Augustana College Physics Laboratories

Hanson Hall of Science, Rock Island, IL

Measurement and Uncertainty

In addition to allowing you to deal with some of the concepts in your physics class firsthand, lab exercises are designed to help you to become a better scientist, investigator, and critical thinker. One aspect of this is learning how to properly use measuring tools, from things as simple as a ruler to computer-interfaced sensors. It also involves developing good data-taking practices, such as recording data in a complete, organized way, and providing uncertainty estimates for measurements whenever feasible. It also means learning to draw sound conclusions from results and data, using statistics, well-designed experiments, and logical thinking.

Objectives/Tasks (What You’ll Do):

  1. Record data on a computer with a motion sensor
  2. Measure the speed of a cart moving at constant velocity using both a motion sensor and a stopwatch.
  3. Compare propagation of uncertainty, standard deviation, and fit parameter error as measures of experimental uncertainty of the cart’s velocity.

Resources (What You’ll Use):

Background:

Experimental work often involves repeated measurements, both to confirm your own results and those of others. A way to combine multiple measurements of something into a single number that expresses something about all the measurements is to simply use the arithmetic mean (μ). The idea is that all measurements will tend to cluster around some value, and the more measurements we take, the closer the mean gets to this theoretical “actual” value (although whether the actual value exists is a subject of philosophy – we can try using more accurate instruments to get the “true” value of a measurement or take huge numbers of measurements, but there will always be uncertainty in a measurement to some degree).

With the mean, we can combine measurements, but it is also useful to express the uncertainty in those measurements. With a single measurement, we just have to estimate based on things such as our knowledge of the experimental setup, or the accuracy of our measuring instrument. With multiple measurements, we can again use the idea that the measurements will cluster around some value. Numbers that cluster like this will tend to form a bell-shaped distribution, called a Gaussian, around the mean. The standard deviation (\sigma) is a measure of how spread-out the distribution is. Mathematically, future measurements have about a 68% probability of being within one standard deviation of the mean of all current measurements, so this is often used as a way to express the uncertainty in the mean. However, filling in the shape of the Gaussian requires a very large number of measurements, which is not practical in most experiments. Instead, experimenters often use the standard deviation of the mean, which takes into account the fact that increasing the number of measurements should decrease the overall uncertainty. Formulas for the mean and standard deviation of the mean are:

\mu=\frac{x_1+x_2+\cdots+x_N}{N} \sigma=\frac{1}{N}\sqrt{(x_1-\mu)^2+(x_2-\mu)^2+\cdots+(x_N-\mu)^2}

Often we use two or more experimental values to calculate a third. The amount of uncertainty in the input values should have an effect on the uncertainty of the result. To do this, we use propagation of error. The error propagation formula for some function f of variables x_1, x_2, x_3, … is

\sigma_f=\sqrt{\left(\frac{\partial f}{\partial x_1}\right)^2\sigma_{x_1}^2 + \left(\frac{\partial f}{\partial x_2}\right)^2\sigma_{x_2}^2 + \left(\frac{\partial f}{\partial x_3}\right)^2\sigma_{x_3}^2 + \cdots}

One of the measurement devices you will use in this lab exercise is called a motion sensor, though it might be more accurately called a position sensor. This device uses ultrasonic pulses (you can hear a clicking noise from the speaker when it is operating) to measure the distance to objects by timing the echo. It can detect objects in a cone-shaped zone that widens with distance from the sensor. The sensor has both narrow-beam and standard modes, selectable with a switch, that can detect objects at a range of 15cm to 2m and 15cm to 8m, respectively.

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