Review: Uncertainty Types and Notation
Type B Uncertainties
There are two types of uncertainties that arise when taking measurements.
The first is a Type B uncertainty. Type B uncertainties occur due to the limited precision of the equipment used for taking the measurement. For example, a scale to measure the weight of a sample might present the value to one decimal place.
Assuming the scale is rounding, the weight of the sample is anywhere between 3.15 g and 3.25 g. A more precise scale weighing the same sample reads 3.17 g. Now, we know the weight is between 3.165 g and 3.175 g.
All types of measurement tools include Type B uncertainties. We can see this with a ruler as well.
Here the length is between 22.50 to 22.60 cm.
Type A Uncertainties
[latex]\sigma = \frac{a}{\sqrt{6}}[/latex]
[latex]\sigma = \sqrt{\frac{\Sigma_i (x_i - \bar{x})^2}{\sqrt{N-1}}}[/latex]
Uncertainties on the Mean
Typically, when conducting experiments, we average several measurements. We do this to improve the accuracy of our measurement. This means the error on our mean will be smaller than the error on each individual measurement. The error on the mean is calculated using the error on each individual measurement [latex]\sigma[/latex] and the number of measurements used [latex]N[/latex]:
[latex]S_m =\frac{\sigma}{\sqrt{N}}[/latex]
Correct Notation for Writing a Measurement with it’s Uncertainty
There are two important things to remember when writing a measurement along with its uncertainty.
1. ONE significant digit for the uncertainty
2. SAME precision between the uncertainty and the measurement
For example,
[latex]2.4 \pm 0.3[/latex] cm is written CORRECTLY.
[latex]2.40 \pm 0.33[/latex] cm is written INCORRECTLY.
[latex]2.4 \pm 0.03[/latex] cm is written INCORRECTLY.
Can you spot the mistakes?
You should be using this notation every time you write a measurement with uncertainty.
You can access the more detailed notes and activities in the Physics 1C03 Lab Manual.
Propagation of Uncertainties
Suppose we have two measurements with uncertainties:[latex]x \pm s_x[/latex] and [latex]y \pm s_y[/latex]. If we are adding or subtracting these two measurements, we use the sum/difference rule:
[latex]z = x + y[/latex]
[latex]z = x - y[/latex]
[latex]s_z = \sqrt{s_x^2 + s_y^2}[/latex]
For example, if we have the two measurements: 3.2 [latex]\pm[/latex] 0.3 g + 5.1 [latex]\pm[/latex] 0.2 g:
[latex]x = 3.2[/latex] [latex]y = 5.1[/latex]
[latex]s_x = 0.3[/latex] [latex]s_y = 0.2[/latex]
[latex]s_z = \sqrt{0.3^2 + 0.2^2} = 0.4[/latex]
[latex]z = 8.3 \pm 0.4 g[/latex]
If we are multiplying or dividing the two numbers, we use the product/quotient rule:[latex]z = xy[/latex]
[latex]z = \frac{x}{y}[/latex]
[latex]s_z = z\sqrt{(\frac{s_x}{x})^2 + (\frac{s_y}{y})^2}[/latex]
For adding/subtracting we deal with absolute uncertainties (ex. [latex]s_x[/latex]) For multiplying/dividing we deal with fractional uncertainties (ex. [latex]s_x/x[/latex]). To determine the number of decimal places in our final answer, use the absolute uncertainty which is kept to one significant digit, and match the precision. In this course, please report your final ansers with their absolute uncertainty unless otherwise specified.
If we are multiplying a measurement by a constant value, we use the following equation:
[latex]z = cx[/latex]
[latex]s_z = cs_x[/latex]
If we need to take a measurement to some power [latex]n[/latex], we use the equation:
[latex]z = x^n[/latex]
[latex]s_z = nx^{n-1}s_x[/latex]
These rules can all be extended to include more than two variables as be put together to handle more complicated equations.
A summary of the error propagation rules can be found in the Appendix for reference.