Propagation of Error

Propagating error can be tricky. The general formula for the error eᵧ of a quantity Y that cannot be measured directly but is a function of one or more directly measured quantities yᵢ, each with their own error eᵢ, is

where N is the number of measured quantities, and the errors eᵢ were assumed to be small and independent. You can always do the math yourself (that's usually what I do) but there are a few standard forms that come up regularly, or you can ask MATLAB to do the math for you using the Symbolic Math Toolbox.

In our pump calibration example the volumetric flow rate Y (in mL/s) can be calculated as the ratio of the measured volume V (in mL) to the discharge time t (in s), or

Applying the propagation of error form to this expression yields

and this is what we can use to determine the uncertainties eᵧ of each flow rate. The following MATLAB code performs the necessary calculations and produces an errorbar plot (you might want to begin a similar MATLAB script as we'll be using the results in subsequent examples).

% Values

k = [0; 1; 2; 3; 4; 5]; % knob ticks

V = [0; 23; 43; 71; 83; 120]; % mL

t = [10.1; 10.0; 10.2; 9.9; 9.9; 10.1]; % s

Y = V./t; % mL/s

% Uncertainties

ek = 0.25*ones(size(Y)); % knob ticks

eV = 0.5; % mL

et = 0.05; % s

eY = Y.*sqrt((eV./V).^2 + (et./t).^2); % prop error

% Plot using ERRORBAR()

errorbar(k, Y, eY, eY, ek, ek, 'ko')

axis square

xlabel('Knob Setting')

ylabel('Discharge Rate (mL/s)')