Lecture 4 Random Errors in Chemical Analysis

Uncertainty in multiplication and division Uncertainty in addition and subtraction

Gaussian Distribution

Gaussian curve: negative st.dev mean

ab x y Uniform or rectangular probability distribution 0

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

Gaussian curve: negative st.dev mean 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: