Facets of Thinking about Measurements

The following Facets are an extension of the list developed by Jim Minstrell, et. al. Below is an excerpt from the Facets website:

This coding scheme was begun in 1987. It is based on a "Knowledge in Pieces" perspective derived from our earlier attempts to construct frameworks to organize students' conceptual understanding and from a theoretical view developed by A. diSessa. The Facet codes are slight abstractions of what students say or do when confronted with a situation in which they are asked to predict or explain a physical phenomenon. Although our research has investigated students' conceptual understanding, many of the Facets have been identified in the research done by others. We can not possibly acknowledge all those researchers who have contributed. The Facet Codes are our attempt to organize the phenomena of students' conceptual understanding.
Within a Facet Cluster, the codes ending in 9 are associated with the Facets that seem to be the most problematic. They are typically the ideas we choose to address first in our instruction. The codes ending in 0 or 1 represent understandings that are probably "OK" at this introductory level. The others are roughly rank ordered between the 9 and 1. These facets are descriptions of characteristic understanding and reasoning and are not intended to be used directly as a numerical scoring scheme.

The Facets listed here are based on the research conducted by Duane Deardorff on assessing introductory physics students' understanding about measurement uncertainty. These Facets include many of the common student difficulties documented by this research, beliefs held by instructors, and intermediate conceptions that lie within the continuum between novice and expert understanding.

* Indicates acceptable statements that are consistent with expert beliefs; higher numbers denote novice beliefs.

Best Value in Measurement

*0 Use a procedure appropriate to the context to obtain a representative value of the desired quantity.
*1 Average several measures of the desired quantity, eliminating outliers if justified.
2 The median is the best measure of central tendency.
5 Average all the measures obtained, including outliers.
6 Eliminate the high and low values and average the remaining values (trimmed mean).
8 "Mine" (measure made by the student doing the reporting) is the most valid measure.
9 A value provided by an authority is the most accurate, best, or "actual" value.

The Nature of Uncertainty in Measurements

*0 All measurements have some degree of uncertainty, no matter how carefully the measurement was obtained
*1 The uncertainty in a measurement is limited by the precision of the measuring device and the capability of the experimenter
8 "Theoretical" values are exact and do not have uncertainty
9 Exact measurements can be made if high-quality equipment is used and there is no "human error" (mistakes made by the person taking the measurement)

Rationale for Reporting the Uncertainty in a Measurement

* 0 A scientist has an ethical obligation to clearly communicate to others how well a measured value is known.
* 1 Reasonably accurate uncertainty estimates are necessary for deciding if two values agree with each other. (hypothesis testing)
9 Calculating and reporting uncertainties is a waste of time and not important.

Determining and Reporting Measurement Uncertainty

*0 Report the combined standard uncertainty (U) from Type A (statistical) and Type B (other) evaluations of standard uncertainty (per ISO guidelines, 1993). The combined standard uncertainty roughly corresponds to a 68% confidence interval (one sigma) and combines estimates of both the accuracy and precision of the measured value from all suspected sources of error, some of which may be reasonable guesses.
*1 A measured value should be reported with the standard uncertainty (u) associated with this measurement, along with an explanation of what this uncertainty means and how it was determined.
*2 Report the measured value and an uncertainty range equivalent to a 68% confidence interval, or some other expanded uncertainty from a coverage factor that is explicitly defined.
*3 Report the variability in the data in some way appropriate to the context.
5 The reported uncertainty should cover the entire range of the variation in the measured value.
8 The uncertainty is always half the smallest division on the measuring tool. (instrument precision)
9 There is no uncertainty (zero error) in measurement if one is knowledgeable, uses a good instrument, and is careful.

Reporting of Significant Figures

*0 The last digit in a reported value is the least significant digit.
*1 Significant figures show an approximate degree of relative uncertainty (precision).
*2 The uncertainty of a measured value should be used to determine the number of significant figures reported for that value; insignificant digits should not be reported.
*3 Uncertainty values should be reported to only one or two significant figures to be consistent with the inherent uncertainty of these error estimates.
*4 Round-off error can result from using too few significant figures in intermediate calculations.
6 All the digits given by calculation should be reported to avoid losing information.
8 All the digits given by a measurement instrument are significant and should be reported.
9 All the digits given by calculation are significant and should be reported.

Best Value in Calculations

*1 Appropriately calculate the best value using the best values of the individual parts of the calculation.
4 Calculate the average of all the extremes, using the uncertainties for each measure involved.
5 Calculate the average of the two calculated extremes.
9 The value calculated from single measures of each involved quantity.

Propagation of Uncertainty in Calculations

*0 The uncertainty in a calculated value depends on the uncertainties associated with each term used to compute the result.
*1 The combined uncertainty should be found by adding the variances of the input quantities according to the propagation of uncertainty equation.(adding errors in quadrature or RMS method)
*2 A conservative but simple estimate of the uncertainty in a calculated value can be found by computing the maximum and minimum values based on the uncertainties of each term. (max/min method)
*3 The "rules" of significant figures for multiplication, division, addition, and subtraction are an efficient means of propagating uncertainty and estimating the appropriate degree of precision in a calculated result, but these rules are often invalid for other mathematical functions like exponentials, logorithms, and trigonometric functions.
*4 Uncertainty calculations can often be simplified by ignoring small errors that contribute to the total uncertainty.
5 The overall uncertainty is the largest uncertainty of the terms involved in a calculation.
6 The uncertainty in a multiplication calculation is the sum of the individual uncertainties.
7 The uncertainty in a calculation is found by applying the same mathematical function to the uncertainties as is done for the result. (e.g. When finding the mean, compute the average of the individual uncertainties.)
8 Propagation of uncertainties is time consuming, difficult, and not worth the effort.
9 There is no uncertainty in the calculation if no mistakes are made.

Identifying and Reducing Sources of Error

*0 Measurement errors can arise from a variety of sources.
*1 Errors can be classified as "random" or "systematic", "Type A" or "Type B."
*2 Measurement errors are unavoidable and should not be confused with procedural mistakes or blunders that can be avoided or corrected if discovered.
*3 Averaging over multiple measurements reduces random error, but not systematic errors.
*4 An attempt should be made to rank the sources of error in order to determine which factor contributes most to the overall uncertainty in a result. (error budget) This analysis can be used to design or improve an experimental procedure.
9 "Human error" is often a significant source of error.

Evaluating Agreement between Measured Values

*0 Comparison of measured values requires knowledge about their uncertainty. Two values agree if the difference between them is less than their combined uncertainty.
*1 A statistical t-test is used to determine if there is a significant difference between two measured values.
*2 Values agree if their uncertainty ranges overlap; error bars plotted on a number line help to visualize this.
3 When comparing results, a percent difference or percent error should be calculated and reported.
9 Agree if percent error is less than some arbitrary value (e.g. OK if less than 10% error), without considering the amount of uncertainty associated with the measured value.

Last revised: 2/2/01 by Duane Deardorff