Experimental Error or Uncertainty: Data Analysis and Presentation Data Presentation with Uncertainty (Chapter 3). We will then see in Chapter 4 how to measure that uncertainty which is presented in Chapter 3. More on unceratinty and certainty and quality analysis and quality control in Chapter 5 Q: How would you determine uncertainty (error ) in your direct and indirect (function) measurement?

Question: qualitative answer True /False concept

Measurements Direct: MASS, VOLUME Functions: DENSITY

Measurements Quantitative values: scales linear and log

Measurement procedure: more details Calibration curve of a 50ml buret. You can read to the nearest 0.1ml Exam[le : for 29.43 mL the correction is -0.03ml

Measurement procedure See textbook: Systematic error in ozone measurements

Uncertainty in Measurement IMPORTANT Uncertainty in Measurement Precision and Accuracy RED is “correct” or “true” value, BLUE is “error” or uncertainty

What is origin of error/uncertainty?

3-4 What are these types of uncertainties, errors measured, and true values in chemical analysis that we want to present ?  Systematic error/uncertainty, also known as determinate error can be large, one-sided, can be corrected- we do not present them !!!!!!!! (eliminate them ) Random error/uncertainty, also known as indeterminate error, depends on the instrument and its sensitivity, positive and negative, inherent to the method Measured values are what we measure. True values: measured and /or established, often standards or references. In an analytical sense the true value has been established by reliable methods using accepted standards, reproducible standard procedures, and multiple measurements.

Data Presentation: how to present uncertainty  Quantitative data with units are always presented in a similar way as: “target” + “circle around the target”: magnitude of measurement +- uncertainty 1.0000 +/- 0.0001 NOTE: the LAST digit in magnitude of measurement and uncertainty is THE SAME!!!!!! The uncertainty is found from experiment. The uncertainty determinates the number of significant figures

Uncertainty in Measurement IMPORTANT 3-1 Significant Figures  The number of digits reported in a measurement reflect the precision of the measuring device 4.0kg versus 4.00kg. All the figures known with certainty plus one extra figure are called significant figures: 4.00 +- 0.01kg. In any calculation, the results are reported to the fewest significant figures (for multiplication and division) or fewest decimal places (addition and subtraction) See the rules.

Uncertainty in Measurement: Significant Figures – complicated way Guidelines Nonzero digits are always significant–457 cm (3 significant figures); 2.5 g (2 significant figures). Zeros between nonzero digits are always significant–1005 kg (4 significant figures); 1.03 cm (3 significant figures). Zeros at the beginning of a number are never significant; they merely indicate the position of the decimal point–0.02 g (one significant figure); 0.0026 cm (2 significant figures). Zeros that fall at the end of a number or after the decimal point are always significant–0.0200 g (3 significant figures); 3.0 cm (2 significant figures). When a number ends in zeros but contains no decimal point, the zeros may or may not be significant–130 cm (2 or 3 significant figures); 10,300 g (3, 4, or 5 significant figures).

Scientific (exponential) notation: Sig Figs easy way !!!!  2.50 *104 cm 3 significant figures 1.03 *104 cm 3 significant figures 1.030 *104 cm 4 significant figures 1.0300 *104 cm 5 significant figures The number of digits remaining is the number of SIG FIGS Simple!!

How to measure uncertainty? Direct measurement uncertainty: see average and standard deviation X+-s Function measurement uncertainty : ?

Q: How do we treat uncertainty when presenting a function of observables F(x1, x2,…xn)? Example: density is a function of mass and volume A: We treat random uncertainty/error as any small (infinitesimal) deviation from functions value: “dx” F(x+dx)

3-5 Propagation of uncertainties from variables in(to) the function Derivation of general analytical expression Derivation of expressions for selected functions General rules for: sum/difference, product/division,…. and any function you need!

Derivation of general analytical expression for the propagation of uncertainty for a function of n independent measured variables What is the meaning and how to get results in Table 3-1 and more, propagation of uncertainty for any function!!!

Definitions: We have seen absolute uncertainty , or “e”, as error : e= +-0.03 We can also express relative uncertainty as : e/magnitude of measurement: +- 0.03/1.76 Or percent relative uncertainty, %e: (e/value) * 100%: %e= +- (0.03/1.76)*100 We can apply this to the propagation of uncertainty rules too and And express the propagated total uncertainty of a function efunction in terms of e and %e..

Rules Addition and subtraction Multiplication and division %

Density =Mass/Volume Mass = 2.012+-0.002 g Volume = 1.01 +-0.03mL First : Density = 2.012/1.01 =1.9921 Second: =(0.002/2.012)*100 = .0994 =(0.03/1.01)*100= 2.97 = 2.97 Result is : 1.99+- %e = 1.99 +- 3% Or 1.99+- e = 1.99 +- 0.06

The real rule for significant figures: IMPORTANT The first uncertain figure of the answer is the last significant figure. (That is the one you keep) IMPORTANT : Propagation of uncertainty is very important, as it will show what is the major contribution to the uncertainty

Propagation of uncertainty : dependent and independent