# 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:

**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.**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.**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.**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 = a^{x} |
\( \frac{s_y}{y} = x\Big(\frac{s_a}{a}\Big) \) |

Logarithmos | y = log_{10}a |
\( s_y = 0.434\Big(\frac{s_a}{a}\Big) \) |

Antelogarithmos | y = antilog_{10}a |
\( \frac{s_y}{y} = 2.303s_a \) |