2 Experimental Error Experimental Error The uncertainty obtained in a measurement of an experimentResults can from systematic and/or random errorsBlundersHuman ErrorInstrument LimitationsRelates to the degree of confidence in an answerPropagation of uncertainties must be calculated and taken into account
3 Experimental ErrorIt is impossible to make an exact measurement. Therefore, all experimental results are wrong. Just how wrong they are depends on the kinds of errors that were made in the experiment.As a science student you must be careful to learn how good your results are, and to report them in a way that indicates your confidence in your answers.
4 Types of Errors Systematic Errors These are errors caused by the way in which the experiment was conducted. In other words, they are caused by the design of the system or arise from flaws in equipment or experimental design or observerSometimes referred to as determinate errorsReproducible with precisionCan be discovered and corrected
5 Systematic Error Examples: The cloth tape measure that you use to measure the length of an object had been stretched out from years of use. (As a result, all of your length measurements were too small)The electronic scale you use reads 0.05 g too high for all your mass measurements (because it was improperly zeroed at the beginning of your experiment).
6 Detection of Systematic Errors Analyze samples of known compositionUse standard Reference materialDevelop a calibration curveAnalyze “blank” samplesVerify that the instrument will give a zero resultObtain results for a sample using multiple instrumentsVerifies the accuracy of the instrument
7 How to Eliminate Systematic Errors Elimination of systematic error can best be accomplished by a well planned and well executed experimental procedureHow would you measure the distance between two parallel vertical linesmost would pull out a ruler, align one end with one bar, read of the distance.You should put ruler down randomly (as perpendicular as you can). Note where each mark hits the ruler, then subtract the two readings. Repeat a number of times and average the result.Minimize the number of human operations you can
8 Types of Errors Random Errors Sometimes referred to as indeterminate errors or noise errorsArises from things that cannot be controlledVariations in how an individual or individuals read the measurementsInstrumentation noiseAlways present and cannot always be corrected for, but can be treated statisticallyThe important property of random error is that it adds variability to the data but does not affect average performance for the data
9 Random Errors Examples: You measure the mass of a ring three times using the same balance and get slightly different values: g, g, gThe meter stick that is used for measuring, slips a little when measuring the object
10 Accuracy or Precision Precision Accuracy Reproducibility of results Several measurements afford the same resultsIs a measure of exactnessAccuracyHow close a result is to the “true” value“True” values contain errors since they too were measuredIs a measure of rightness
11 Accuracy vs Precision = Accuracy Precision 3 NO 7.18281828 YES 3.14
12 Calculating Errors Terminology Significant Figures – minimum number of digits required to express a value in scientific notation without loss of accuracyAbsolute Uncertainty – margin of uncertainty associated with “a” measurementRelative Uncertainty – compares the size of the absolute uncertainty with the size of its associated measurement (a percent)Propagation of Uncertainty – The calculation to determine the uncertainty that results from multiple measurements
13 Significant Figures How to determine which digits are Significant Write the number as a power of 10Zero’s are significant and must be included when they occurIn the middle of a numberAt the end of a number on the right hand side of the decimal pointThis implies that you know the value of a measurement accurately to a specific decimal pointThe significant figures (digits) in a measurement include all digits that can be known precisely, plus a last digit that is an estimate.
14 Significant Figures Scientific Notation Let’s look at 123.45 1.2345x102We have 5 significant digitsLet’s look at1.23x10-4We have 3 significant digits
15 Significant FiguresDetermine the number of significant digits in the following numbers:
20 Significant FiguresThe last significant digit in a measured quantity is the first digit of uncertainty
21 Significant FiguresDetermine the significant figures from the diagram belowCertain values1 degree of uncertaintyTrue expressionAbsorbance0.230.2340.234 ± 0.001% Transmittance5858.358.3 ± 0.1
22 Significant FiguresWhen adding or subtracting, the last digit retained is set by the first doubtful number.When multiplying or dividing, the number of significant digits you use is simply the number of significant figures as is in the term with the fewest significant digits.
23 Adding Significant Digits =50903 significant digits5 is the first doubtful number0 is the first doubtful number3 is the first doubtful numberVia Calculator:The 87.9 are the doubtful numbers
27 RoundingRounding is the process of reducing the number of significant digits in a number. The result of rounding is a "shorter" number having fewer non-zero digits yet similar in magnitude. The result is less precise but easier to use. There are several slightly different rules for rounding.
28 Rounding Common method This method is commonly used, for example in accounting.Decide which is the last digit to keep.Increase it by 1 if the next digit is 5 or more (this is called rounding up)Leave it the same if the next digit is 4 or less (this is called rounding down)Example: rounded to hundredths is 7.15 (because the next digit  is 5 or more).
29 Rounding Decide which is the last digit to keep. This method is also known as statistician's rounding . It is identical to the common method of rounding except when the digit(s) following to rounding digit start with a five and have no non-zero digits after it. The new algorithm is:Decide which is the last digit to keep.Increase it by 1 if the next digit is 6 or more, or a 5 followed by one or more non-zero digits.Leave it the same if the next digit is 4 or lessRound up or down to the nearest even digit if the next digit is a five followed (if followed at all) only by zeroes. That is, increase the rounded digit if it is currently odd; leave it if it is already even.Examples:7.016 rounded to hundredths is 7.02 (because the next digit (6) is 6 or more)7.013 rounded to hundredths is 7.01 (because the next digit (3) is 4 or less)7.015 rounded to hundredths is 7.02 (because the next digit is 5, and the hundredths digit (1) is odd)7.045 rounded to hundredths is 7.04 (because the next digit is 5, and the hundredths digit (4) is even)rounded to hundredths is 7.05 (because the next digit is 5, but it is followed by non-zero digits)
30 Increasing Precision with Multiple Measurements One way to increase your confidence in experimental data is to repeat the same experiment many times.When dealing with repeated measurements, there are three important statistical quantitiesMean (or average)Standard DeviationStandard Error
31 Mean What is it: An estimate of the true value of the measurement Statistical Interpretation:The central valueSymbol:
32 Standard Deviation What is it: A measure of the spread in the data Statistical Interpretation:You can be reasonably sure (about 70% sure) that if you repeat the same experiment one more time, that the next measurement will be less than one standard deviation away from the averageSymbol:Use you calculator or computer to determine the Standard Deviation.
33 Standard Error What is it: An estimate in the uncertainty in the average of the measurementsStatistical Interpretation:You can be reasonably sure (about 70% sure) that if you repeat the entire experiment again with the same number of repetitions, the average value from the new experiment will be less than one standard deviation away from the average value of this experimentSymbol:
34 Standard Error Example Measurements: 0.32, 0.54, 0.44, 0.29, 0.48Use this technique to determine the uncertainty if you do not know the uncertainty of a measurement, but have multiple measurements of the value.Calculate the Mean:0.41Calculate the Standard Deviation:0.09Calculate the Standard Error:0.04Therefore: 0.41±0.04
35 Propagation of Uncertainty Since measurements commonly will contain random errors that lead to a degree of uncertainty, arithmetic operations that are performed using multiple measurements must take into account this propagation of errors when reporting uncertainty values
36 Systematic Errors Errors calculated from data are Random Errors Errors from the instrument are called System Errors (usually labeled on instrument or told by instructor as a percent)or
37 Error PropagationThere are 3 different ways of calculating or estimating the uncertainty in calculated resultsSignificant digits (The easy way out)Useful when a more extensive uncertainty analysis is not needed.Error Propagation (Not as bad as it looks)Useful for limited number or single measurementsStatistical Methods (When you have lots of numbersUseful for many measurements
38 Dependent Error Propagation Adding and SubtractingMultiplying and DividingAverage
39 Dependent (approx) (121)+(52)-(73) (121)*(52)*(73) If the Average is 25, then 25 5
40 Propagation of Errors Basic Rule Adding and SubtractingIf x and y have independent random errors and , then error in z=x+y isTherefore we have
42 Multiplying and Dividing Propagation of Errors Basic RuleMultiplying and DividingIf x and y have independent random errors and , then error in z=xy isTherefore we have
43 Multiplying and Dividing [1.76(0.03) x 1.89(0.02)] / 0.59(0.02) =Z=Therefore z=5.6 0.2
44 Putting it Together x=200 2 Y=50 2 z=40 2 x, y, z are independent, find qLet d=y-zTherefore q=20 6
45 What about Functions of 1 Variable Find error for with s=20.02We cannot use becauses, s, s are not independentWhat to the rescue???Calculus
46 V=s3 Let’s take the derivative of V with respect to s Think of dV and ds as a small change (error) in V and sTherefore the value for V is V=80.2
47 x=100 6 then find V when A function of one variable… CALCULUS Therefore V=10.0 0.3
48 What about a Function with a Constant? You measure the diameter of a circle to be 20.02Determine the area of the circleCalculusThe area is 3.14 0.06
49 If q=f(x1, x2, x3, …xn)thenPrevious ruleLet q=x1+x2PROOF
50 If q=f(x1, x2, x3, …xn)Let q=x1*x2Previous rulePROOF
51 The Atwood Machine consists of two masses M and m attached to the ends of a light, frictionless pulley. When the masses are released, the mass M is show to accelerate down with an acceleration:Suppose the M and m are measured as M=100 1g and m=50 1 g. Find the uncertainty in aThe Partial Derivatives are:
53 Focal LengthDetermine the focal length plus uncertainty when p=100±2 cm and q=30±1 cm
54 Focal LengthThe focal length is (23.1±0.6) cm or (23±1) cm
55 Ugly Trig Problem Determine q and error is x=10±2, y=7±1, Ø=400±30 =-0.732=0.963=9.813
56 Still the Ugly Trig Problem Trig should be in radiansTherefore q=3.5±2
57 Max-Min TechniqueIf you do not have a calculus background, then you can use this technique to determine the uncertainties in a complicated equation.Determine the actual value.Make the largest possible value.Make the smallest possible value.Average the difference between the actual and the largest value and the actual and the smallest value.This average is the uncertainty.
58 Ugly Trig Problem again Determine q and error is x=10±2, y=7±1, Ø=400±30Ave=5.0236Therefore: 25±5
59 Using Percent Errors Two Simple Rules: When Adding or Subtracting add the Absolute Errors like you would normally do, then convert to Percent (Relative) Error.When Multiplying or Dividing add the Percentage Errors
60 Percent Error Exmple For wx: For y2: To find Error for z, we need to convert Percent Errors to AbsoluteTherefore: