Algebra 1 Summer Institute 2014
Data Measurement and Variation Uncertainty
Summary
Goals
Participant Handouts
Participants will make
 Distinguish possible
measurements using
errors in measurement
different measurement
 Understand the
devices. Variation, or
difference between
differences in measured
random error and
data, occurs for a number of
systematic error
reasons. Participants will
 Realize there are human
conjecture on the possible
errors when measuring
reasons for variation
Materials
Technology
Source
Poster paper
Markers
12-inch ruler
Tape Measure
LCD Projector
Facilitator Laptop
GeoGebra file
“adjusting lengths”
1. Data Measurement and
Variation Uncertainty
2. Measuring Reaction
Time and Error Analysis
Annenberg Learner
website
Estimated Time
90 minutes
Mathematics Standards
Common Core State Standards for Mathematics
MAFS.6.SP.1: Develop understanding of statistical variability
1.1: Recognize that a statistics question as one that anticipates variability in the
data related to the question and accounts for it in the answers.
MAFS.6.SP.2: Summarize and describe distributions
2.5: Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations
b. Describing the nature of the attribute under investigation, including how it
was measured and its units of measurement.
Standards for Mathematical Practice
1. Make sense of problems and persevere in solving them
2. Reason abstractly and Quantitatively
3. Construct viable arguments and critique the reasoning of others
5. Use appropriate tools strategically
6. Attend to precision
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Algebra 1 Summer Institute 2014
Instructional Plan
Variation, or differences in measured data, occurs for a number of reasons. Examining
variation is a crucial part of data analysis and interpretation. In fact, explaining the
variation in data is as important as measuring the data itself.
1. Place the participants in groups of two and ask them to find out the length of the
room they are in. Participants will record their answers in inches using three
forms of measurements: their stride, a 12-inch ruler, a tape measure. (Slide 2)
2. After all measurement are taken, participants will share their results, record the
information given in a chart:
Group
Stride
Ruler
Tape
3. After all results have been recorded, have a class discussion about the difference
in results. The following questions are suggested for the discussion: (Slide 3)
a. Are the measurements obtained with the strides exactly the same? Can you
explain why there may be differences?
b. Are the measurements obtained with the ruler exactly the same? Can you
explain why there may be differences?
c. Are the measurements obtained with the tape exactly the same? Can you
explain why there may be differences?
d. Did you get similar answers using the different measuring tools? Why or
why not? Did you get identical answers using the different measuring
tools? Why or why not?
e. Which measuring tool do you think gave you more accurate results? Why?
f. Do you think a tape measure would be more or less accurate than a ruler
or a tape measure? Why? If you have a tape measure available, use it to
measure the same room five times and see how the results compare with
your previous measurements.
4. Measurement is never perfect, and we can always expect measurement errors in
our data. Our goal, of course, is to keep these errors to a minimum. For this
reason, we need to be aware of the various sources and causes of measurement
error.
Participants should measure the room again with the device they feel is the most
accurate. Record their results. Is their second measurement the same as the first
measurement? If there is a difference, why might be some of the reasons for it?
Have a class discussion, write down on the board or laptop their given reasons.
(Slide 4)
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Algebra 1 Summer Institute 2014
Random error is when variations in the measurements occur without a predictable
pattern. If repeated measurements are made, random errors cause the measured
value to vary, sometime above and below the actual measured value. A source of
random errors comes from reading the scale on a measurement tool like a
thermometer or tape measure. The errors are random rather than biased: They
neither understate nor overstate the actual measurement.
5. In contrast, measurement bias, or systematic error, favors a particular result. A
measurement process is biased if it systematically overstates or understates the
true value of the measurement. Consider our scale example again. If a scale is not
properly calibrated, it might consistently understate weight. In this case, the
measuring device -- the scale -- produces the bias. The important thing to keep in
mind is that biased measurements invariably produce unreliable results.
Ask the participants to compare their tape measures and rules to verify if they
have the same calibration
6. In any statistical investigation, we can always attribute some of the variation in
data to measurement error, part of which can result from the measurement
instrument itself. But human mistakes, especially recording errors (e.g.,
misreading a dial, incorrectly writing a number, not observing an important event,
misjudging a particular behavior, position of observer), can also often contribute
to the variability of the measurement and thus to the results of a study.
Ask participants for suggestions on how to change the experiment to improve
their results. Are those suggestions reducing systematic or random errors? (Slide
5, 6)
7. The difference between what individuals think they see (and how they see it) and
the objective reality of what they have observed is called "visual error." These
errors in perception, which can significantly misrepresent reality, are a natural
consequence of being human. (Slide 7)
In this Interactive Activity, participants will have an opportunity to see how well
they make two visual judgments. Ask them to open the GeoGebra filed called
“adjusting lengths”.
Would you say that the errors made in the visual judgment of the interactive
activity were due to random error, or to bias? Why or why not?
Optional: in order to reinforce the difference between errors and the impact of
human error, the “Measuring Reaction Time and Error Analysis” could be done.
Participants should be in groups of 2. Have a partner hold the ruler vertically with
the “0 cm” mark just above your fingers and release it with no warning. Try to
close your fingers as quickly as possible to catch the falling ruler. We can use the
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Algebra 1 Summer Institute 2014
distance the ruler falls before you catch it to calculate how long it takes you to
react to the motion and close your fingers to stop the falling ruler. We’ll find
several sources of uncertainty and we’ll practice using our new understanding of
uncertainty during these calculations. (Slide 8)
8. Conclusion: review the different types of errors. What type could be more easily
fixed? (Slide 9)
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