1. Accuracy and Precision
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Accuracy and Precision
of Measurement
Introduction to Engineering Design
© 2012 Project Lead The Way, Inc.

2. Recording Measurements
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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 useful if we can 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. Sources of Error in Measurement
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Sources of Error in Measurement
Potential errors create uncertainty
Two sources of error in measurement
Random Error
Errors without a predictable pattern
E.g., reading scale where actual value is between marks and value is estimated
Can be determined by repeated measurements
Systematic Error
Errors that consistently cause measurement value to be too large or too small
E.g., reading from the end of a meter stick instead of from the zero mark
[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.

4. Uncertainty in Measurements
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Uncertainty in Measurements
Scientists and engineers often use significant digits to indicate the uncertainty of a measurement
A measurement is recorded such that all certain digits are reported and one uncertain (estimated) digit is reported
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.

5. 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 certain the true measurement lies between 3.79 cm and 3.89 cm
[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.

6. Uncertainty in Measurement
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Uncertainty in Measurement
In some cases the uncertainty from a digital or analog instrument is greater than indicated by the scale or reading display
Resolution of the instrument is better than the accuracy
Example: Speedometers
[click] Sometimes using significant digits to indicate uncertainty can be misleading in terms of the closeness of the reading to the actual measurement value.
[click] The resolution (indicated by the smallest increment of the measurement scale) of the instrument may lead the user to believe that the measurement reading is a better estimate of the actual measurement than the instrument can reliably produce.
[click] For example, speedometers are often incremented to 1 mph (that is, the resolution of the scale is 1 mph). However, speedometers typically do not display the speed within 1 mph of the actual speed.
Sometimes the manufacturer specifications will give a maximum error that would indicate uncertainty; however, it is sometime difficult to determine the meaning of specifications when terms such as accuracy, precision, error, and uncertainty are often used interchangeably. When a measurement is critical, the precision and accuracy of the equipment should be checked often to ensure that errors are within acceptable limits.
[click] How can we determine, with confidence, how close a measurement is to the true value?
How can we determine, with confidence, how close a measurement is to the true value?

7. Uncertainty in Measurement
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Uncertainty in Measurement
Uncertainty of single measurement
How close is this measurement to the true value?
Uncertainty dependent on instrument and scale
Uncertainty in repeated measurements
Random error
Best estimate is 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.

8. Accuracy and Precision
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Accuracy and Precision
Accuracy = the degree of closeness of measurements of a quantity to the actual (or accepted) value
Precision (repeatability) = the degree to which repeated measurements show the same result
There are two distinct terms that are related to uncertainty in measurement – accuracy and precision. Although precision and accuracy are often confused, there is a difference between the meanings of the two terms in the fields of science and engineering.
[click] Accuracy indicates how close measurements are to the actual quantity being measured. For example, if you put a 50 pound weight on a bathroom scale, we would consider the scale to be accurate if it reported a weight of 50 pounds.
[click] Precision indicates how close together repeated measurements of the same quantity are to each other. A precise bathroom scale would give the same weight each time you stepped on the scale within a short time (even if it did not report your true weight). So, if you placed a 50 pound weight on the bathroom scale five times and the scale displayed a weight of 47 pounds each time, the scale could be considered precise (but not accurate).
A target analogy is sometimes used to differentiate between the two terms. Consider the “arrows” or dots on the targets to be repeated measurements of a quantity.
[click] The first target shows that the arrows (or repeated measurements) are “centered” around the center of the target (the actual value), so on the whole, the measurements are fairly close to the target (actual) measurement, making the measuring device accurate. But the repeated measurements are not close to each other, so the precision of the measuring device is low.
[click] The second target shows that the arrows (or repeated measurements) are close together, so the precision is high. But the “center” of the measuremens is not close to the target (actual) value of the quantity.
What should the target look like if the measurement is both highly accurate and highly precise? [allow student to answer, then click]. The third target shows both precision (because the measurements are close together) and accuracy (because the “center” of the measurements is close to the target value).
High Accuracy
Low Precision
Low Accuracy
High Precision
High Accuracy
High Precision

9. Accuracy and Precision
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Course Name
Unit # – Lesson #.# – Lesson Name
Accuracy and Precision
Ideally, a measurement device is both accurate and precise
Accuracy is dependent on calibration to a standard
Correctness
Poor accuracy results from procedural or equipment flaws
Poor accuracy is associated with systematic errors
Precision is dependent on the capabilities of the measuring device and its use
Reproducibility
Poor precision is associated with random error

10. Presentation Name
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Unit # – Lesson #.# – Lesson Name
Your Turn
Two students each measure the length of a credit card four times. Student A measures with a plastic ruler, and student B measures with a precision measuring instrument called a micrometer.
Student A
Student B
85.1mm mm
85.0 mm mm
85.2 mm mm
84.9 mm
To illustrate the difference between these two terms, let’s look at an example. [click]
[The following slides align with exercises in the corresponding activity. You may wish to have students complete the exercises as you progress through the slides.]

11. Your Turn Plot Student A’s data on a number line
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Course Name
Unit # – Lesson #.# – Lesson Name
Your Turn
Plot Student A’s data on a number line
Plot Student B’s data on a number line
Student A
Student B
85.1mm mm
85.0 mm mm
85.2 mm mm
85.1 mm
[click] Plot student A’s data on the number line [click].
[click] Plot student B’s data on the number line [click].

12. Your Turn Accepted Value
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Your Turn
Student A’s data ranges from 85.0 mm to 85.2 mm
Student B’s data ranges from mm to mm
The accepted length of the credit card is mm
Accepted
Value
[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
85.105

13. Your Turn Which student’s data is more accurate?
Presentation Name
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Unit # – Lesson #.# – Lesson Name
Your Turn
Which student’s data is more accurate?
Which student’s data is more precise?
Student A
Student B
[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. Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Quantifying Accuracy
The accuracy of a measurement is related to the error between the measurement value and the accepted value
mean of
Error = measured value – accepted value s
Student A:
x A = mm
Student A
Student B
85.1mm mm
85.0 mm mm
85.2 mm mm
85.1 mm
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].
Student B:
x B = mm

15. Quantifying Accuracy Calculate the error of Student A’s measurements
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Course Name
Unit # – Lesson #.# – Lesson Name
Quantifying Accuracy
Calculate the error of Student A’s measurements
Error A = mean of measured values – accepted value
Error A = mm – mm = − mm
Error
x A =
Accepted
Value
85.105

16. Quantifying Accuracy Calculate the error of Student B’s measurements
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Quantifying Accuracy
Calculate the error of Student B’s measurements
Error B = mean of measured values – accepted value
Error B = mm – mm = mm
Error
x A =
Error
0.1948
x B=
Accepted
Value
85.105

17. Quantifying Accuracy Calculate the error of Student B’s measurements
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Quantifying Accuracy
Calculate the error of Student B’s measurements
Error B = mean of measured values – accepted value
Error B = mm – mm = mm
Error
| |
= 0.005
Error
x A =
Error
|0.1948| =
Error
0.1948
x B=
When we compare errors to determine the most accurate measurement data set, we compare the absolute value of the error. In other words, the direction of error is unimportant.
Accepted
Value
85.105

18. Quantifying Accuracy Calculate the error of Student B’s measurements
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Quantifying Accuracy
Calculate the error of Student B’s measurements
Error B = mean of measured values – accepted value
Error B = mm – mm = mm
Error
| |
= 0.005
Error
x A =
Error
|0.1948| =
Error
0.1948
x B=
Student A
MORE ACCURATE
When we compare errors to determine the most accurate measurement data set, we compare the absolute value of the error. In other words, the direction of error is unimportant.
Accepted
Value
85.105

19. 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.

20. 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
Student A:
sA= mm
Student A
Student B
85.1mm mm
85.0 mm mm
85.2 mm mm
85.1 mm
[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.
Student B:
sB = mm

21. Quantifying Precision
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Unit # – Lesson #.# – Lesson Name
Quantifying Precision
Use the empirical rule to express precision
True value is within one standard deviation of the mean with 68% confidence
True value is within two standard deviations of the mean with 95% confidence
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 (using plus/minus notation) and a level of confidence that the true measurement value is within that range.
[click] So, for example, you can state with 68% confidence that the true value of a measurement is within one standard deviation based on multiple measurements of that quantity.
[click] Likewise, you can state with 95% confidence that the true value of a measurement is within two standard deviations of the mean.

22. Quantifying Precision
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Course Name
Unit # – Lesson #.# – Lesson Name
Quantifying Precision
Express the precision indicated by Student A’s data at the 68% confidence level
True value is ± 0.08 mm with 68% confidence
85.10 − 0.08 mm ≤ true value ≤ mm
85.02 mm ≤ true value ≤ mm 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.
Student A:
x A= mm
sA= mm

23. Quantifying Precision
Presentation Name
Course Name
Unit # – Lesson #.# – Lesson Name
Quantifying Precision
Express the precision indicated by Student A’s data at the 95% confidence level
True value is ± 2(0.08) mm with 95% confidence
85.10 − 0.16 mm ≤ true value ≤ mm
84.94 mm ≤ true value ≤ mm with 95% confidence
Follow the same procedure to write a statement indicating the precision with 95% confidence. In this case you will add/subtract two standard deviations from the mean to obtain the minimum and maximum values in the range.
Student A:
x A= mm
sA= mm

24. The Statistics of Accuracy and Precision
Introduction to Summary Statistics
The Statistics of Accuracy and Precision A B
Low Accuracy
High Precision
High Accuracy
High Precision
Students used four different measuring devices to measure the same length and recorded the data. This slide represent the measurement data collected from each device (A through D) with a normal curve. The scale for each graph is the same.
Which graph represents relatively high accuracy and high precision? [Allow students time to answer and then click.] The answer is B. The data here shows high accuracy because the mean of the data is the same as the true value. The data recorded implies that the instrument is precise because the standard deviation is small compared to the graphs for C and D. This is indicated by the fact that the normal curve in graph B is narrower than the curves in C and D.
Which graph represents relatively low accuracy and low precision? [Allow students time to answer and then click.] The answer is C. Graph C indicates that the mean of the data is offset from the true value of the measurement; therefore the C instrument appears to be less accurate than A and B. In addition, the normal curve represented in C is wider that those shown in A and B indicating a larger standard deviation for the data collected using instrument C than the instruments A and B, and therefore less precision.
Describe the relative accuracy and precision of instrument A. [Allow students time to answer then click.]
Describe the relative accuracy and precision of instrument D. [Allow students time to answer then click.] C D
Low Accuracy
Low Precision
High Accuracy
Low Precision

25. Gauge Blocks (Gage Blocks)
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Gauge Blocks (Gage Blocks)
A block whose length is precisely and accurately known
Standard = basis of comparison
Precision measuring devices are often calibrated using gauge blocks
Calibrate = to check or adjust by comparison to a standard
[click] A gauge block can be used to help determine the accuracy of a measuring device. A gauge block is a block made of metal or ceramic with two opposing faces that are precisely ground (to be flat and parallel) at a precise distance apart.
[click] A gauge block is referred to as a standard because it is used as a basis of comparison. [Show students an example of a gage block]
[click] Precision measuring devices, such as a caliper, can be calibrated using a gauge block.
[click] Calibrate means to check and/or adjust by comparison to a standard. The measuring device measurement reading can be adjusted to agree with the precise size of the gauge block.
You will calibrate a dial caliper in a following activity using a gauge block. Every student in the class will measure the gauge block using the same dial caliper. You will analyze the data and then assess the accuracy and precision of the instrument.