Chapter 3
Oops, I Did It Again: Errors in Measurement
Overview
Understanding the precision and accuracy of
measurements is a crucial skill that scientists
must develop to do their work. All measurements
involve some degree of error. Knowing how to
assess and work with error is an essential part
of making sense of scientific observations.
What is accuracy? What is precision? How do
we make our measurements more accurate or
precise? Why would two measurements of the
same object yield different values? How can
error in measurement be reduced? Students
investigate these questions through a series of
measurement tasks involving heights, mass,
and different scales of measurement.
Background
Objectives
• To develop skills in measuring.
• To become familiar with properties of materials that can be
measured.
• To develop an understanding
of accuracy, precision, and
measurement error.
• To develop an understanding
of how measurement error
changes with the instrument
used to make measurements.
Process Skills
• Observing
• Measuring
• Communicating
Activity Duration
60 minutes
If you ask 30 people to measure the width
of a room, it is likely that there will be many
different answers. Variation in measurement is always present and is a part of doing
science. Why does this variation in measurement occur? How much measurement
variation is tolerable? These questions are considered as students make measurements and compare their measurements with those of other students.
It is important for students to recognize how errors in measurement occur and
to be able to describe how to minimize both random and systematic errors. Ultimately there will always be some error or level of uncertainty. Students must also
learn that this is a fundamental part of science. Nothing is known with quantitative
exactness. A level of uncertainty in measurement is usually referred to as the “error.” A sophisticated understanding of the occurrence of random errors develops
after students have many opportunities to make repeated measurements and see the
patterns of data that emerge. Encouraging students to check their own measurements, as well as to compare them with those made by others, is a fundamental part
of this developmental process.
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Ch a p t e r 3
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Materials
Each group will need:
• 1 thermometer (inexpensive alcohol thermometers work best)
• Metric measuring tape
• Scale for measuring mass
Class materials needed:
• Shoe box
• Egg (hard-boiled)
• Metric ruler
• 2 meter sticks
• Centimeter ruler
• Yarn
• Meter tape
Engage
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Pass out one thermometer to every two students. For this activity, very inexpensive
thermometers work best. Ask the students to record the temperature of the room.
Then record the temperatures from the different groups on the board in a table.
Look across the data and ask the students to explain why there is variation in the
temperature measurements. Make note of the different reasons that they provide.
Possible explanations include readings to the nearest half-degree versus the nearest
whole degree, thermometers that have various calibrations, inaccurate thermometers, variations in temperature throughout the room, thermometers being under
a vent or in the sunlight, and readings of thermometers from different angles.
Explore how the different thermometers give different readings by laying
the thermometers side by side on a table and having the students view them as
a group. Ask, Why do different thermometers show different temperatures?
Explain that in this lesson they will investigate the different sources of errors in measurement and the different types of errors that occur. In the case
of the thermometers, the differences are due in part to their manufacturing,
which results in variable readings. However, other human sources of error are
inherent in the act of measurement.
Explore
Divide the class into groups of two students each. Explain that the focus is to
make accurate measurements for the following: (a) a student’s height in meters;
(b) a shoe box length measured in centimeters; (c) a shoe box length measured in
millimeters; (d) a shoe box length measured in meters; and (e) the mass of an egg
in grams. Each measurement should be recorded on the Student Data Sheet.
NATIONAL SCIENCE TEACHERS ASSOCIATION
IN TR O D U C Measurement
TIO N TO S C A LE
A. How Tall Am I? Select a student volunteer who is willing to have his or
her height measured. The first measurement task is to measure the student’s
height in centimeters. Provide Group 1 with a meter stick, Group 2 with meter
tape, Group 3 with a ruler marked in inches and centimeters, Group 4 with
yarn and a meter stick taped down to a table, and Group 5 with a small centimeter ruler (approximately 25 cm). Remind them to convert units to centimeters if they use a different unit for their initial measurement.
After the students make measurements, have the groups report their measurements to the class. Record the different measurements on the board on
a chart (see Figure 1). Note the variation in students’ measurements. Discuss
possible sources of error. Students should note that some of the differences in
the answers are the result of different measurement tools. Ask, Which tool is the
most accurate? Why? Discuss the ways that students made the measurements and
how they could be more consistent in their measurement strategy.
Extend the discussion by taking the mean of the different values and asking the students if the mean is more likely to be accurate than an individual
measurement.
B, C, and D. Shoe Box Measurements. Each group should measure the same
shoe box length (long side) in centimeters, millimeters, and meters, using either
a ruler or metric tape. Record the measurements on the Student Data Sheet.
Ask, Which unit gave the most consistent measurements across the different groups—the
centimeter, millimeter, or meter units? Which unit of measurement gave the most precise measurement? Why would the smaller unit of measurement be more precise?
E. Egg Mass. Each group should measure the egg in grams using a scale.
Note: Use hard-boiled eggs to avoid possible breakage and mess. Why did different groups get different measurements? What are the possible sources of
errors that students could make when measuring with a scale? How could they
increase both the precision and the accuracy of the measurement?
Figure 1.
Measurement Tool
Height in Centimeters
Group 1- meter stick
Group 2- meter tape
Group 3- metric ruler
Group 4- yarn/meter stick
Group 5- centimeter ruler
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Explain
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Accuracy in Measurement. Most Figure 2.
of the time we want our mea- Archery Analogy
surements to be as “true” or The relationship between precision and
accurate as possible. When we accuracy in target shooting.
make a measurement we want
to know that the value we determine is close to the real
value. If our bathroom scale
at home consistently measures
our weight as higher than we
know it to be, the measurement
imprecise/inaccurate
imprecise/accurate
is inaccurate. Students may
have noticed that when they
are weighed in the doctor’s office the weight there is heavier
or lighter than the value they
get on their bathroom scales.
This inconsistency exemplifies
precise/accurate
precise/inaccurate
differences in accuracy due to
the measurement tool.
Calibration. How do we know if our measurements are accurate? With any
instrument we use for measurement, whether it is a ruler, a weight scale, or a
thermometer, we need a method of calibration. Calibration is a process in which
a known standard is measured with the instrument in question. If the instrument doesn’t agree with the standard, we adjust the instrument, or calibrate
it, so that it does agree as closely as possible. For example, for a thermometer
we know that at standard pressure, water freezes at 0° C and boils at 100° C.
If the thermometer is designed to withstand these temperatures, we dunk it
in a glass of ice water and in boiling water to determine if it reads accurately
(ice water and boiling water, for thermodynamic reasons, maintain a constant
temperature very near the freezing and boiling temperatures respectively). A
second option is to calibrate our thermometer against a thermometer that
we are reasonably sure is calibrated (if such an instrument is available). For
a scale or balance, we need weights of known mass (e.g., 100 g, or 5 lbs.) for
calibration. For length measurements, there are fewer options for standards.
One option is simply to check our ruler or meter stick against other rulers and
meter sticks. Have students check the rulers and see if they are all in reasonable agreement. Can they detect any variation in the rulers or meter sticks?
You may be surprised to find real variations. How large are they?
NATIONAL SCIENCE TEACHERS ASSOCIATION
C hapter 3
O o p s , I did it again: Er r or s inMeasurement
Measur ement
Precision in Measurement. Precision is a term that refers to the degree of fineness
or coarseness of a measurement. If we use a meter stick that just has centimeter
tick marks, then our measurement will have less precision than if we use a
meter stick with millimeter marks. Precision is independent of accuracy. We
can have cases of high precision and low accuracy and vice versa. The figure
on page 38 illustrates this point with a target-shooting analogy. Precision in
this case is how close together the shots are (independent of whether they
are on target). In the case of measurement, high precision means that the
measurement technique is very fine grained. It provides more significant digits
than a lower-precision measurement.
In some cases, high precision is not necessarily desired or meaningful. If
I want to know how much I weigh, it may be enough to say I weigh 125 lbs. I
may not care to know that I weigh 125.255 lbs. In fact, these extra digits imply
a precision that may be misleading. Our instrument may provide weight readings to the nearest thousandth of a pound, but it is unlikely that this precision
is meaningful (especially in the case of a bathroom scale). However, if I wanted
to know how much gold there was in my gold nugget, I would want very high
precision. First I would select an instrument that can accurately weigh objects
with high precision (not a bathroom scale, but a high-precision balance). If I
simply measured my nugget to the nearest ounce, I might find it weighed 1 oz.
However if I used a tool that accurately measured to the nearest thousandth
of an ounce, I would find that my nugget weighed 1.458 oz. At the time of
writing, gold is selling for $871.16 per ounce. If I used my low-precision measurement of 1 oz, instead of my high-precision measurement of 1.458 oz, I
would loose $398.99 of value. Precision is often noted by the last digit of the
measurement value. In the case of my weight measurement, the level of precision is the nearest pound, whereas the level of precision for my gold nugget
weight measurement is the nearest thousandth of an ounce.
Types of Error. Virtually all measurements have some degree of error. There
are a number of sources of error, some that are random and some that are
systematic. Random error results from fluctuations that occur in making measurements; it is related to the inability of the person making the measurements
to do it in exactly the same way each time. By definition, random errors are
neither biased high nor low; you are as likely to make a measurement that is
too high as you are to make one that is low. A measurement is considered to
have high precision if it has low random error.
Systematic error happens when the same error is made over and over again
in the process of measuring. An example of systematic error would be using
a ruler by measuring from the end of the ruler, rather than from the first tick
mark, or zero on the ruler. By using the ruler improperly the person measuring
would make the same mistake consistently, over and over again with different
measurements. As this example indicates, a systematic error tends to bias a set
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Ch a p t e r 3
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of measurements either high or low. Reducing systematic error can produce
measurements that are more accurate.
Measurements are likely to involve a degree of both kinds of error. Again
using the ruler analogy, if a student was making a set of measurements using
the end of the ruler rather than the zero mark, there would be a variation
around an average in that set of measurements that would be called the random
error. There would also be an offset (due to the extra length between the end of
the ruler and the zero tick mark being erroneously added) to the whole set of
measurements that is called the systematic error.
Extend
Extend these ideas by exploring how measurements have evolved over time.
Earlier measurement standards used the span of a person’s hand or the length
of the foot. How might these types of measurements decrease the accuracy
of the measurement? Today many commercial scales are inspected to ensure
that they weigh purchased items consistently and accurately. Have students research how scales were altered in past times to allow merchants to overcharge
for products being sold.
Hubble Space Telescope. There have been a number of very public and highly significant errors in measurement that students can research. The $1.5 billion Hubble Space Telescope was sent into space with a significant flaw in the telescope’s
main mirror. This flaw resulted from a 1.3 mm spacing error in the assembly of an
instrument used to guide the grinding of the mirror. As a result of this error, the
mirror was deformed and the telescope did not produce precise images.
For more information on the Hubble Telescope:
• www.newscientist.com/article/mg12717301.000-the-testing-error-that-led-tohubble-mirror-fiasco-.html
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Mars Climate Orbiter. Another significant error in measurement caused the
crash of NASA’s Mars Climate Orbiter in 1999. The spacecraft entered a
much lower orbit around Mars than intended, was captured by the Martian atmosphere, and likely crashed. An independent investigation found that NASA
engineers did not convert English measurements of the rocket thrusts to metric measurements in Newtons. The difference in the two values was enough
to cause the spacecraft rocket to misfire, allowing the craft to be destroyed by
entering the atmosphere of Mars instead of going into orbit.
For more information on the Mars Climate Orbiter:
• www.cnn.com/TECH/space/9911/10/orbiter.02
• http://mars.jpl.nasa.gov/msp98/orbiter
NATIONAL SCIENCE TEACHERS ASSOCIATION
Measurement
What would happen? After discussing famous errors in measurement that
have been reported to the public, ask students to think about different technologies in our world and imagine what would happen if an error in measurement had occurred in the design or manufacture of the product. Ask the
students to write a story about the source of error and the possible impact that
the error could have on the people who use the technology.
Evaluate
Check for student understanding:
1. What is error of measurement?
2. How can measurements be made more accurate?
3. How can measurements be made more precise?
4. What are potential sources of error when using a metric stick?
5. Does precision increase as the unit of measurement gets larger or smaller?
Explain.
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Student Data Sheet
O ops , I Did It Again
Name ________________________________________
Measure Up!
In this investigation you will make a series of measurements of different objects.
Record your measurements below.
Procedure
A. How Tall Am I?
Measure the height of the student volunteer in centimeters. Make sure that everyone in the group agrees with the measurement before recording the height.
Student height: ___________ cm
What are possible sources of error that could influence the measurement?
B. Shoe Box Length—Centimeters
Measure the long side of the shoe box. How many centimeters long is the box?
Shoe box length: ___________ cm
C. Shoe Box Length—Millimeters
Measure the long side of the shoe box. How many millimeters long is the box?
Shoe box length: _____________ mm
D. Shoe Box Length—Meters
Measure the long side of the shoe box. How many meters long is the box?
Shoe box length: _____________ m
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E. Egg Mass
Measure the mass of the egg in grams.
Egg mass: _____________ grams
Compare your measurements with those of other students.
Did you agree or disagree?
NATIONAL SCIENCE TEACHERS ASSOCIATION
Student Data Sheet
Oops , I Did It Again
Questions
1. When you measured the shoe box, which unit do you think gave you the
most precise measurement?
2. When you measured the shoe box, what were possible sources of error
in your measurement?
3. Another teacher had five of her classes measure the mass of the egg.
Each class reported a different mass. What could explain these different
measurements?
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