Error Analysis

Experimental Error Experimental Error The uncertainty obtained in a measurement of an experiment Results can from systematic and/or random errors Blunders Human Error Instrument Limitations Relates to the degree of confidence in an answer Propagation of uncertainties must be calculated and taken into account

Experimental Error It 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.

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 observer Sometimes referred to as determinate errors Reproducible with precision Can be discovered and corrected

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).

Detection of Systematic Errors Analyze samples of known composition Use standard Reference material Develop a calibration curve Analyze “blank” samples Verify that the instrument will give a zero result Obtain results for a sample using multiple instruments Verifies the accuracy of the instrument

How to Eliminate Systematic Errors Elimination of systematic error can best be accomplished by a well planned and well executed experimental procedure How would you measure the distance between two parallel vertical lines most 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

Types of Errors Random Errors Sometimes referred to as indeterminate errors or noise errors Arises from things that cannot be controlled Variations in how an individual or individuals read the measurements Instrumentation noise Always present and cannot always be corrected for, but can be treated statistically The important property of random error is that it adds variability to the data but does not affect average performance for the data

Random Errors Examples: You measure the mass of a ring three times using the same balance and get slightly different values: 12.74 g, 12.72 g, 12.75 g The meter stick that is used for measuring, slips a little when measuring the object

Accuracy or Precision Precision Accuracy Reproducibility of results Several measurements afford the same results Is a measure of exactness Accuracy How close a result is to the “true” value “True” values contain errors since they too were measured Is a measure of rightness

Accuracy vs Precision = Accuracy Precision 3 NO 7.18281828 YES 3.14 3.1415926

Calculating Errors Terminology Significant Figures – minimum number of digits required to express a value in scientific notation without loss of accuracy Absolute Uncertainty – margin of uncertainty associated with “a” measurement Relative 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

Significant Figures How to determine which digits are Significant Write the number as a power of 10 Zero’s are significant and must be included when they occur In the middle of a number At the end of a number on the right hand side of the decimal point This implies that you know the value of a measurement accurately to a specific decimal point The significant figures (digits) in a measurement include all digits that can be known precisely, plus a last digit that is an estimate.

Significant Figures Scientific Notation Let’s look at 123.45 1.2345x102 We have 5 significant digits Let’s look at 0.000123 1.23x10-4 We have 3 significant digits

Significant Figures Determine the number of significant digits in the following numbers:

Significant Figures 142.7 1.427x102 4 significant digits 142.70

Significant Figures 0.000006302 6.302x10-6 4 significant digits 0.003050 3.050x10-3 4 significant digits

Significant Figures 9.250x104 9.250x104 4 significant digits

Significant Figures 9000 9x103 1 significant digit 9000. 9.000x103 4 significant digits

Significant Figures The last significant digit in a measured quantity is the first digit of uncertainty

Significant Figures Determine the significant figures from the diagram below Certain values 1 degree of uncertainty True expression Absorbance 0.23 0.234 0.234 ± 0.001 % Transmittance 58 58.3 58.3 ± 0.1

Significant Figures When 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.

Adding Significant Digits 4503+34.90+550= 5090 3 significant digits 5 is the first doubtful number 0 is the first doubtful number 3 is the first doubtful number Via Calculator: 5087.9 The 87.9 are the doubtful numbers

Adding Significant Digits 2456.2345+23.21= 2479.4445 2479.44 23400.00+111.49= 23511.49 23511.49 23400+111.49= 23511.49 23500 234000-2340= 231660 232000

Multiplying Significant Digits 2.7812x1.7= 4.72804 4.7 Rounded to 4.7 because 1.7 only has 2 significant digits

Multiplying Significant Digits 48.008 14.200x3.2400= 48.008 150 1.00x150.03= 150.03 1500 1200x1.234= 1480.8 8.65 8.654121… 45.35.2345=

Rounding Rounding 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.

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: 7.146 rounded to hundredths is 7.15 (because the next digit [6] is 5 or more).

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 less Round 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) 7.04501 rounded to hundredths is 7.05 (because the next digit is 5, but it is followed by non-zero digits)

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 quantities Mean (or average) Standard Deviation Standard Error

Mean What is it: An estimate of the true value of the measurement Statistical Interpretation: The central value Symbol:

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 average Symbol: Use you calculator or computer to determine the Standard Deviation.

Standard Error What is it: An estimate in the uncertainty in the average of the measurements Statistical 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 experiment Symbol:

Standard Error Example Measurements: 0.32, 0.54, 0.44, 0.29, 0.48 Use 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: 0.41 Calculate the Standard Deviation: 0.09 Calculate the Standard Error: 0.04 Therefore: 0.41±0.04

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

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

Error Propagation There are 3 different ways of calculating or estimating the uncertainty in calculated results Significant 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 measurements Statistical Methods (When you have lots of numbers Useful for many measurements

Dependent Error Propagation Adding and Subtracting Multiplying and Dividing Average

Dependent (approx) (121)+(52)-(73) (121)*(52)*(73) If the Average is 25, then 25 5

Propagation of Errors Basic Rule Adding and Subtracting If x and y have independent random errors and , then error in z=x+y is Therefore we have

Adding and Subtracting 1.76 (0.03) + 1.89 (0.02) – 0.59 (0.02) = Z=1.76+1.89-0.59=3.06 Therefore Z=3.06 0.04

Multiplying and Dividing Propagation of Errors Basic Rule Multiplying and Dividing If x and y have independent random errors and , then error in z=xy is Therefore we have

Multiplying and Dividing [1.76(0.03) x 1.89(0.02)] / 0.59(0.02) = Z= Therefore z=5.6 0.2

Putting it Together x=200 2 Y=50 2 z=40 2 x, y, z are independent, find q Let d=y-z Therefore q=20 6

What about Functions of 1 Variable Find error for with s=20.02 We cannot use because s, s, s are not independent What to the rescue??? Calculus

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 s Therefore the value for V is V=80.2

x=100 6 then find V when A function of one variable… CALCULUS Therefore V=10.0 0.3

What about a Function with a Constant? You measure the diameter of a circle to be 20.02 Determine the area of the circle Calculus The area is 3.14 0.06

If q=f(x1, x2, x3, …xn) then Previous rule Let q=x1+x2 PROOF

If q=f(x1, x2, x3, …xn) Let q=x1*x2 Previous rule PROOF

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 a The Partial Derivatives are:

Uncertainty Therefore a=(3.3 0.1) m/s2

Focal Length Determine the focal length plus uncertainty when p=100±2 cm and q=30±1 cm

Focal Length The focal length is (23.1±0.6) cm or (23±1) cm

Ugly Trig Problem Determine q and error is x=10±2, y=7±1, Ø=400±30 =-0.732 =0.963 =9.813

Still the Ugly Trig Problem Trig should be in radians Therefore q=3.5±2

Max-Min Technique If 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.

Ugly Trig Problem again Determine q and error is x=10±2, y=7±1, Ø=400±30 Ave=5.0236 Therefore: 25±5

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

Percent Error Exmple For wx: For y2: To find Error for z, we need to convert Percent Errors to Absolute Therefore: