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Accuracy, Precision Measurement
Accuracy refers to how close a measurement is to the true value, while precision refers to the reproducibility of measurements. Accuracy is determined by calculating percentage error compared to the accepted value. Precision depends on the number of significant figures in a measurement as determined by the measuring tool. Random and systematic errors can affect accuracy, while random errors affect precision. The uncertainty of a measurement combines its precision and accuracy errors and is reported with the mean value and at a given confidence level, typically 95%. Propagation of error calculations allow determining the total uncertainty when a value depends on multiple measurements.
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Accuracy, Precision Measurement
- 1. Accuracy & Precision inMeasurement Nagendra Bahadur Amatya Associate Professor nbamatya@ioe.edu.np 9851072341
- 2. Accuracy & Precision •Accuracy: • How close you are to the actual value • Depends on the person measuring • Calculated by the formula: % Error = (YV – AV) x 100 ÷ AV Where: YV is YOUR measured Value & AV is the Accepted Value • Precision: • How finely tuned your measurements are or how close they can be to each other • Depends on the measuring tool • Determined by the number of significant digits
- 3. Accuracy & Precision •Accuracy & Precision may be demonstrated by shooting at a target. • Accuracy is represented by hitting the bulls eye (the accepted value) • Precision is represented by a tight grouping of shots (they are finely tuned)
- 4. Accuracy & Precision Precisionwithout Accuracy No Precision & No Accuracy Accuracy without Precision
- 5. Accuracy - Calculating% Error How Close Are You to the Accepted Value (Bull’s Eye)
- 6. • Accuracy isthe closeness of a measured value to the true value. • For example, the measured density of water has become more accurate with improved experimental design, technique, and equipment. Density of H2O at 20° C (g/cm3) 1 1.0 1.00 0.998 0.9982 0.99820 0.998203
- 7. Accuracy - Calculating% Error • If a student measured the room width at 8.46 m and the accepted value was 9.45 m what was their accuracy? • Using the formula: % error = (YV – AV) x 100 ÷ AV • Where YV is the student’s measured value & AV is the accepted value
- 8. Accuracy - Calculating% Error • Since YV = 8.46 m, AV = 9.45 m • % Error = (8.46 m – 9.45 m) x 100 ÷ 9.45 m • = -0.99 m x 100 ÷ 9.45 m • = -99 m ÷ 9.45 m • = -10.5 % • Note that the meter unit cancels during the division & the unit is %. The (-) shows that YV was low • The student was off by almost 11% & must remeasure
- 9. •percent error isused to estimate the accuracy of a surement. •percent error will always Positive What is the percent error if the measured density of titanium (Ti) 45 g/cm3 and the accepted density of Ti is 4.50 g/cm3?
- 10. remeasure -5% 5% remeasure •Acceptableerror is +/- 5% •Values from –5% up to 5% are acceptable •Values less than –5% or greater than 5% must be remeasured
- 12. Significant Digits How toCheck a Measurement for Precision
- 13. Significant Digits &Precision • The precision of a measurement is the smallest possible unit that could be measured. • Significant Digits (sd) are the numbers that result from a measurement. • When a measurement is converted we need to make sure we know which digits are significant and keep them in our conversion • All digits that are measured are significant
- 14. Significant Digits &Precision • How many digits are there in the measurement? • All of these digits are significant • There are 3 sd Length of Bar = 3.23 cm cm 1 2 3 40 •What is the length of the bar?
- 15. Significant Digits &Precision • If we converted to that measurement of 3.23 cm to µm what would we get? • Right! 32 300 µm • How many digits in our converted number? • Are they all significant digits (measured)? • Which ones were measured and which ones were added because we converted? • If we know the significant digits we can know the precision of our original measurement
- 16. Significant Digits &Precision • What if we didn’t know the original measurement – such as 0.005670 hm. How would we know the precision of our measurement. • The rules showing how to determine the number of significant digits is shown in your lab manual on p. 19. Though you can handle them, they are somewhat complex.
- 17. Types of Errors Differencebetween measured result and true value. •Illegitimate errors •Blunders resulting from mistakes in procedure. You must be careful. •Computational or calculational errors after the experiment. •Bias or Systematic errors •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). •Systematic errors can be reduced through calibration •Faulty equipment--such as an instrument which always reads 3% high •Consistent or recurring human errors-- observer bias •This type of error cannot be evaluated directly from the data but can be determined by comparison to theory or other experiments.
- 18. Types of Errors(cont.) • Random, Stochastic or Precision errors: • An error that causes readings to take random-like values about the mean value. • Effects of uncontrolled variables • Variations of procedure • The concepts of probability and statistics are used to study random errors. When we think of random errors we also think of repeatability or precision.
- 19. Bias, Precision, andTotal Error •X True •Total error •Bias Error •Precision Error •X measured
- 20. Accuracy and Precision •Accuracyis the closeness of a measurement (or set of observations) to the true value. The higher the accuracy the lower the error. •Precision is the closeness of multiple observations to one another, or the repeatability of a measurement.
- 21. Uncertainty Analysis •The estimateof the error is called the uncertainty. •It includes both bias and precision errors. •We need to identify all the potential significant errors for the instrument(s). •All measurements should be given in three parts •Mean value •Uncertainty •Confidence interval on which that uncertainty is based (typically 95% C.I.) •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) •We will use a 95 % confidence interval throughout this course (20:1 odds).
- 22. How to EstimateBias Error • Manufacturers’ Specifications • If you can’t do better, you may take it from the manufacturer’s specs. • Accuracy - %FS, %reading, offset, or some combination (e.g., 0.1% reading + 0.15 counts) • Unless you can identify otherwise, assume that these are at a 95% confidence interval • Independent Calibration • May be deduced from the calibration process
- 23. Use Statistics toEstimate Random Uncertnaty • Mean: the sum of measurement values divided by the number of measurements. • Deviation: the difference between a single result and the mean of many results. • Standard Deviation: the smaller the standard deviation the more precise the data • Large sample size • Small sample size (n<30) • Slightly larger value
- 25. The Population • Population:The collection of all items (measurements) of the group. Represented by a large number of measurements. • Gaussian distribution* • Sample: A portion of (or limited number of items in) a population. • *Data do not always abide by the Gaussian distribution. If not, you must use another method!!
- 26. Student t-distribution (smallsample sizes) • The t-distribution was formulated by W.S. Gosset, a scientist in the Guinness brewery in Ireland, who published his formulation in 1908 under the pen name (pseudonym) “Student.” • The t-distribution looks very much like the Gaussian distribution, bell shaped, symmetric and centered about the mean. The primary difference is that it has stronger tails, indicating a lower probability of being within an interval. The variability depends on the sample size, n.
- 27. Student t-distribution • Witha confidence interval of c% • Where α=1-c and ν=n-1 (Degrees of Freedom) • Don’t apply blindly - you may have better information about the population than you think. n txX n tx ss σσ νανα ,2/,2/ +<<−
- 28. Example: t-distribution • Sampledata • n = 21 • Degrees of Freedom = ν = 20 • Desire 95% Confidence Interval ∀α = 1 - c = 0.05 ∀α/2 = 0.025 • Student t-dist chart • t=2.086 Reading Number Volts, mv 1 5.30 2 5.73 3 6.77 4 5.26 5 4.33 6 5.45 7 6.09 8 5.64 9 5.81 10 5.75 11 5.42 12 5.31 13 5.86 14 5.70 15 4.91 16 6.02 17 6.25 18 4.99 19 5.61 20 5.81 21 5.60 Mean 5.60 Standard dev. 0.51 Variance 0.26 Reading Number Volts, mv 1 5.30 2 5.73 3 6.77 4 5.26 5 4.33 6 5.45 7 6.09 8 5.64 9 5.81 10 5.75 11 5.42 12 5.31 13 5.86 14 5.70 15 4.91 16 6.02 17 6.25 18 4.99 19 5.61 20 5.81 21 5.60 Mean 5.60 Standard dev. 0.51 Variance 0.26
- 29. Estimate of PrecisionError is Then: 23.060.5 21 51.0 086.260.5 ,2 ± ⋅± ⋅± n tx sσ ν α • Precision error is • ±0.23 Volts
- 30. How to combinebias and precision error? • Rules for combining independent uncertainties for measurements: • Both uncertainties MUST be at the same Confidence Interval (95%) • Precision error obtained using Student’s-t method • Bias error determined from calibration, manufacturers’ specifications, smallest division. 22 xxx PBU +=
- 31. Propagation of Error •Used to determine uncertainty of a quantity that requires measurement of several independent variables. • Volume of a cylinder = f(D,L) • Volume of a block = f(L,W,H) • Density of an ideal gas = f(P,T) • Again, all variables must have the same confidence interval.
- 32. RSS Method (RootSum Squares) • For a function y = f(x1,x2,...,xN), the RSS uncertainty is: • First determine uncertainty of each variable in the form ( xN ± ∆xN) • Use previously established methods, including bias and precision error. ∆++ ∆+ ∆=∆ 22 2 2 2 1 1 ... N N RSS x x f x x f x x f u ∂ ∂ ∂ ∂ ∂ ∂
- 33. Example Problem: Propagationof Error • Ideal gas law: • Temperature • T±∆T • Pressure • P±∆P • R=Constant ρ = P RT •How do we estimate the error in the density?
- 34. ( ) 2 2 222 1 + = + = TΔ TR P PΔ TR TΔ T ρ pΔ p ρ ρΔRSS ∂ ∂ ∂ ∂ •Apply RSS formula to density relationship: 22 ∆ + ∆ = ∆ T T p p ρ ρ •Apply a little algebra: ρ = P RT