# General Resources

Notes on measurement error estimation.

In all experiments it is very important to give a numerical value for the error in a measurement. In this page, we describe shortly the methods for error estimation. A detailed explanation (in Hebrew) can be found here as PDF or doc:

1. Errors when Reading Scales – The error is the smallest tick size. For example, if your ruler has 1mm ticks on it than you can say that your pen is 15cm long with a 1mm error.
2. Errors of Digital Instruments – The error is half the smallest digit. For example, if you measure the weight of salt in a four digit semi analytical scale, you can say that you have 0.5682 gr salt with a 0.00005 gr error.
3. Statistical Errors – When you measure the same variable several times, you get a statistical scatter of the result. In order to estimate the error, we assume that the results are distributed normally and take their standard deviation to be the measurement error.
4. Propagating Errors – If you are interested in the error of a function of a measured variable, or a function of several measured variables you need to know how to propagate the error. The general function is error(F)= error(Fx)+error(Ft) = sqrt( (|df/dx|*error(x))^2+(|df/dt|*error(t))^2). For example, if I want to know the error in the velocity of running man that ran for 1 min (error 1 s) a distance of 200 meters (error 20 cm) then:v = x/t -> error(v)/v = sqrt((error(x)/x)^2+(error(t)/t)^2) -> v = 200 m/min error(v) = 200*sqrt((0.2/200)^2+(1/60)^2) = 3.3393 m/min.
Mathematical operation Example Mathematical notation
Addition and subtraction y = a + b − c $s_y = \sqrt{s_a^2 + s_b^2 + s_c^2}$
Multiplication and division y = a · b / c $\frac{s_y}{y} = \sqrt{\Big(\frac{s_a}{a}\Big)^2 + \Big(\frac{s_b}{b}\Big)^2 + \Big(\frac{s_c}{c}\Big)^2}$
Exponentiation y = ax $\frac{s_y}{y} = x\Big(\frac{s_a}{a}\Big)$
Logarithmos y = log10a $s_y = 0.434\Big(\frac{s_a}{a}\Big)$
Antelogarithmos y = antilog10a $\frac{s_y}{y} = 2.303s_a$