1. Attend to Precision Introduction to Engineering Design
© 2012 Project Lead The Way, Inc.

2. Precision Engineers communicate precisely to others
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Precision
Engineers communicate precisely to others
Use clear definitions
State the meaning of variables
Are careful about specifying units
Calculate accurately
Express answers with a degree of precision appropriate for the problem context
Engineers must be able to clearly communicate their solution process and their proposed solution. When problem solving requires measurements and/or calculations, engineers must take care to …. [ read bullet list].
In this presentation, we will concentrate on the last item on the list, “express answers with a degree of precision appropriate for the problem context.”

3. Precision of Answers 6.33 cm
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Precision of Answers
How precisely should you report a measured value?
Digital Instruments—Read and record all digits, including zeros after the decimal point, exactly as displayed.
Decimal Scaled Instruments—Record all digits that you can certainly determine from the scale markings and estimate one more digit.
Remember that we have already talked about how to report a measured value. If you are using a measuring device with a digital scale, record all of the digits displayed. If you are using a measuring device that requires reading a scale, record all of the digits that you can certainly determine from the scale markings and estimate one more digit.
Once you have the original measurements, how do you record solutions to problems that require using the measured values to make calculations?
6.33 cm

4. Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Precision of Answers
How precisely should you report a calculated value?
Specified precision
Common sense
Accepted Convention
Least precisely measured value used
A common misunderstanding is that you should report a calculated quantity to as many digits as possible. But part of solving problems is understanding that most numeric solutions are simply approximations of the true answer. And giving lots of digits, especially lots of decimal places, implies that you know the answer very precisely. But, your answer is only as precise as the information you use to find it.
[click]
If the precision of your answer is specified, report your answer to that level of precision.
If the required precision of the answer is not specified (or given), there are three things to consider when deciding the precision (or number of digits) to report. [click]
First [click], does the answer make sense?
Second [click], don’t violate accepted conventions. For instance, if you are asked how much it will cost to ship a load of bricks, don’t report the cost as $ Money is commonly reported in dollars and cents, not fractions of cents.
Third [click], consider the least accurate measured value. Your estimate of the answer cannot be more precise than the least precise measured value you used to calculate the area.

5. Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Precision of Answers
How precisely should you report a calculated value?
Specified precision
A cylinder has a height of 6 inches and a diameter of 3 inches.
What is the volume of the cylinder? Precision = 0.00
If the cylinder weights 4.5 pounds, what is the density of the material?
In some cases, the precision will be specified—always follow this specification, if given.
In this case, the precision for the calculated value of volume is specified and should be reported to the nearest hundredth cubic inch. Here, the volume result is used later in a calculation for density and so is an intermediate calculation. Best practice is to keep the volume result in the memory of your calculator to use in the later calculation. But rather than report the intermediate value to eight or more decimal places, you should report it to the precision specified.
If the precision of the answer is not specified, you must determine the appropriate level of precision.

6. Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Precision of Answers
How precisely should you report a calculated value?
Common sense
How many people on a train?
9.3, say 9 people
If the precision of the answer is not specified, you must choose an appropriate level of precision [click]
First question, does the answer make sense?
Does it make sense to report the number of people on a train as 9.3? No, there probably won’t be 0.3 of a person on the train.
Does it makes sense to say that you will buy 4.4 gallons of paint if you are buying gallon cans? No, you will have to buy 5 gallons.
How many gallons of paint should you buy?
4.4? Buy 5 gallons

7. Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Precision of Answers
How precisely should you report a calculated value?
Accepted Convention
Money $
Lumber
3.4 in. x 5.2 in.
$237.90
Second [click], don’t violate accepted conventions.
For instance, let’s say you are asked how much it will cost [click] to ship a load of bricks. You use your calculator to make the calculation and the result reads $ [click]. How do you report the cost? Money is commonly reported in dollars and cents, not fractions of cents [click].
Lumber [click] is typically specified by its nominal sizes such as 2 x 4. So, if you determine based on a calculation for minimum cross sectional area that you need a beam that is 3.4 in. x 5.2 in. [click], you should report the size as 4 x 6 using the accepted convention [click].
4 x 6
Can you think of any other accepted conventions?

8. Precision of Answers Accepted Convention—Rounding Statistics
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Course Name
Unit # – Lesson #.# – Lesson Name
Precision of Answers
Accepted Convention—Rounding Statistics
General Rule: Don’t round until the final answer.
Mean—Round to one more decimal place than the original data.
Standard Deviation—Round to one more decimal place than the original data.
Remember that we have already talked about rounding statistics. To review, the general rule is to keep all numbers in memory until the final statistic is reported. For both the mean and the standard deviation, the accepted convention is to report the statistic to one more decimal place than the original data.

9. Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Precision of Answers
How precisely should you report a calculated value?
Least precisely measured value used
A third consideration when deciding how precisely to report a calculated answer (if the precision is not specified) is to consider the least accurate measured value that you used in your calculation. Your estimate of the answer cannot be more precise the least precise measured value that you used to calculate the answer.
Does it make sense to report the area of a circle to the nearest square micrometer if the radius is measured to the nearest cm? No, you can’t possibly know the area to the nearest square micrometer if the radius is only measured to the nearest centimeter.
So, an appropriate area would be 30 square centimeters (one digit in the measured value and one significant digit in the answer) and would be a good estimate of the true area. However, in most circumstances, it would be acceptable to report this value as 28 square centimeters. But using any more digits would imply that you know the radius of the circle to a much high level of precision than you really do.
3 cm
A = π r 2 = π (3 cm) 2
= cm2
= m2
30 cm2
OR 28 cm2

10. Think About Precision It is only an ESTIMATE.
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Course Name
Unit # – Lesson #.# – Lesson Name
3 cm
Think About Precision
It is only an ESTIMATE.
Assume a minimum radius of 2.5 cm
A = π r 2 = π (2.5 cm) 2
= cm2
Assume a maximum radius of 3.5 cm
Remember, every measurement is only an estimate of the true value of the quantity.
Let’s look at why 30 might be the best estimate for the area of a circle with a diameter of 3 cm.
If the radius of the circle is represented as 3 cm, we might assume that the measurement was recorded to the nearest whole centimeter. Otherwise, the person recording the measurement should have written 3.0 cm or 3.00 cm to indicate a more precise measurement.
So, 3 cm could represent any true value for the radius between 2.5 centimeters and 3.5 centimeters. If the true measurement were less than 2.5 cm the measurement should have been reported as 2 cm, for instance.
Assuming the smallest possible radius of 2.5 cm, the area of the circle would be … cm^2, let’s say 19.6.
Assuming the largest possible radius of 3.5 cm, the area of the circle would be … cm^2, lets say 38.5.
So, the true area of the circle should be between 19.6 and 38.5 cm^2. So 30 cm^2, or even 28 cm^2 both seem like reasonable, middle-of-the-road answers. But cm^2 would imply that you are pretty sure that the area is just over 28 cm^2.
In reality, you can never be sure of the true area, and your answer should reflect that uncertainty based on the precision of the measurements you used to calculate the answer.
A = π r 2 = π (3.5 cm) 2
= cm2
The true value of the area is
19.6 cm2 < A < 38.5 cm2
30 cm2 OR 28 cm2

11. How precise? BUT, if you will use your answer in further calculations
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Course Name
Unit # – Lesson #.# – Lesson Name
How precise?
BUT, if you will use your answer in further calculations
It is best to keep an intermediate value in your calculator without rounding.
If you record an intermediate value, use a level of precision greater than the final answer dictates to eliminate compounding errors.

12. How precise? In general (if it makes sense) use the following rules:
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Unit # – Lesson #.# – Lesson Name
How precise?
In general (if it makes sense) use the following rules:
If you are adding or subtracting measured values, use the precision of the least precise measured value.
Example: Find the perimeter of a rectangle with side lengths 3.2 in. and 4.76 in.
Perimeter = 2(3.2 in in.) = in.
If you are multiply or dividing measured values, use the least number of digits in any of the measured values.
Example: Find the area of a rectangle with sides lengths 13 in. and 4.76 in.
Area = (13 in.)(4.76 in.) = in.2
15.9 in.
tenths place
As a general rule, if it makes sense, the precision of the least precisely measured value dictates the precision of the calculated value.
If you are adding or subtracting values, use the least number of decimal places/place value in any of the values for the resulting answer.
In this example, which measured value is least precise? 3.2 is measured to the tenths place and is least precise. Therefore the answer should be reported to the tenths place.
If you are multiplying or dividing measured values, use the least number of digits in any of the measured values as the number of digits in the resulting answer.
In this example, which measured value has the least number of digits? 13 has only two digits. Therefore the answer should be reported with two significant digits.
61 in.2
2 digits

13. Calculation Precision—Example
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Course Name
Unit # – Lesson #.# – Lesson Name
Calculation Precision—Example
If a carpet, represented by the below parallelogram, was measured by pacing out steps, calculate the area. h
b = 3 feet
h = 6 feet
A = bh = 3 ft x 6 ft
A = 18 ft2
Discuss with students the importance of knowing the context of a problem to report the answer to an appropriate degree of precision. Either 18 square feet or 20 square feet might be acceptable as an answer for this problem. But square feet would not be appropriate if the dimensions were measured to the nearest foot. You could never be confident that the calculated area was accurate to the tenths place (with an estimated digit in the hundredths place).
The significant digits for the measured dimensions is only 1 place value which would suggest that an answer should have only one significant figure; so using this reasoning 20 square feet (with 2 being the only significant digit) is the most appropriate answer.
Think about this. What if, for example, a better estimate of the carpet dimensions was 3.4 ft x 6.4 ft (which rounds to 3 ft x 6 ft), giving an area of approximately 22 square feet? In this case 20 square feet is a closer estimate of the area. But, if the carpet dimensions were actually 2.6 ft x 5.6 ft (which also rounds to 3 ft x 6 ft) results in an area of approximately 15 square feet. Each of these estimated areas could be rounded to 20 ft2. However, it would generally be acceptable to report the area as 18 square feet (the product of 3 ft and 6 ft).
= ft2
= 20 ft2
OR 18 ft2
Which answer is appropriate?

14. Calculation Precision—Example
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Course Name
Unit # – Lesson #.# – Lesson Name
Calculation Precision—Example
The same carpet has now been measured using a ruler. Calculate the area.
b = 3.48 feet
h = 6.21 feet h
A = bh = (3.48 ft)(6.21 ft)
A = ft2
= 21.6 ft2
Note that the given values would be rounded to the values on the previous slide (3 ft x 6 ft) according to rounding rules. Rounding at different points in a calculation can result in compounding errors. Best practice is to leave the resultant of intermediate calculations in your calculator and round only the final answer to an appropriate level of precision.
In this case, if we are interested in the area of the carpet only and assume that the dimensions of the carpet were recorded such that the final digit, (the hundredths place) is estimated, it seems unreasonable to give an area that indicates we are confident of the area value to the thousandths place (three decimal places) and adding an estimated fourth decimal place.
It also seems inappropriately imprecise to record the area to only the tens (20 square feet) or ones (22 square feet) place. Intuitively, we know that we can confidently give a closer estimate of the area if the dimensions are given to the nearest hundredth of a foot.
Depending on how you will use the final answer, an area estimate of 21.6 square feet is appropriate.
However, if you will use the area value in additional calculations, say to calculate the weight of the carpet per square foot, you should use the unrounded value. Leaving intermediate resultants in your calculator and rounding only at the completion of a calculation is best practice.
= 22 ft2
= 20 ft2
Which answer is appropriate?

15. It is just an estimate! All measured values are estimates.
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Unit # – Lesson #.# – Lesson Name
It is just an estimate!
All measured values are estimates.
And therefore, all calculated values resulting from measured values are estimates.
You can’t be CERTAIN of the answer to a level of precision greater than the information you use to calculate it, so don’t give an answer that implies that you are.

16. Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Precision of Answers
How precisely should you report a calculated value?
Specified precision
Common sense
Accepted Convention
Least precisely measured value used
And don’t forget units!
In the end, think about your answer.
Is the precision of the answer specified? If so, use that precision.
If not,
Does your answer make sense?
Should you use an accepted convention?
Have you considered the precision of the measured values you have used in your calculation?