Measurement and Error

Physics is an experimental science, meaning any theoretical predictions made must be tested thoroughly by researchers in the lab. Usually these predictions are mathematical equations and they are compared quantitatively to experimental data. The precision of an experiment becomes important since it can determine if the technique is sensitive enough to distinguish between multiple predictions. For instance, one theory may predict that volume depends on the side length squared of a cube while another may predict it depends on the side length cubed. (This may seem like a trivial problem, but surface area was rigorously mathematically defined in the early twentieth century.) How could you determine which theory is correct? Describe an experiment to do so.

If the cubes used in an experiment have a side length of near one, the volumes calculated using the two different theories will be very similar. The figure below shows some possible data and the two theoretical predictions. Considering the error bars, which theory is supported by the data, or is neither validated by the theory? Justify your answer thoroughly using the information in the graph after discussing with your group.

Now consider the following graph. Is the answer more obvious?

When presenting experimental data at a scientific conference or in a publication, error bars are necessary to indicate to the audience if the measurements can distinguish between possible theories. Error bars represent a range of values that are consistent with the experimental results. No experiment yields one exact number. Instead, experiments give a spread represented by error bars. These error bars arise from the precision of the measurement tools and the “noise” in the data. Noise comes from variations in the environment or conditions when the experiment is repeated. It can be quantified by taking the same measurement many times and calculating the standard deviation of the multiple trials. This semester we will not measure noise since the time required to take enough samples to calculate a standard deviation is more than the lab period allows.

The other contribution to error is the precision of the measuring device. Each instrument is limited by its size, shape, and mechanism. Big graduate cylinders are less precise because a small difference in height of a liquid represents a large volume. Thinner cylinders are more precise. In this class, as a rule of thumb, the precision of a measuring tool is half of the smallest increment. For instance, the smallest unit on a meter stick is typically a millimeter. The precision is therefore 0.5mm. If the tick marks on a graduate cylinder represent 2mL, the precision is 1mL.

When reporting data, numbers should be given as the reading plus/minus error. Using the above cylinder, a volume of 22mL is determined. This number would be reported as 22+/-1mL. Note the inclusion of units. The last number in a measurement could be an estimation if the indicator falls between tick marks and the scientist conducting the experiment is confident in his/her lab technique. A reading of 21mL could be obtained from the even-number-only graduated cylinder. Note that the last significant digit in a reading is either the same as or one less than the error. For instance, a measurement could be 22.1+/-0.05 (range is 22.15 to 22.05) or 22.1+/-0.1 (range is 22.2 to 22.0) but never 22.1+/-1. Look at the measuring tools available to you today. Record the precision of each and give an example measurement from the device.

Now use graduate cylinder and some water to determine the volume of the metal cylinder and rectangular solid. Remember to record both the reading and the error.

What happens when the quantity of interest is calculated from measurements, rather than measured directly? The calculated quantity must also be represented by a range. Look at the following graph. It shows the dependence of volume on side length for a cube.

The green bar on the x-axis represents the range indicated by the measurement of the side length (1+/1 0.1). The bar on the y-axis represents the corresponding range of volume. How can we compute this range? We could take the average reading, then add the error to it and calculate volume, then subtract the error and calculate volume. For a quantity calculated from one measurement, this is not unreasonable.

However, if multiple measurements are used to calculate something this process breaks down. Instead, we take a linear approximation, meaning we model the function as a straight line with slope equal to the magnitude of the derivative at the average reading. The red line above is the linear approximation of the graph. As you can see, it gives a pretty good estimation of the range of the volume for the given range of side length. We then use this line to calculate the length of the green bar on the y-axis. We do this same process for all the measurements which go into a calculation. The length of the y-axis bars are then added together like the legs of a right triangle to give a total range for the calculated quantity. Mathematically this is expressed as:

Here, the delta () means “error in..” So, F is the error in F, x is the error in the measured quantity x, or the precision of the instrument used to measure x. ∂F/∂x represents the derivative of F with respect to x. All other variables are held constant and treated as numbers except for x. The ellipses (…) mean you keep adding terms until all measured quantities used to calculate F are represented.

Let’s look at an example:

The volume of a cylinder is given by πr2h where r is the radius and h is the height. The formula for the error in volume is then:

The derivatives are:

Record on your paper the complete formula, with the derivatives inserted in to the above equation.

The volume of a rectangular solid is given by: L×W×H, where L is length, W is width, and H is height. Find the equation for the error in the volume of a rectangular solid and record it on your paper.

Now use at least two different tools to measure the radius and height of the cylinder and the length, width, and height of the rectangular solid. This should give you two measurements of each. Then calculate the volume of each and the error in the volume using the two different measurements. This should give you two different volumes with different errors for each.

Compare the three measured (with cylinder) or calculated volumes and error in volumes. Do they agree? To check agreement, draw a line segment that represents one calculation and its range on a number line. Then draw line segments to represent the other two volumes. Do the lines overlap? If so, the volumes are in agreement. Here is an example of measurements in agreement. The horizontal lines represent the volume and the vertical lines the error in the reading. Each has an average volume of 3 with different errors. Note the error in the caliper calculation is too small to see in the graph.

Now measure the mass of the objects. Calculate the density of each and the error in density. Show the formula you used to calculate the error. It should depend on the error in volume and you can use the error in volume calculated previously.

Using the internet, find at least two different metals or alloys that fall within the range of densities you measured with this experiment. Explain how you could design an experiment to differentiate which of these two substances your samples are.

Follow up questions.

  1. Which method for measuring the volume was “the best”? Justify your answer. Include in your analysis the ease at which the measurement could be made. Is the increase in precision worth the effort?
  2. Have each person in your group use the calipers to take the same measurement, such as the width of the rectangular solid. Don’t tell each other what your readings are until after you each have taken them. Are all the readings exactly the same? Is this an example of random error? Why or why not? Remember you cannot use the phrase “human error” in this class.
  3. Systematic error is another kind of error in experiments. It denotes an equal skew to all data and changes all measurements in the same way. An example of systematic error is putting one end of an object to be measured at the 1cm mark on a meter stick rather than the 0 mark. All the lengths measured would then be 1cm too long. What are some sources of systematic error in this lab (other than the one given)? Again, no “human error”. You should explain in detail how the error would change your measurements and ay quantities calculated from them. For instance, if all the lengths were 1cm too long, the volume calculated would be larger than it really is, and therefore the density calculated would be less than the actual density of the object.