Part I of Fundamental Measurements: Uncertainties in addition to Error Propagation Simple Content 1. Systematic & R in addition to om Errors

Part I of Fundamental Measurements: Uncertainties in addition to Error Propagation Simple Content 1. Systematic & R in addition to om Errors www.phwiki.com

Part I of Fundamental Measurements: Uncertainties in addition to Error Propagation Simple Content 1. Systematic & R in addition to om Errors

Kim, Gina, Features Reporter has reference to this Academic Journal, PHwiki organized this Journal Part I of Fundamental Measurements: Uncertainties in addition to Error Propagation http://www.rit.edu/~uphysics/uncertainties/Uncertaintiespart2.html Vern Lindberg, Copyright July 1, 2000 Contents-1 of Uncertainties in addition to Error Propagation 1. Systematic vs R in addition to om Errors () 2. Determining R in addition to om Errors Instrument Limit of Error, least count (, ) Estimation () Average Deviation () & St in addition to ard deviation () Conflicts St in addition to ard Error in the Mean () 3. What does uncertainty tell me Range of possible values 4. Relative in addition to Absolute error () Contents-2 of Uncertainties in addition to Error Propagation 5. Propagation of errors () add/subtract (/) multiply/divide (/) powers () mixtures of +-/ () other functions 6. Rounding answers properly () 7. Significant figures () 8. Problems to try 9. Glossary of terms (all terms that are bold face in addition to underlined)

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Simple Content 1. Systematic in addition to r in addition to om errors. 2. Determining r in addition to om errors. 3. What is the range of possible values 4. Relative in addition to Absolute Errors 5. Propagation of Errors, Basic Rules 1. Systematic & R in addition to om Errors No measurement made is ever exact. The accuracy (correctness) in addition to precision (number of significant figures) of a measurement are always limited: by the degree of refinement of the apparatus used, by the skill of the observer, in addition to by the basic physics in the experiment. In doing experiments we are trying to establish the best values as long as certain quantities, or to validate a theory. We must also give a range of possible true values based on our limited number of measurements. Why should repeated measurements of a single quantity give different values Mistakes on the part of the experimenter are possible, but we do not include these in our discussion. A careful researcher should not make mistakes! (Or at least she or he should recognize them in addition to correct the mistakes.)

Accuracy (correctness) & Precision (number of significant figures) Uncertainty, error, or deviation – the synonymous terms to represent the variation in measured data. Two types of errors are possible: 1. Systematic error: 2. R in addition to om errors Systematic Errors The result of A mis-calibrated device, or A measuring technique which always makes the measured value larger (or smaller) than the “true” value. Example: Using a steel ruler at liquid nitrogen temperature to measure the length of a rod. The ruler will contract at low temperatures in addition to there as long as e overestimate the true length. Careful design of an experiment will allow us to eliminate or to correct as long as systematic errors. R in addition to om Errors These remaining deviations will be classed as r in addition to om errors, in addition to can be dealt with in a statistical manner. This document does not teach statistics in any as long as mal sense. But it should help you to develop a working methodology as long as treating errors.

2. Determining r in addition to om errors Several approaches are used to estimate the uncertainty of a measured quantity. (a) Instrument Limit of Error (ILE) in addition to Least Count Least count: the smallest division that is marked on the instrument. – A meter stick will have a least count of 1.0 mm, – A digital stop watch might have a least count of 0.01 s. Instrument limit of error (ILE): the precision to which a measuring device can be read, in addition to is always equal to or smaller than the least count. – Very good measuring tools are calibrated against st in addition to ards maintained by the National Institute of St in addition to ards in addition to Technology. Instrument Limit of Error, ILE Be generally taken to be the least count or some fraction (1/2, 1/5, 1/10) of the least count. Which to choose, the least count or half the least count, or something else. No hard in addition to fast rules are possible, instead you must be guided by common sense. If the space between the scale divisions is large, you may be com as long as table in estimating to 1/5 or 1/10 of the least count. If the scale divisions are closer together, you may only be able to estimate to the nearest 1/2 of the least count, in addition to if the scale divisions are very close you may only be able to estimate to the least count. For some devices the ILE is given as a tolerance or a percentage. – Resistors may be specified as having a tolerance of 5%, meaning that the ILE is 5% of the resistor’s value. Problem: For each of the following scales (all in centimeters) determine the least count, the ILE, in addition to read the length of the gray rod.

Problem: to determine the least count, the ILE, in addition to read the length of the gray rod as long as each of the following scales (all in centimeters). (b) Estimated Uncertainty Often other uncertainties are larger than the ILE. Try to balance a simple beam balance with masses that have an ILE of 0.01 grams, but find that the mass on one pan vary by as much as 3 grams without seeing a change in the indicator. To use half of this as the estimated uncertainty, thus getting uncertainty of ±1.5 grams. Another good example is determining the focal length of a lens by measuring the distance from the lens to the screen. The ILE may be 0.1 cm, however the depth of field may be such that the image remains in focus while we move the screen by 1.6 cm. the estimated uncertainty would be half the range or ±0.8 cm. Problem: I measure your height while you are st in addition to ing by using a tape measure with ILE of 0.5 mm. Estimate the uncertainty. Include the effects of not knowing whether you are “st in addition to ing straight” or slouching. There are many possible correct answers to this. However the answer h = 0.5 mm is certainly wrong. Here are some of the problems in measuring. As you st in addition to , your height keeps changing. You breath in in addition to out, shift from one leg to another, st in addition to straight or slouch, etc. I bet this would make your height uncertain to at least 1.0 cm. Even if you do st in addition to straight, in addition to don’t breath, I will have difficulty measuring your height. The top of your head will be some horizontal distance from the tape measure, making it hard to measure your height. I could put a book on your head, but then I need to determine if the book is level. I would put an uncertainty of 1 cm as long as a measurement of your height.

Average Example 1 Problem Find the average, in addition to average deviation as long as the 5 following data on the length of a pen, L. To get the average : sum the values in addition to divide by the number of measurements. To get the average deviation L, Find the absolute values of the deviations, L – Lave Sum the absolute deviations, Get the average absolute deviation by dividing by the number of measurements To get the st in addition to ard deviation Find the deviations in addition to square of them Sum the squares Divide by (N-1), (here it is 4) Take the square root. The pen has a length of (12.22 + 0.14) cm or (12.2 + 0.1) cm [use average deviations] Or (12.22 + 0.22) cm or (12.2 + 0.2) cm [use st in addition to ard deviations]. Average Example 2 Problem: Find the average in addition to the average deviation of the following measurements of a mass. This time there are N = 6 measurements, so as long as the st in addition to ard deviation we divide by (N-1) = 5. The mass is (4.342 + 0.022) g or (4.34 + 0.02) g [using average deviations] or (4.342 + 0.023) g or (4.34 + 0.02) g [using st in addition to ard deviations].

(c) Average Deviation – Statistical method: Estimated Uncertainty by Repeated Measurements To repeat the measurement several times, find the average, in addition to find either the average deviation or the st in addition to ard deviation. Suppose we repeat a measurement several times in addition to record the different values. We can then find the average value, here denoted by a symbol between angle brackets, , in addition to use it as our best estimate of the reading. How can we determine the uncertainty Let us use the following data as an example. Column 1 shows a time in seconds. Average: the sum of all values (7.4+8.1+7.9+7.0) divided by the number of readings (4), which is 7.6 sec. Column 2 of Table 1 shows the deviation of each time from the average, (t – ). A simple average of these is zero, in addition to does not give any new in as long as mation. Average deviation, t: To get a non-zero estimate of deviation we take the average of the absolute values of the deviations, as shown in Column 3. St in addition to ard deviation: Column 4 has the squares of the deviations from Column 2, making the answers all positive. The sum of the squares is divided by 3, (one less than the number of readings), in addition to the square root is taken to produce the sample. An explanation of why we divide by (N-1) rather than N is found in any statistics text. The sample st in addition to ard deviation is slightly different than the average deviation, but either one gives a measure of the variation in the data.

Built-in Functions in Excel use a spreadsheet such as Excel there are built-in functions that help you to find these quantities. To round the uncertainty to one or two significant figures (more on rounding in Section 7), in addition to To round the average to the same number of digits relative to the decimal point. Thus the average length with average deviation is either (15.47 ± 0.13) m or (15.5 ± 0.1) m. If we use st in addition to ard deviation we report the average length as (15.47±0.18) m or (15.5±0.2) m. Follow your instructor’s instructions on whether to use average or st in addition to ard deviation in your reports.

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Problem Find the average, in addition to average deviation as long as the following data on the length of a pen, L. We have 5 measurements, (12.2, 12.5, 11.9,12.3, 12.2) cm. Solution Problem: Find the average in addition to the average deviation of the following measurements of a mass. (4.32, 4.35, 4.31, 4.36, 4.37, 4.34) grams. Solution (d) Conflicts in the above In some cases we will get an ILE, an estimated uncertainty, in addition to an average deviation in addition to we will find different values as long as each of these. Be pessimistic in addition to take the largest of the three values as our uncertainty. For statistics, a more correct approach involving adding the variances. For example we might measure a mass required to produce st in addition to ing waves in a string with an ILE of 0.01 grams in addition to an estimated uncertainty of 2 grams. Using 2 grams as the uncertainty. The proper way to write the answer is Choose the largest of (i) ILE, (ii) estimated uncertainty, in addition to (iii) average or st in addition to ard deviation. Round off the uncertainty to 1 or 2 significant figures. Round off the answer so it has the same number of digits be as long as e or after the decimal point as the answer. Put the answer in addition to its uncertainty in parentheses, Then put the power of 10 in addition to unit outside the parentheses. Problem: I measure a length with a meter stick with a least count of 1 mm. I measure the length 5 times with results (in mm) of 123, 123, 123, 123, 123. What is the average length in addition to the uncertainty in length Answer

Example One make several measurements on the mass of an object. The balance has an ILE of 0.02 grams. The average mass is 12.14286 grams, the average deviation is 0.07313 grams. What is the correct way to write the mass of the object including its uncertainty What is the mistake in each incorrect one Answer 12.14286 g (12.14 ± 0.02) g 12.14286 g ± 0.07313 (lack of unit) 12.143 ± 0.073 g (12.143 ± 0.073) g (12.14 ± 0.07) (12.1 ± 0.1) g 12.14 g ± 0.07 g The correct answer is (12.14 ± 0.07) g. (e) Why make many measurements St in addition to ard Error in the Mean (SEM,) Is there any point to measuring a quantity more often than this Based on statistics, the st in addition to ard error in the mean is affected by the number of measurements made. It is defined as the st in addition to ard deviation divided by the square root of the number of measurements. Notice that the average in addition to st in addition to ard deviation do not change much as the number of measurements change. But that the st in addition to ard error does dramatically decrease as N increases. Finding St in addition to ard Error in the Mean For this introductory course in addition to most cases, we will not worry about the st in addition to ard error, but only use the st in addition to ard deviation, or estimates of the uncertainty.

Short rule as long as multiplication & division The answer will contain a number of significant figures equal to the number of significant figures in the entering number having the least number of significant figures. How many significant figures is 2.3 x 3.413 = 2.3 had 2 significant figures while 3.413 had 4, so the answer is given to 2 significant figures. It is important to keep these concepts in mind as you use calculators with 8 or 10 digit displays if you are to avoid mistakes in your answers in addition to to avoid the wrath of physics instructors everywhere. A good procedure: is to use all digits (significant or not) throughout calculations, in addition to only round off the answers to appropriate “sig fig.” Problem How many significant figures are there in each of the following (1) 0.00042 (2) 0.14700 (3) 4.2 x 106 (4) -154.090 x 10-27 8. Problems on Uncertainties in addition to Error Propagation. 8. Seven Problems on Uncertainties in addition to Error Propagation

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