1. Accuracy and Precision
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Course Name
Unit # – Lesson #.# – Lesson Name
Accuracy and Precision
of Measurement
Introduction to Engineering Design
© 2012 Project Lead The Way, Inc.

2. Recording Measurements
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Unit # – Lesson #.# – Lesson Name
Recording Measurements
A measurement always includes a value
A measurement always includes units
A measurement always involves uncertainty
A measurement is the best estimate of a quantity
A measurement is USELESS if we can’t quantify the uncertainty
[click] An important part of a measurement is the numerical value of the measurement; however, the value is meaningless without units.
[click] A measurement must always include units. Be sure to always include units when recording measurements.
[click] A measurement is never certain. There are always errors in measurements, even if they are very small. It is important to know the level of error that may be inherent in a measurement.
A measurement is only useful if a value is associated with units and the uncertainty of the value is understood.

3. Two Sources of Error in Measurement
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Two Sources of Error in Measurement
Random Error
Errors without a predictable pattern
Average random error = zero
So, can be eliminated by repeated measurements
Affects PRECISION, not accuracy
Example: Errors in reading a thermometer scale
[click] Potential Error in a measurement creates uncertainty.
[click] There are two types of errors that can occur in measurement.
[click] Random errors are typically associated with the measuring equipment or the person performing the measurement. For example,
When measuring the volume of a liquid in a graduated cylinder, the measurement value may fall between two graduations on the cylinder. The reader must estimate the value, which may sometimes be too large or too small.
Random errors can be identified by repeating the same measurement.
[click] Systematic Errors are more difficult to identify.
Can you think of other examples of systematic errors?
A person timing heats of a track and field race stands 100 meters away from the starting gun. Since it takes some time for the sound to travel to the person timing the race at the finish line, the timer consistently depresses the button to start the stopwatch a fraction of a second later for every race.
A scale is not zeroed before taking measurements and is later found to read 0.2 grams with no mass on the scale. Therefore, all of the measurements consistently show a mass of .2 grams greater than the mass of the object measured.
Systematic errors can be eliminated if they are identified, but often systematic error are undetected.
High Accuracy
Low Precision

4. Two Sources of Error in Measurement
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Two Sources of Error in Measurement
Systematic Error
Errors with a predictable pattern
Average systematic error ≠ zero
So, can NOT be eliminated by repeated measurements
Affects ACCURACY, not precision
Example: Error in placement of 0° on thermometer scale
[click] Potential Error in a measurement creates uncertainty.
[click] There are two types of errors that can occur in measurement.
[click] Random errors are typically associated with the measuring equipment or the person performing the measurement. For example,
When measuring the volume of a liquid in a graduated cylinder, the measurement value may fall between two graduations on the cylinder. The reader must estimate the value, which may sometimes be too large or too small.
Random errors can be identified by repeating the same measurement.
[click] Systematic Errors are more difficult to identify.
Can you think of other examples of systematic errors?
A person timing heats of a track and field race stands 100 meters away from the starting gun. Since it takes some time for the sound to travel to the person timing the race at the finish line, the timer consistently depresses the button to start the stopwatch a fraction of a second later for every race.
A scale is not zeroed before taking measurements and is later found to read 0.2 grams with no mass on the scale. Therefore, all of the measurements consistently show a mass of .2 grams greater than the mass of the object measured.
Systematic errors can be eliminated if they are identified, but often systematic error are undetected.
Low Accuracy
High Precision

5. Two Sources of Error in Measurement
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Course Name
Unit # – Lesson #.# – Lesson Name
Two Sources of Error in Measurement
EVERYTHING WE’RE ABOUT TO COVER DEALS WITH RANDOM ERRORS …

6. Uncertainty in Measurements
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Unit # – Lesson #.# – Lesson Name
Uncertainty in Measurements
Scientists and engineers often use significant digits to indicate the uncertainty of a measurement
We know that errors always exist in measurement. Therefore, there is always some uncertainty associated with a measurement. We never know the true value of a measured quantity. However, we can quantify the uncertainty. If we know how close to (or how far from) the true measurement value we may be, the measurement is useful.
[click] One way to indicate uncertainty is through the use of significant digits. We used this method in previous activities. Using significant digits, the measurement value is recorded such that all of the certain digits are recorded and the last significant digit represents a somewhat uncertain digit. In this case, if the measurement was recorded with a metric scale, the person recording this measurement is certain that the value is between 3.8 and 3.9 cm and estimates that the reading is just under half way between 3.8 and 3.9 on the scale.

7. Uncertainty in Measurements
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Uncertainty in Measurements
Another (more definitive) method to indicate uncertainty is to use plus/minus notation
Example: ± .05 cm
3.79 ≤ true value ≤ 3.89
This means that we are “confident” the true measurement lies between 3.79 cm and 3.89 cm
Measured value
Uncertainty
[click] We often use the plus/minus designation to indicate uncertainty of a value.
[click] In this case, if we are certain that our measurement falls within 0.05 cm of the our reading, then we are certain the true value falls between 3.84 cm and 3.89 cm.

8. Uncertainty in Measurement
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Course Name
Unit # – Lesson #.# – Lesson Name
Uncertainty in Measurement
Uncertainty of single measurement
Assumption: Dependent on instrument and scale
Example: Assume meter stick “accurate” to ± 0.2 mm because that’s what its scale says
Uncertainty in repeated measurements
Single measurements at mercy of random error, regardless of scale
Random error can be reduced by repeated measurement
Best estimate is (usually) the mean of the values
[click] In our previous activities, we have discussed uncertainty in a single measurement. The question has been:
[click] How close to the actual value is this particular measurement?
[click] We have answered this question by identifying a range within which we feel confident that the true value lies based on the scale markings of the measuring instrument.
[click] But we can use repeated measurements of the same quantity to get a better estimate of the true value. With repeated measurements, we can begin to better quantify the uncertainty of a measurement taken with a specific instrument.
[click] Of course, we can not eliminate random error – so when repeated measurements are taken, some measurements will be larger and some measurements will be smaller than the true value.
[click] Our best estimate is the mean of the measurements.

9. Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Your Turn
4 students measure length of credit card with (a) ruler and (b) digital caliper (do not adjust!)
Record your measurements in your notebook.
RULER
CALIPER
85.1mm mm
85.0 mm mm
85.2 mm mm
85.1 mm
USE DIGITAL CALIPER MIS-ZEROED (about 1 mm off)

10. Your Turn Plot ruler & caliper data on a number line RULER CALIPER
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Course Name
Unit # – Lesson #.# – Lesson Name
Your Turn
RULER
CALIPER
85.1mm mm
85.0 mm mm
85.2 mm mm
85.1 mm
Plot ruler & caliper data on a number line
[click] Plot student A’s data on the number line [click].
[click] Plot student B’s data on the number line [click].

11. Your Turn Accepted Value Ruler data ranges from 85.0 mm to 85.2 mm
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Course Name
Unit # – Lesson #.# – Lesson Name
Your Turn
Ruler data ranges from 85.0 mm to 85.2 mm
Caliper data ranges from mm to mm
The accepted length of the credit card is mm
Accepted
Value
85.105
[click] Student A’s data ranges from 85.0 to 85.2 mm.
[click] Student B’s data ranges from to mm.
[click] The accepted value of the measurement is

12. Your Turn Which student’s data is more accurate?
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Your Turn
Which student’s data is more accurate?
Which student’s data is more precise?
85.105
[Allow students to answer the questions, then click.] Student A’s data is more accurate because, on the whole, the data points are closer to the accepted value.
[Click and allow students to answer the question, then click.] Student B’s data is more precise because the recorded values are closer to each other than Student A’s data points.
[Instruct students to complete Activity 3.8 Precision and Accuracy number 1 parts a - c.]

13. Your Turn Which student’s data is more accurate?
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Your Turn
Which student’s data is more accurate?
Which student’s data is more precise?
Student A
Student B
85.105
[Allow students to answer the questions, then click.] Student A’s data is more accurate because, on the whole, the data points are closer to the accepted value.
[Click and allow students to answer the question, then click.] Student B’s data is more precise because the recorded values are closer to each other than Student A’s data points.
[Instruct students to complete Activity 3.8 Precision and Accuracy number 1 parts a - c.]

14. Quantifying PRECISION
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Course Name
Unit # – Lesson #.# – Lesson Name
Quantifying PRECISION
Precision is related to the variation in measurement data due to random errors that produce differing values when a measurement is repeated
Precision is related to the variation of the data. How much do the measurements of the same quantity vary?
There are several ways to indicate precision.
[click] This type of data is very often assumed to be normally distributed. [click] If we assume that the measurement data from each student is normally distributed, one method to quantify precision involves the standard deviation of the repeated measurement data.

15. Quantifying Precision
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Course Name
Unit # – Lesson #.# – Lesson Name
Quantifying Precision
The precision of a measurement device can be related to the standard deviation of repeated measurement data
RULER
CALIPER
85.1mm mm
85.0 mm mm
85.2 mm mm
85.1 mm
Ruler s:
sR=
[Click] Precision can be represented numerically using the standard deviation to provide a measure of variation.
Find the (sample) standard deviation of the measurements of each student. [Allow students time to calculate the standard deviation for each data set.]
[click] The standard deviation of the credit card measurements taken by Student A is 0.07 mm.
[click] The standard deviation of the credit card measurements performed by Student B is mm.
Caliper s:
sC =

16. Introduction to Summary Statistics
Empirical Rule
If the data are normally distributed:
68% of the observations fall within 1 standard deviation of the mean.
95% of the observations fall within 2 standard deviations of the mean.
99.7% of the observations fall within 3 standard deviations of the mean.
Many quantities tend to follow a normal distribution – heights of people, test scores, errors in measurement, etc.
Given normally distributed data, 68% of the data values should fall within 1 standard deviation of the mean, 95% should fall within 2 standard deviations of the mean and 99.7 % should fall within 3 standard deviations of the mean. This is referred to as the Empirical Rule.
Of course, with small samples/populations, these percentages may not hold exactly true because the number of values will not allow divisions to this precision.

17. Quantifying Precision
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Course Name
Unit # – Lesson #.# – Lesson Name
Quantifying Precision
Express the precision indicated by RULER data at the 68% confidence level
≤ true value ≤
with 68% confidence
[click] You can use the empirical rule to quantify precision based on a set of measurement data for a specific quantity. Precision can be indicated by giving a range of values and a level of confidence that the true measurement value is within that range. The range of values can be written using plus/minus notation as shown here.
Student A’s data has a mean of and a standard deviation of Therefore we can write a statement to indicate the precision of the measurements as / mm.
[click] This range can also be written as a compound inequality that specifies the minimum and maximum value of the quantity at a given confidence level. [click] Simply subtract the standard deviation from the mean for the minimum value and add the standard deviation to the mean for the maximum value.

18. Quantifying Precision
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Quantifying Precision
Express the precision indicated by RULER data at the 95% confidence level
≤ true value ≤
with 95% confidence
[click] You can use the empirical rule to quantify precision based on a set of measurement data for a specific quantity. Precision can be indicated by giving a range of values and a level of confidence that the true measurement value is within that range. The range of values can be written using plus/minus notation as shown here.
Student A’s data has a mean of and a standard deviation of Therefore we can write a statement to indicate the precision of the measurements as / mm.
[click] This range can also be written as a compound inequality that specifies the minimum and maximum value of the quantity at a given confidence level. [click] Simply subtract the standard deviation from the mean for the minimum value and add the standard deviation to the mean for the maximum value.

19. Quantifying Precision: Assignment
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Course Name
Unit # – Lesson #.# – Lesson Name
Quantifying Precision: Assignment
Predict with 95% confidence how far any cotton ball launched from your Fling Machine will travel. Show your work …

20. Two Sources of Error in Measurement
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Course Name
Unit # – Lesson #.# – Lesson Name
Two Sources of Error in Measurement
NOW WE’LL CONSIDER SYSTEMATIC ERRORS …

21. Presentation Name
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Unit # – Lesson #.# – Lesson Name
Quantifying ACCURACY
The accuracy of a measurement is related to the error between the mean measurement value and the accepted value
Error = Mean measured– Accepted value
RULER
CALIPER
85.1mm mm
85.0 mm mm
85.2 mm mm
85.1 mm
Ruler Error =
Caliper Error =
You determined the precision and accuracy of the credit card measurements intuitively by looking at the data. Is there a way we can determine accuracy and precision with numbers? [click]
[Click] Accuracy can be represented numerically by the error in the measurement. The error is the difference between the measured value and the accepted value. As a measured value gets further and further from the accepted value, the error gets larger and larger.
It is common practice to take several measurements of important quantities to help reduce the potential for avoidable errors in the measurements. If a measurement is obviously much different than the other more consistent results, the “outlier” can be eliminated from the data or new measurements can be taken to verify the results.
[click] When multiple measurements are taken, the mean of the measurements is used as the measured value.
Find the mean of the measurements of each student. [Give students time to calculate the mean of each data set.]
The mean of the credit card measurements by Student A is [click].
The mean of the credit card measurements performed by student B is [click].

22. Quantifying ACCURACY Error = ? A = 85.10 85.105 Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Quantifying ACCURACY
Error = ?
A =
85.105

23. Quantifying ACCURACY Error = ? B= 85.2998 85.105 Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Quantifying ACCURACY
Error = ? B=
85.105

24. Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Quantifying ACCURACY
So, where the heck do you get an “accepted value”???
(1) You buy it.
Example: Gage Block — a piece of metal or ceramic manufactured to insanely high tolerances to a certain dimensions. Sometimes traceable to reference blocks kept by the NIST.
SHOW/MEASURE inch GAGE BLOCK & CALIBRATION MASSES

25. Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Quantifying ACCURACY
So, where the heck do you get an “accepted value”???
(2) You specify it.
Example: Wooden blocks are supposed to be ¾ inch (“accepted value”). Are they? Calculate the error between the specified length and the actual length (in cm).
SHOW/MEASURE inch GAGE BLOCK & CALIBRATION MASSES