Measurements and error in experiments
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Every measurement has some level of uncertainty that should be reported along with the measured value. There are several ways to determine the uncertainty, including making repeated measurements and reporting the standard deviation, or estimating uncertainty based on the precision of the measuring instrument. Accuracy refers to how close a measurement is to the true value, while precision refers to how reproducibly a measurement can be made. When reporting measurements, the number of significant figures should be based on the precision of the measuring device and include an estimated uncertainty.
Measurements and error in experiments
- 1. 1 Measurements and Error in Experiments Source: http://www.cis.rit.edu/class/simg215/SIMG-215-lab-reports.pdf The Certainty of Uncertainty Every measurement exhibits some associated uncertainty. A good experimentalist will try to evaluate the level of uncertainty, which is specified along with the measured value (e.g., 10.4 ± 0.1mm), which are then graphed as “error bars” along with the measured numerical value). In these labs, it is very important for you to estimate the uncertainty. In particular, it is helpful to decide which of the three possible categories mentioned in the introduction that applies to your data. How can you do this? One way is to make what you think is a reasonable decision about the uncertainty. For example, if you measure a length with a ruler, maybe you can only measure the length to some fraction of the smallest rule division. Another way is to measure the uncertainty. That is, take the same measurement multiple times. The average value is then used as the final measurement, and the uncertainty is related to the standard deviation of the individual values. We will discuss these ideas more thoroughly as the class goes on, but you should be prepared to estimate uncertainties for as many measurements as you can. The measurements are usually expressed as the mean value μ plus or minus (±) the uncertainty, which generally is standard deviation σ of the measurement. Be sure to specify the units used in any measurements! Also watch the number of significant figures; just because your spreadsheet calculates to 7 or 8 places does not mean that this level of precision is merited! Accuracy vs. Precision The “accuracy” of an experiment refers to how closely the result came to the “true” value. The “precision” measures how exactly the result was determined, and thus how reproducible the result is. Significant Figures and Round-Off Error You have probably already learned the definition of the number of significant figures: 1. The MOST significant digit is the leftmost nonzero digit in the number 2. If there is no decimal point, then the LEAST significant digit is the rightmost nonzero digit 3. If there is a decimal point, then the LEAST significant digit is the rightmost digit, even if it is a zero. 4. All digits between the most and least significant digit are “significant digits.”
- 2. 2 In the result of a measurement, the number of significant digits should be one larger than the precision of any analog measuring device. In other words, you should be able to read an analog scale with more precision than given by the scale; if the scale is labeled in millimeters, you should be able to estimate the measurement to about a tenth of a millimeter. Retain this extra digit and include an estimate of the error, e.g., for a measurement of a length s, you might report the measurement as s = 10.3mm± 0.2mm Rounding Rules Source: http://www.nrel.gov/biomass/pdfs/42626.pdf 1. When the digit following the last the last significant digit is less than 5, the number remains unchanged (round down.) Example: 1.63 is rounded to 1.6 2. When the digit following the last the last significant digit is greater than 5, the digit to be retained is increased by 1. Example: 2.6 is rounded to 2.7 3. When the digit following the last significant digit equals 5, the number is rounded off to the nearest even number. Example: 3.85 is rounded to 3.8, 4.55 is rounded to 4.6 Note: Some automated systems such as spreadsheets, calculators and data management software always round up a five. 4. When calculations are complete and two or more figures are to the right of the last the last significant digit, rounding begins at the least certain digit and continues until the correct number of digits remains. Example: Rounding to two significant figures: 2.4501 is viewed as 2.4(501) and is rounded to 2.5 2.5499 is viewed as 2.5(499) and is rounded to 2.5 It follows that the reported value of the mean (or other value) should be of the same order of magnitude as the experimental uncertainty. A very useful reference book on this subject is Data Reduction and Error Analysis for the Physical Sciences (Third Edition), by Philip Bevington and D. Keith Robinson (McGraw-Hill, 2002, ISBN0072472278). It is not unreasonable to say that this classic book should be on the shelf of every scientist.