Lecture 4 Random Errors in Chemical Analysis Uncertainty

Lecture 4 Random Errors in Chemical Analysis

Uncertainty in addition and subtraction Uncertainty in multiplication and division

Gaussian Distribution

Gaussian curve: negative mean st. dev

Uniform or rectangular probability distribution y 0 a b x

triangular probability distribution y 0 a (a+b)/2 b x

Gaussian curve: negative mean st. dev If you know and , you know everything! Our goal: and

Case 1: We know: Real value of a number Standard deviation Nothing left, we know everything about this random number Example: Concentration of Cr in steel is 21. 23± 0. 07 % This material was analyzed by numerous labs, so we have hundreds of measurements to support these numbers Sometimes we can even estimate standard deviation theoretically

Case 2: We know: Real value Standard deviation ? Take N measurements; Calculate standard deviation S as Example: I need to use a new method. I know the real value of concentration but I want to check my method performance

Case 3: We know: Standard deviation Real value ? Take N measurements; calculate average as With increase of the number of measurements N, we expect that will be close to the real value Example: I am using the same procedure for a long time; it always gives me the same standard deviation ± 0. 03%. Now I have my readings for average: 1. 37%. Therefore, the result is 1. 37 ± 0. 03% - I already had a better estimate for standard deviation than I can receive from this particular measurement

Case 4 We know nothing: Mean - ? Standard deviation -? Take N measurements; calculate average as Calculate standard deviation as Now N-1 !

I know the result: I have measured the value myself: