.
This makes it prone to high levels of uncertainty, which propagate to a large uncertainty in
distance. In astronomy, measuring distances is extraordinarily difficult!
Figure 2. The method of stellar parallax to determine the distances to nearby stars.
1.3 The Difficulty of a Scientific Measurement – A Direct Measurement Example
Scientific measurement is a deceptively difficult task with a large amount of nuance.
Consider the simple task of measuring exactly how long a room is. For simplicity, let us
imagine that the room’s floor print is a square, so that the length and width are equal. How
would you go about measuring the length of the room? Likely, you would get a measuring
tape, a meter/yard stick, or some other standard of length measurement and just see how
long the room is. Since we are working in the world of science, let us imagine that you chose
to measure the length in meters. Unfortunately, you only have a meter stick in your room,
but nevertheless, it seems like a good one with centimeters (cm) and millimeters (mm)
clearly marked. So, you set to the task with confidence that you can determine the length of
the room. You measure 5 full meters and have a bit leftover where the meter stick is too
long to measure the remaining distance to the wall. Being clever, you mark the 5-meter
mark on the floor and then measure from the wall out to that point to and find that length to
be 35.6 centimeters. Proudly, you declare the length of the room to be 5.356 meters (m) in
length!
How certain are you that 5.356 m is the true length of the room? You recall that the
uncertainty for any measurement is half the smallest division, so +/- 0.5 mm, or 0.0005 m.
That seems wildly accurate and precise for a single measurement, and you suspect that your
own measurement method prevented such high precision. Thinking on your method of
measurement, first you begin to wonder how good of a job you did when you moved the
meter stick from one position to the next. Did you truly put it exactly where it left off? Did
that amount vary each time you did it? You certainly weren’t perfect, so how much were you
off by? A few millimeters? Over the multiple measurements that could easily add up to being
off by a centimeter or more. You also are unsure if it was always extra length added, or did
you underestimate sometimes? Worried, you repeat the task again and this time you get 5
full meters, but the extra measurement this time comes out to be 34.3 cm given a total
length of 5.343 m. A different value by 5.356 m – 5.343 m = 0.013 m! Clearly, you can’t be
certain about the length of the room from these measurements alone. A fundamental rule of
measurement dawns on you, … All measurement has some level of uncertainty in it. A
scientist’s goal is to reduce the uncertainty as much as possible. Entire theories may
depend on it!
A few days later, you are still curious about how long your room really is. You ask a friend to
come over and measure the room. You have shared your concern with not being able to
perfectly move the meter stick and how you measured that last remaining bit and aren’t
sure if that was the best method. Your friend decides that the best way to avoid this is to lay
out a string from one end of the room. She then measures the length of the string, and she
is very careful to mark each meter with marker before moving the meter stick. Using this
method your friend determines the room to be 5.352 m long. You double check and
measure the length of the string as well and get 5.348 meters long. This time only a
difference of 5.352 m – 5.348 m = 0.004 m, or 4 mm!
You are beginning to feel more confident in the length of the room, but your friend has a few
concerns. She noticed that there is a bit of extra material at the start and end of the meter
stick, so the meter stick is likely slightly longer than a full meter. Being an acute observer,
she noticed that the meter stick is a bit worn on the ends, and that she measured along the
2. Collecting the Data
I hope you have gained an appreciation for how difficult and important measurement is to
science. It requires incredible care and often relies on clever indirect measurement
techniques. Today in lab, you will be engaging with some of those concepts, but in a very
simplified, and hopefully fun, way that demonstrates core concepts of measurement
methodology, accuracy, precision, and the how uncertainty is quantified. Measurements will
be made, and the data for the whole class will be collected and examined to reinforce the
concepts learned in this lab.
For this lab, you will work in student pairs.
Lab Activity 1 – Playing the Game and Taking Measurements
For our exploration this labs core concepts, you will collect x- and y-position data of bean
bags you will toss onto a board. You will also measure each bag’s angular location on the
board with respect to a reference line. To increase interest, and hopefully have fun while
doing so, your tossed bean bag locations on the board will be made into a game. The goal of
the game is to get the highest score possible while also carefully measuring the (x, y)-
positions. The general layout of the board and the rules of the game are shown in Figure 3.
Figure 3. The rules of the game. Two competitions will be taking place while you play. You will
directly compete against your lab partner, and as a team, you and your lab partner will be
competing with all other lab pairs. The goal is to get the highest total score. While one person
throws the bean bags, their lab partner will record the x- and y-positions. The right-hand side of
the board is positive x and radius. The top of the board is positive y.
The Rules of the Game
Each lab partnership will have a turn at the game. You may want to take a couple practice
tosses before starting. One person of the pair will throw their five bean bags while the other
makes and records the following measurements: x-position, y-position, and of course, the
score of each toss. The partners will then switch locations and repeat the process. At the
end of each lab partnerships turn at the game, the pair should have 10 tosses with all the
measurements recorded.
There are two competitions going on during this game. One is between partners. Highest
score wins. The second is your combined team score versus all the other teams. Again,
highest score wins. A bullseye is worth 100 points, and then from inner-most to outer-most
rings, the scores for a bag in that ring are: 50, 30, and 15 points. Bags on the board, but not
in a ring, are worth 5 points. Bags off the board are worth 0 points, and you have the
additional challenge of still trying to get accurate measurements for the (x, y) position.
In cases when a bag lands on a boundary, the score is determined by the ring that
containing the majority of the bag. This is roughly equivalent to the what ring the center of
the bag is in.
Measurement Data Collection
After all the matches have been played, and every team has a score, you will need to enter
your data into our online system using the provided Google Form – “Measurement Lab Data
Entry Form.” Every team will have to enter their measurements into the form 10 times: one
for each toss. In addition to the (x, y)-position, you will also have to calculate the radial
distance away from the center. The radius, r , is calculated using the equation
B
B
X- and y-values range from - 5 to 5 using the grid on the board. Do your best to estimate
fractional distances. The right-hand side of the board will be assigned positive radius values,
and the left-hand side will be assigned negative radius values. An example of the
measurements to be taken for each bag is shown in Figure 4.
Figure 4. An example of the
data to be recorded in the
Google Form. (x,y)-positions
are determined using the
grid on the board. Do your
best to estimate fractional
distances. The angle is
measured using the provided
protractors. The radius is
calculated using Equation 3.
The score is determined by
the bags location on the
board. Positive r values are
the right-hand side of the
board.
Lab Activity 2 – Examining the Distributions of Measurements
With all the data collected, it is time to calculate the radial distance away from the center of the
board. Use Equation 3 (𝑟 = @𝑥
B
B
) to determine r. Once you have calculated all the radial
distances, access the Google Form “Measurement Lab Data Entry Form” and enter your
measurements. The form will prompt you to input the x-position, the y-position, the radial
distance, and the score earned. Once finished with an entry, choose the prompt to do
another entry. You will do this repeatedly until all the data on all 10 tosses is entered.
Once all the data is entered, your instructor will graphically display the results on the
projector. The distributions for the x, y, angle, radius, and score will be displayed as
histograms, which shows how frequently each measurement was made within a certain
range. Using this graphical information, you will need to answer questions relating to what
the distributions look like, the accuracy, and the precision of each distribution.
You will also calculate the averages of your bag trials and compare that to the overall class
distribution.
Measurement & Data Lab Student Worksheet
Name:
Lab Instructor:
Lab Meeting Time:
Measurement Data Records
Input your measured data in the table below. Put the name of each player under Player 1
and Player 2. Below the all the data entry, calculate your average x, y, radius, and score for
all ten of your team’s measurements. If you need to save calculating the averages for the
take-home portion of this lab, then that is okay. You can skip them for now if you do not
have time.
Measurement Data Table / Game Scorecard
Toss # x y Angle Radius Score
Player 1:
Total Score
Player 2:
Total Score
Player 1 + Player 2 Scores = Team Score
Team
Averages x average y average
Average
angle
Average
radius
Average
score
- Describe at least 2 ways you would improve the accuracy and precision of the
distance and height if you had more time than this lab allowed. Assume you really
want to know the distance and height.
Questions Relating to Distributions of the Measurements
- Using the provided axes below, sketch the overall distributions of all the
measurements from today’s lab. Your instructor should have these distributions
displayed on the projector.
- Do all the distributions have roughly the same shape? If not, which one(s) is(are) the
different, and how are they different?
- The radial distance distribution combines information from the x-positions and the y-
positions. Therefore, characteristics of the x-position distribution and y-position
distribution should be present in the radial distance distribution.
a. Describe the shape of the radial distribution with respect to the shape of the
x- and y-position distributions.
b. Are there any systematic offsets present in the radial distances? If so, provide
your hypothesis on what caused the offset.
c. Describe how you could use bean bags and the target board to test your
hypothesis. What experiment would you run to either confirm there is no
systematic offset, or to confirm your explanation of what caused the offset.
- Any measurement has some inherent error in it. You certainly had some when
measuring the x- and y-positions of the bean bags. List and describe at least TWO
sources of error you encountered when measuring a bag’s x- and y-positions. That is,
what things that you can think of made it difficult to measure the bags exact (x, y)-
location?
Lab Follow-up Questions
[Complete the following questions at home if you do not have time to during class.
- In your own words, write definitions for accuracy , precision , and uncertainty.
- Use the triangulation method to determine
the distance d using the image to the right.
The baseline distance is 5 meters, and the
measured angle is 70°.
- Describe the difference between a direct and an indirect measurement.
