Uncertainty & Error “Science is what we have learned about how to keep from fooling ourselves.” ― Richard P. FeynmanRichard P. Feynman

Types of Errors Error u Uncertainty is NOT error! u Difference between measured result and true value. u Illegitimate errors u Blunders resulting from mistakes in procedure. You must be careful. u Computational or calculational errors after the experiment. u Not paying attention! u Bias or Systematic errors u An offset error; one that remains with repeated measurements (i.e. a change of indicated pressure with the difference in temperature from calibration to use). u Systematic errors can be reduced through calibration u Faulty equipment--such as an instrument which always reads 3% high u Consistent or recurring - observer bias u This type of error cannot be evaluated directly from the data but can be determined by comparison to theory or other experiments.

“Human Error” u Giraffes don’t do science!! u Of course you’re a human. u You probably mean “systematic error” u OR... You mean uncertainty, which isn’t an error at all. It is you being honest. u Explain what you mean! (Example: Parallax on Meter Stick) u Letting it fall isn’t human error, it is not following the procedure.

4 Types of Uncertainties AKA- “Plus/Minuses”, +/-, Tolerance, Standard Deviations Random Uncertainties: result from the randomness of measuring instruments. They can be dealt with by making repeated measurements and averaging. One can calculate the standard deviation of the data to estimate the uncertainty. Systematic Uncertainties: result from a flaw or limitation in the instrument or measurement technique. Systematic uncertainties will always have the same sign. For example, if a meter stick is too short, it will always produce results that are too long.

Accuracy and Precision u Accuracy is the closeness of a measurement (or set of observations) to the true value. The higher the accuracy the lower the error. u Precision is the closeness of multiple observations to one another, or the repeatability of a measurement.

6 Accuracy vs. Precision Accurate: How close a measurement is to an accepted / “true” value. An accurate measurement correctly reflects the size of the thing being measured. Must know the correct answer beforehand! Precise: How close a measurement is to another. repeatable, reliable, getting the same measurement each time. A measurement can be precise but not accurate.

Accuracy versus Precision Precise and AccurateAccurate and NOT Precise Precise and NOT Accurate Not Accurate or Precise

Bias, Precision, and Total Error Bias Error Total Error Precision Error X True X measured

Uncertainty9 Percent Difference: It’s Accuracy! Calculating the percent difference is a useful way to compare experimental results with the accepted value, but it is not a substitute for a real uncertainty estimate. Example: Calculate the percent difference if a measurement of g resulted in 9.4 m / s 2.

10 Absolute and Percent Uncertainties If x = 99 m ± 5 m then the 5 m is referred to as an absolute uncertainty and the symbol σ x (sigma) is used to refer to it. You may also need to calculate a percent uncertainty/fractional uncertainty ( %σ x ): NO UNITS!

Uncertainty Analysis u The estimate of the error is called the uncertainty. u It includes both bias and precision errors. u We need to identify all the potential significant errors for the instrument(s). u All measurements should be given in three parts u Mean value u Uncertainty u Confidence interval on which that uncertainty is based (typically 95% C.I.) u Uncertainty can be expressed in either absolute terms (i.e., 5 Volts ±0.5 Volts) or in percentage terms (i.e., 5 Volts ±10%) (relative uncertainty =  V / V x 100) u We will use a 95 % confidence interval throughout this course (20:1 odds).

Use Statistics to Estimate Random Uncert. u Mean: the sum of measurement values divided by the number of measurements. u Deviation: the difference between a single result and the mean of many results. u Standard Deviation: the smaller the standard deviation the more precise the data  Large sample size  u Small sample size (n<30)  Slightly larger value 

The Population u Population: The collection of all items (measurements) of the group. Represented by a large number of measurements. u Gaussian distribution* u Sample: A portion of (or limited number of items in) a population. u * Data do not always abide by the Gaussian distribution. If not, you must use another method!!

Uncertainty14 Standard Deviation

Uncertainty15 Standard Deviation The average or mean of a set of data is The formula for the standard deviation given below is the one used by Microsoft Excel. It is best when there is a small set of measurements. The version in the book divides by N instead of N-1. Unless you are told to use the above function, you may use the Excel function ‘=stdev(B2:B10)’

Uncertainty16 Expressing Results in terms of the number of σ In this course we will use σ to represent the uncertainty in a measurement no matter how that uncertainty is determined You are expected to express agreement or disagreement between experiment and the accepted value in terms of a multiple of σ. For example if a laboratory measurement the acceleration due to gravity resulted in g = 9.2 ± 0.2 m / s 2 you would say that the results differed by 3σ from the accepted value and this is a major disagreement To calculate N σ

Uncertainty17 Uncertainty resulting from averaging N measurements If the uncertainty in a single measurement of x is statistical, then you can reduce this uncertainty by making N measurements and averaging. Example: A single measurement of x yields x = 12.0 ± 1.0, so you decide to make 10 measurements and average. In this case N = 10 and σ x = 1.0, so the uncertainty in the average is This is not true for systematic uncertainties- if your meter stick is too short, you don’t gain anything by repeated measurements.

Propagation of Error u Used to determine uncertainty of a quantity that requires measurement of several independent variables. u Volume of a cylinder = f(D,L) u Volume of a block = f(L,W,H) u Density of an ideal gas = f(P,T) u IB Does this on a worst case scenario!

Uncertainty19 Uncertainty when a number is raised to a power Example: If z = 12 ± 1.0 = 12.0 ± 8.3 % then If z = x n then %σ z = n ( % σ x )

Uncertainty20 Uncertainty when calculation involves a special function Example: If θ = 12 0 ± sin(14 0 ) = sin(12 0 ) = sin(10 0 ) = For a special function, you add and subtract the uncertainties from the value and calculate the function for each case. Then plug these numbers into the function. And thus sin(12 0 ± 2 0 ) = ±