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

Recording Measurements Presentation Name Course Name 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.

Two Sources of Error in Measurement Presentation Name Course Name Unit # – Lesson #.# – Lesson Name 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

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

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

Uncertainty in Measurements Presentation Name Course Name 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.

Uncertainty in Measurements Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Uncertainty in Measurements Another (more definitive) method to indicate uncertainty is to use plus/minus notation Example: 3.84 ± .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.

Uncertainty in Measurement Presentation Name 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.

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 85.301 mm 85.0 mm 85.298 mm 85.2 mm 85.299 mm 85.1 mm USE DIGITAL CALIPER MIS-ZEROED (about 1 mm off)

Your Turn Plot ruler & caliper data on a number line RULER CALIPER Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Your Turn RULER CALIPER 85.1mm 85.301 mm 85.0 mm 85.298 mm 85.2 mm 85.299 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].

Your Turn Accepted Value Ruler data ranges from 85.0 mm to 85.2 mm Presentation Name Course Name Unit # – Lesson #.# – Lesson Name Your Turn Ruler data ranges from 85.0 mm to 85.2 mm Caliper data ranges from 85.298 mm to 85.301 mm The accepted length of the credit card is 85.105 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 85.298 to 85.301 mm. [click] The accepted value of the measurement is 85.105

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.]

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.]

Quantifying PRECISION Presentation Name 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.

Quantifying Precision Presentation Name 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 85.301 mm 85.0 mm 85.298 mm 85.2 mm 85.299 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 0.0013 mm. Caliper s: sC =

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.

Quantifying Precision Presentation Name 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 85.10 and a standard deviation of 0.08. Therefore we can write a statement to indicate the precision of the measurements as 85.10 +/- 0.08 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.

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 85.10 and a standard deviation of 0.08. Therefore we can write a statement to indicate the precision of the measurements as 85.10 +/- 0.08 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.

Quantifying Precision: Assignment Presentation Name 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 …

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

Presentation Name Course Name 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 85.301 mm 85.0 mm 85.298 mm 85.2 mm 85.299 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].

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

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

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 2.000 inch GAGE BLOCK & CALIBRATION MASSES

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 2.000 inch GAGE BLOCK & CALIBRATION MASSES