significant figures and unit

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Feb 20, 2024

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Experimental Error and Uncertain No measurement is perfect so how do you know how good the measurement is? Can you trust the value if you continue to calculate with it? These are questions that prompted the study of experimental error. There are several topics involved in our study today – Determine Random or Statistical Error vs. Systematic Error Know the difference between Accuracy vs. Precision and use them correctly Explain the Accuracy of Measuring Devices Use Percent Error and Percent Difference in correct circumstances Find the Average or Mean Take the Standard Deviation from the Mean Review and use Significant Digits Correctly Random vs. Systematic Error (also referred to as indeterminate error and determinate error) Observational error (or measurement error ) is the difference between a measured value of quantity and its true value. In statistics , an error is not a "mistake". Variability is an inherent part of things being measured and of the measurement process. When either randomness or uncertainty modeled by probability theory is attributed to such errors, they are "errors" in the sense in which that term is used in statistics ; see errors and residuals in statistics . Every time we repeat a measurement with a sensitive instrument, we obtain slightly different results. The common statistical model we use is that the error has two additive parts: 1. systematic error which always occurs, with the same value, when we use the instrument in the same way and in the same case, and 2. random error which may vary from observation to observation. Systematic error is sometimes called statistical bias. It may often be reduced by very carefully standardized procedures. Part of the education in every science is how to use the standard instruments of the discipline. The random error (or random variation ) is due to factors which we cannot (or do not) control. It may be too expensive or we may be too ignorant of these factors to control them each time we measure. It may even be that whatever we are trying to measure is changing in time (see dynamic models ), or is fundamentally probabilistic (as is the case in quantum mechanics—see Measurement in quantum mechanics ). Random error often occurs when instruments are pushed to their limits. For example, it is common for digital balances to exhibit random error in their least significant digit. Three measurements of a single object might read something like 0.9111g, 0.9110g, and 0.9112g. The section on Random and Systematic Errors is taken from Wikipedia, page https://en.wikipedia.org/wiki/Observational_error . Accuracy vs. Precision

Accuracy and precision are similar but not the same ideas. Accuracy means that you are close to a target place while precision means that all your shots are located in a small pattern which may or may not be close to the target. Here are some examples taken from National Ocean Service | National Oceanic and Atmospheric Administration | U.S. Department of Commerce | USA.gov http://celebrating200years.noaa.gov/magazine/tct/accuracy_vs_precision.html Accuracy of Measuring Devices The measuring device you use determines the accuracy of the measurement you will get. The smallest unit of marking on the device plus one estimated value beyond that creates the number of significant digits that will be measured by that device. One can estimate to the tenths between the smallest two markings on the measuring scale. As an example a meter stick with mm markings on it can be read to the tenth of a millimeter or to an accuracy of .0001 meters. Percent Error and Percent Difference These values are distinct and used in different situations. Percent error, Percent Error ( % Error ) = | Scientifically Accepted Value − Laboratory Measured Value Scientifically Accepted Value | × 100% Is used when a value that is known and agreed upon by the scientific community is used or compared in an experiment. Percent Difference (% Difference) = | Measured Value 1 − Measured Value 2 1 2 ( MeasuredValue 1 + MeasuredValue 2 ) | × 100 % should be used when comparing two values that were measured or calculated during the experiment and supposed to be for the same

item. An example would be a measurement of the acceleration of the ball down an incline comparing it to the calculated value of the acceleration of the ball down the incline. In this case one of the values is a calculation and one is a measurement. The percent difference is used when no scientifically accepted value is known. Finding the Average (or Mean), the deviation from the mean and the standard deviation Finding the average (the mathematical mean) is no different than what you do to calculate the average of your test scores. If you have 5 tests add up all the individual scores and divide by 5. Finding the Standard Deviation from the mean simply means taking each individual test score from this example and subtracting the average score from it. Here is an example with 5 tests included. T1 = 60; T2=80; T3 = 77; T4=82; T5=74 Test Average = T 1 + T 2 + T 3 + T 4 + T 5 5 = 60 + 80 + 77 + 82 + 74 5 = 74.6 To find the deviation of each test from the mean subtract the mean from the test. d T 1 = T 1 − Mean = 60 − 74.6 =− 10.6 d T 2 = T 2 − Mean = 80 − 74.6 = 5.4 d T 3 = T 3 − Mean = 77 − 74.6 = 2.4 d T 4 = T 4 − Mean = 82 − 74.6 = 7.4 d T 5 = T 5 − Mean = 74 − 74.6 =− 0.6 If you are required to find the standard deviation at this point all you need to do is take the average of the individual deviations that you just calculated. In this case the standard deviation is found taking − 10.6 + 5.4 + 2.4 + 7.4 − 0.6 ¿ 5 = 0.8 ¿ . Rules for Significant Digits The significant figures of a number are digits that carry meaning contributing to its measurement resolution . This includes all digits except :  All leading zeros ;  Trailing zeros when they are merely placeholders to indicate the scale of the number (exact rules are explained at identifying significant figures ); and  Spurious digits introduced, for example, by calculations carried out to greater precision than that of the original data, or measurements reported to a greater precision than the equipment supports. Significance arithmetic are approximate rules for roughly maintaining significance throughout a computation. The more sophisticated scientific rules are known as propagation of uncertainty . Numbers are often rounded to avoid reporting insignificant figures. For example, it would create false precision to express a measurement as 12.34500 kg (which has seven significant figures) if the scales only measured to the nearest gram and gave a reading of 12.345 kg (which has five significant figures). Numbers can also be rounded

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merely for simplicity rather than to indicate a given precision of measurement, for example to make them faster to pronounce in news broadcasts. Concise rules  All non-zero digits are significant  Zeros between non-zero digits are significant.  Leading zeros are never significant.  In a number with a decimal point, trailing zeros, those to the right of the last non-zero digit, are significant.  In a number without a decimal point, trailing zeros may or may not be significant. More information through additional graphical symbols or explicit information on errors is needed to clarify the significance of trailing zeros. Significant figures rules explained Specifically, the rules for identifying significant figures when writing or interpreting numbers are as follows:  All non-zero digits are considered significant. For example, 91 has two significant figures (9 and 1), while 123.45 has five significant figures (1, 2, 3, 4 and 5).  Zeros appearing anywhere between two non-zero digits are significant. Example: 101.1203 has seven significant figures: 1, 0, 1, 1, 2, 0 and 3.  Leading zeros are not significant. For example, 0.00052 has two significant figures: 5 and 2.  Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant figures: 1, 2, 2, 3, 0 and 0. The number 0.000122300 still has only six significant figures (the zeros before the 1 are not significant). In addition, 120.00 has five significant figures since it has three trailing zeros. This convention clarifies the precision of such numbers; for example, if a measurement precise to four decimal places (0.0001) is given as 12.23 then it might be understood that only two decimal places of precision are available. Stating the result as 12.2300 makes clear that it is precise to four decimal places (in this case, six significant figures).  The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if a number like 1300 is precise to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty. As there are rules for determining the number of significant figures in directly measured quantities, there are rules for determining the number of significant figures in quantities calculated from these measured quantities. Only measured quantities figure into the determination of the number of significant figures in calculated quantities . Exact mathematical quantities like the π in the formula for the area of a circle with radius r , π r 2 has no effect on the number of significant figures in the final calculated area. Similarly the ½ in the formula for the kinetic energy of a mass m with velocity v , ½ mv 2 , has no bearing on the number of significant figures in the final calculated kinetic energy. The constants π and ½ are considered to have an infinite number of significant figures. For quantities created from measured quantities by multiplication and division , the calculated result should have as many significant figures as the measured number with the least number of significant figures. For example,

1.234 × 2.0 = 2.468… ≈ 2.5, with only two significant figures. The first factor has four significant figures and the second has two significant figures. The factor with the least number of significant figures is the second one with only two, so the final calculated result should also have a total of two significant figures. For quantities created from measured quantities by addition and subtraction , the last significant decimal place (hundreds, tens, ones, tenths, and so forth) in the calculated result should be the same as the leftmost or largest decimal place of the last significant figure out of all the measured quantities in the terms of the sum. For example, 100.0 + 1.234 = 101.234… ≈ 101.2 with the last significant figure in the tenths place. The first term has its last significant figure in the tenths place and the second term has its last significant figure in the thousandths place. The leftmost of the decimal places of the last significant figure out of all the terms of the sum is the tenths place from the first term, so the calculated result should also have its last significant figure in the tenths place. The rules for calculating significant figures for multiplication and division are opposite to the rules for addition and subtraction. For multiplication and division, only the total number of significant figures in each of the factors matter; the decimal place of the last significant figure in each factor is irrelevant. For addition and subtraction, only the decimal place of the last significant figure in each of the terms matters; the total number of significant figures in each term is irrelevant. The section on Significant Digits was taken from Wikipedia, page https://en.wikipedia.org/wiki/Significant_figures with original citations on that page from 1. Chemistry in the Community ; Kendall-Hunt:Dubuque, IA 1988 2. Giving a precise definition for the number of correct significant digits is surprisingly subtle, see Higham, Nicholas (2002). Accuracy and Stability of Numerical Algorithms (PDF) (2nd ed.). SIAM. pp. 3–5. 3. Myers, R. Thomas; Oldham, Keith B.; Tocci, Salvatore (2000). Chemistry. Austin, Texas: Holt Rinehart Winston. p. 59. ISBN 0-03-052002-9 . 4. Numerical Mathematics and Computing, by Cheney and Kincaid . 5. de Oliveira Sannibale, Virgínio (2001). "Measurements and Significant Figures (Draft)" (PDF) . Freshman Physics Laboratory. California Institute of Technology, Physics Mathematics And Astronomy Division. Archived from the original (PDF) on June 18, 2013. The Wikipedia material is free to use under the Creative Commons Attribution- Share Alike License, information found at this website, http://creativecommons.org/licenses/by-sa/3.0/

Name: ___________Laith Alshaikh________________________________ Date: _August29th _________________________ Purpose: In this lab we learned about significant figures Procedure: for this lab we answered questions Pre-Lab Questions 1. Why can’t measurements be exact? Regardless of precision and accuracy, all measurements are subject to some degree of uncertainty. This is due to two things: the measuring device's limitations (systematic error) and the measurement skills of the experimenter (random error). 2. Give examples of the differences between random and systematic errors. Random errors occur because of random and inherently unpredictable events in the measurement process. Systematic errors occur when there is a problem in the measurement system that affects all measurements in the same way. 3. What is the difference between precision and accuracy? Can you have one without the other? Precision is the degree to which measurements of the same thing agree with one another. Accuracy is not necessary for precision. In other words, it is possible to be accurate without being exact as well as to be highly precise but not particularly accurate. The highest caliber scientific observations are exact and accurate. 4. What determines how many figures are significant in a measurement? What would be the effect of reporting more or fewer figures or digits than are significant? You must submit your measurements' results with the appropriate number of significant digits (also called significant figures.) Whether a digit indicates a real measurement or not determines its importance. With the specific measuring tool you are employing, any digit that can be precisely measured or reasonably guessed is regarded as important. When utilizing a digital measuring instrument, it is simple to figure out how many significant digits there are; just assume that all of the numbers displayed on the display are important. 5. Give an example when percent error would be appropriate and an example in which it would be more appropriate to use percent difference. When comparing two experimental quantities—accepted value 1 and measured value—neither of which may be regarded as the "right" value, the percentage of difference is used. The absolute value of the difference over the mean multiplied by 100 is the percent difference. Applied when contrasting a theoretical amount, initial, with an experimental quantity, final, which is thought to be the "right" value. The absolute difference divided by the "right" value multiplied by 100 yields the percent mistake.

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Experimental Error and Uncertainty Lab Report 1. Using Significant Figures  In the first column of the table below, express the given numbers with three significant figures. Then, express the given numbers in the second column in terms of scientific notation with three significant figures (powers-of-10).  Calculate the volume of a wooden block that measures 11.2 cm × 3.4 cm × 4.10 cm. Indicate doubtful figures by underlining them throughout the calculations. Report the calculated volume with the correct number of significant figures using scientific notation. Show calculations and don’t forget units. Calculations: Calculated Volume ___1.56 x 10^2_____________________ Volume of woden block = lwh=11.2*3.4* 4. 10 =156.128  In an effort to find an experimental value for π, a series of measurements were made on a cylinder. The average measurement for its diameter was 4.25 cm and the average measurement for the circumference was 13.39 cm. 5280 _________5.28e+3_____________ 0.060 ___6.0e- 2__________________ 82.453 _____8.25e+1_________________ 0.524 __________0.524_____________ 15.08 _________15.1______________ 1444 ___________145____________ 0.0254 __________0.0245_____________ 83,909 ___________83900____________

Determine the experiment value of π using the correct number of significant figures. Show calculations and don’t forget units. Calculations: Diameter: 4.25 cm Experimental Value for π __3.1505___________________ radius:2.125 cm circumference: 13.39 cm circumference= 2 pi r therefore pi= 3.1505 Indicating Experimental Error  The accepted value for π is 3.1416. Based on this information, indicate both the fractional error and the percent error of the experimental π value in the previous exercise. Show calculations. Calculations: Percent Error ______0.283%____________________ Accepted value = 3.1416 The fractional error is (3.1505-3.1416/3.1416) = 0.0028329 The percent error is *100= 0.283%  The accepted value for the acceleration g due to gravity is 9.80 m/s 2 . A recent experiment yielded two values for g : 9.96 m/s 2 and 9.72 m/s 2 . Based on these experimental values, determine (a) the percent difference of the measurements, (b) the percent error of each measurement, and (c) the percent error of their average. Show calculations. Calculations: Percent D =100 x (9.96 -9.72)/ ½ (9.96+9.72) = 0.0243 Percent error 1= 100 x 9.80-9.96/9.80= 1.632 Percent error 2 = 100 x 9.80-9.72/9.80 Percent error average= Percent error 1 + percent error 2 /2 =1.224

Percent Difference _________0.0243___________ Percent Error of Value I _________1.632%___________ Percent Error of Value II ___________0.8163%_________ Percent Error of Average __________1.224 _________  The data table below shows the results of a recent free-fall experiment. The results indicate the fall distance ( y ) of an object in a set measure of time. Complete the table by filling in the average fall distance at the given times. In addition, fill in the computed value for time squared. Use the correct number of significant figures. Time t (s) Distance (m) y avg t 2 y 1 y 2 y 3 y 4 y 5 Trial 1 0.50 1.0 1.4 1.1 1.4 1.5 1.28 0.25 Trial 2 0.75 2.6 3.2 2.8 2.5 3.1 2.84 0.56 Trial 3 1.00 4.8 4.4 5.1 4.7 4.8 4.76 1 Trial 4 1.75 8.2 7.9 7.5 8.1 7.4 7.82 3.0  When an object begins to fall from a state of rest, its motion can be calculated with y = (1/2) gt 2 , where y is the distance, g is acceleration due to gravity, and t is time. Since there is only one unknown, it is possible to modify the equation to solve for g . Based on this information, calculate the value of g for each trial and find the average experimental value for g . Don’t forget units. Trial 1 Trial 2 Trial 3 Trial 4 Acceleration due to Gravity g (m/s 2 ) = 10.24 10.14 = 9.52 5.213 Calculations: g= 2y/t^2 Average Experimental Value of g ______8.778_________

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Trial 1: 2 x 1.28/0.25= 10.24 Trial 2: 2 x 2.84/0.56= 10.14 Trial 3: 2 x 4.76/ 1= 9.52 Trial 4: 2 x 7.82/ 3.0= 5.213 Post-Lab Questions 1. Using the figures below, determine the length of the bold lines using the rulers provided and comment on the results for each. First one has a length of 3.5 second one has 3.6 Third one has 3.65 2. Earlier in this lab report, the volume of a wooden block with dimensions 11.2 cm × 3.4 cm × 4.10 cm was calculated. In assessing these measurements, is it possible to determine whether or not the same measuring instrument was used? Please explain. No, Same measurement is not used in measuring because each variable has a different numbers from one another for example width is more than length + height and length is has the same case.

3. Using the analogy of a dart board, sketch an image depicting a dart grouping with good accuracy, yet poor precision. 4. Discuss the possible indications that percent error and percent difference may have regarding accuracy or precision. Discuss each separately. Since percent error compares the experimental value to a standard value, it provides a measure of measurement accuracy. Since the percent difference compares all the trial data to one another, it provides a measure of precision.

significant figures and unit

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