## Measuring Uncertainty

• Lesson
9-12
1

By measuring long jump results, students will discover how to determine the appropriate number of digits they should report when taking a linear measurement. Students will realize that measured numbers are never exact and researcher skill and tools can be analyzed to determine the precision of a measurement.

Pique students' interest by asking, "Have you ever questioned the accuracy of a football official’s placement of the ball after a play? Have you wondered if two officials would ever place the ball at precisely the same spot? What if the play is only 1 inch short of a first down?”

You can ask if there are situations in life when students question the accuracy or precision of the measurement of something. A great example of inaccuracy would be the measurement of the height of Mount Everest. At one time, it was thought that Mount Everest was 29,035 ft tall. Scientists have now decided that the height is actually 29,028 ft. Since the mountain has not shrunk, it is possible that there is error embedded in how the height of the mountain was being measured. Other examples could include the amount of ice cream in a 5 oz cone from Dairy Queen, the length of a pair of pants with a 30nbsp;in inseam, and the time it takes for a musher and his dog team to complete the 1500 mi Iditarod race in Alaska and the Yukon.

Explain to students that in fact, measurements are never exact. The uncertainty of a measured quantity depends on the skill of the researcher and the limitations of the measuring instrument. These two factors determine the accuracy and precision of a measurement.

Describe to students that they will conduct an experiment, collect data, and then perform absolute and mean deviation calculations to determine whether data was reported to an appropriate number of decimals, or significant digits.

Distribute the Measuring Uncertainty activity sheet to each student. This activity will occur outside.

 Measuring Uncertainty Activity Sheet

Before students begin, it is important to distinguish the difference between accuracy and precision:

• Accuracy is the degree to which information matches true or accepted values. Accuracy can be determined for a single measurement.
• Precision refers to the level of measurement and exactness of description. Precise data may measure position to a fraction of a unit. Precise attribute information may specify the characteristics of features in great detail. A set of data points can be analyzed for precision by comparing values to each other and looking at the range of values measured.

It is important to realize, however, that precise data, no matter how carefully measured, may be inaccurate. High precision does not indicate high accuracy nor does high accuracy imply high precision.

Here is an example that shows the difference between accuracy and precision. If you wanted to make an angel food cake, you must be BOTH accurate and precise or the recipe will fail. Imagine that the recipe called for 1 cup of flour.

• An inaccurate baker will use the 1 cup measure but not level the flour off, putting more than 1 cup of flour into the recipe.
• An imprecise baker may mistakenly use the 1/2 cup measure and very carefully level the flour off, putting exactly 1/2 cup of flour into the recipe.
• An accurate and precise baker will use the 1 cup measure and very carefully level the flour off, putting exactly 1 cup of flour into the recipe.

Precision can be described mathematically for a set of numbers. In order to calculate the precision of the data:

• Find the mean for a set of measured data.
• Find the differences between each observed value and the mean of the data. Take the absolute value of each of these differences. These are called absolute deviations. Mathematically:

Absolute Deviation = | Observed Value – Average Value |

• Find the mean of the absolute deviations. This mean is called the mean deviation.

A simple example of a calculation of precision:

 Trial Measurement Absolute Deviation 1 8.76 cm 0.27 cm 2 8.43 cm 0.06 cm 3 8.28 cm 0.21 cm Mean 8.49 cm 0.18 cm

You will notice that the sample calculation had a mean of EXACTLY 8.49 cm. What if this number had been 8.4866666? You would use ROUND the number to the same number of decimals present in the original data to calculate the absolute deviation. When reporting experimental data, a student will learn to report all digits they are certain of plus one digit they are uncertain of. These are known as significant digits. The uncertainty of the last digit can also be indicated, as it is equal to the mean deviation.

For example, the example used earlier found the precision of a measurement to be:

8.49 cm ± 0.18 cm

• Ask students to look at the mean and mean deviation to identify what digits have uncertainty. To answer this question, they can look at the place values in the mean deviation. In the example given, the mean deviation shows a measured uncertainty in both the tenth and the hundredth decimal places.
• Determine what the LAST certain digit is. In the example, the last certain digit would be the ones place, as the mean deviation has no value in the ones place. We can be certain that any measurement would be 8 cm, but because significant digit rules allow us to have ONE uncertain digit, we are also able to estimate one more digit in our number. One thing to note is that ‘rounding rules’ need to be applied to the last digit reported. This would mean that the appropriate reporting for the measured number would be 8.5 cm.
• This would mean that measured number would be reported as 8.5 cm ± 0.2 cm.

Your class is now ready to gather their own data. Once outside, choose one person in the class to perform a long jump into a soft pit of sand (or snow).

Decide on an order for your students to perform their measurements. Every student will take one turn holding the 0 end of a measuring tape on the takeoff board, and one turn measuring the length of the jump. The same jump and measuring tape are to be used by each experimenter. These measurements are NOT to be shared until students return to the classroom.

Back in the classroom, students will share their measurements on the Class Reporting Sheet overhead. Project this summary on an overhead projector or project the data on the Measuring Uncertainty Data spreadsheet. Have students calculate the mean deviation for this set of data. If you are using the spreadsheet, formulas are set up to do this for students.

Students will now look at the precision of their measurements to determine which digits have uncertainty in them.

Have students look at the class data they have collected for the measurement of the long jump.

• According to the mean and mean deviation, what digits are certain?
• What will be the ONE uncertain digit?
• How many decimals is it appropriate to report the long jump measurements to?
• Go back to your initial data and round all of the measurements to the appropriate number of decimals.
• Recalculate the mean and mean deviation to prevent rounding errors.
• What is the final value for the measurement of the long jump? Students should report it in the format:

_________ m ± _________ m

Assessment Options
1. As a journal response, have students write about a generalization that they can make regarding how to determine the number of digits to report when measuring a number.
2. Have students list areas in their lives where precision is important.
3. Provide a data set and have students determine the mean value along with the precision (mean value ± mean deviation) to decide whether the data has been reported to the correct number of significant digits.

Extensions
1. Revisiting the data:
• Create a box and whisker plot of the original data using the Box Plotter tool.
• Predict what will happen to the box and whisker plot of your data once you round the numbers. Re-create the box and whisker plot of this new data set.
• Compare the new whisker plot to the original — is this what you predicted? If not, explain how the graph is different from your prediction and why you think this occurred.

Students will, in fact, discover that the box and whisker plots for the original data and the rounded data are almost identical. Reasons that students may predict that their box and whisker plots will change would be

• the possible misconception that they have changed the value of the data, or
• they don’t have an understanding of what a box and whisker plot represents.

This extension would show them clearly that the magnitude of the data has not been changed, it has simply been reported to an appropriate number of decimals. ie) 5.467 is the same VALUE as 5.47.

2. Measuring zero: Challenge students to find things that can be expressed with a measured zero as the last digit. For example, a desk could be 45.0 cm wide, a window could be 120 cm high. Have a classroom discussion about what a measured zero is.
3. Split your class into two groups. Give one group measuring tools that only have subdivisions every 10 cm. Give the other group measuring tools that have subdivisions every 1.0 cm. Have them measure items in the classroom, such as binders, desks, counters. Find the mean and precision of the measurement for both tools and compare.

Questions for Students
1. What factors influenced each experimenter’s measurement?
[The person holding the zero end of the tape would have to decide where the 0 cm marking should be placed on the board. This would be slightly different for every measurement.
The person actually reading the measuring tape would have to decide where on the long jumper’s footprint they would measure to. This would be a relatively large error.
The angle that the person was viewing the measuring tape at would affect their reading of the tape — this is called a parallax error.
The angle between the measuring tape and the takeoff board could be different for each pair of measurers. This will change the length of the tape.]
2. How many decimals would you expect a world record in long jump (in meters) to be reported to? Why?
[World records are reported to the nearest hundredth of a meter. If the class data could only be reported to the nearest tenth of a meter, you could discuss the fact that people who measure international sporting events are trained to be very precise, which would give them a smaller deviation from the true measure of the jump than we would expect to get in a classroom.]
3. A measurement is reported to be 65.73 m ± 0.17 m. What would you suggest to the researchers? Why?
[In this situation, researchers should round their measured value to the nearest tenth of a meter so that there is only one uncertain digit. The mean deviation should also be rounded. The reported measurement should actually be 65.7 m ± 0.2 m.]
4. A measurement is reported to be 8.7 m ± 0.07 m. What would you suggest to the researchers? Why?
[Researchers should add another digit to their measured number. They have only reported the certain digits, and should include the zero that they would have observed but not reported. Not reporting a measured zero is a very common error. Reporting this zero is as important as reporting any other number. The reported measurement should actually be 8.70 m ± 0.07 m]
Teacher Reflection
• Were students engaged in the activity portion of the lesson? How was this expressed?
• What worked with classroom behavior management? What didn't work? How could you change what didn’t work?
• Were students able to explain how to determine the appropriate number of digits when reporting measured numbers?

### Learning Objectives

Students will:

• Collect data
• Use the data to calculate absolute and mean deviations to determine measurement precision
• Determine the appropriate number of significant digits when reporting measurements

### NCTM Standards and Expectations

• Analyze precision, accuracy, and approximate error in measurement situations.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.