Errors and Statistics

Error Propagation

When dealing with uncertainties based on a large collection of numbers, the manipulation of measured quantities and the error associated with each quantity will contribute to the error in the final answer. The following is an informal discussion describing how to make reasonable approximations of errors associated with measuring physical quantities.

Error analysis is the study of error propagation within an experiment and the full treatment is quite complicated and detailed. Up until now, we have compared our results using percentage error and percentage difference, but have largely ignored the error inherent in our measurements. Let’s now discuss the topic of error propagation.

A. Uncertainty of an object’s mass. Say you use a triple-beam balance to measure the mass of some object and you find that value to be 156.28g. So is this the actual mass? We say this is the measured or nominal mass value, but it is not the actual mass, for we are somewhat uncertain about the measurement due to our instrument’s imperfections. Let’s find out how uncertain.

Since the least count of the triple-beam balance is 0.1g, the uncertainty of any measurement made with this instrument (no matter how carefully the measurement is performed) is 0.05g, or 50% of the least count. Therefore we would report the mass of this object to be 156.28g ± 0.05g. This implies that we are certain that the mass of the object is somewhere between 156.23g and 156.33g. Of course, this assumes the instrument is properly zeroed and is in proper working order.

B. Uncertainty of a sphere’s volume. Say you use a Vernier caliper to measure the diameter of solid sphere and your find that value to be 4.22cm. Therefore the radius of that sphere is 2.11cm. Using the formula we can calculate the volume of the sphere to be 39.349206cm3. Of course we must take into account significant figures and round our answer to 39.35cm3. How certain are we that this measurement is correct? Let’s calculate the uncertainty of our calculation using a "brute-force" method.

We know from Section IV above that the uncertainty of any measurement taken by a Vernier caliper is ± 0.01 cm. Therefore our original measurement of 2.11cm really falls within the range of 2.11 ± 0.01cm. That is, we can say with a great degree of certainty that the actual value of the sphere’s radius falls between 2.10cm and 2.12cm. For example, if we plug in 2.10cm as the radius, we find the lower-limit of the sphere’s volume to be 38.79cm3. This is a difference of –0.56cm3 from the nominal measurement of 39.35cm3. Plugging in 2.12cm we calculate the upper-limit of the sphere’s volume to be 39.91cm3, this time a difference of +0.56cm3 from the nominal measurement. Here, both the lower and upper limits differ by 0.56cm3 from the nominal measurement; therefore the uncertainty of this measurement is ± 0.56cm3. We are finally able to say that the volume of the sphere is measured to be 39.35cm3 ± 0.56cm3. Sometimes this is written 39.35 ± 0.56cm3.

Figure 14. The radius of a sphere is measured to be 2.11, and its volume is calculated to be 39.35cm3

 With every measurement, however, there is an associated uncertainty. Because a Vernier caliper was used here, the uncertainty in the radius measurement is ± 0.01cm. Using the lower-limit of the sphere’s radius of 2.10cm, the lower-limit of the sphere’s volume is calculated to be 38.79cm3. Using the radius’ upper-limit of 2.12cm, the upper-limit of the sphere’s volume is calculated to be 30.91cm3. The difference between the lower and upper limits and the nominal value is ± 0.56cm3, so we report that the volume of the sphere is 39.35cm3 ± 0.56cm3.

While this method does not treat error propagation from a theoretical point of view, the practical treatment here will suffice at present.

C. Uncertainty of the volume of a rectangular block. Let’s quickly take a look at one more example of error propagation. Say we use a Vernier caliper to measure the length, width, and height of a rectangular block as 1.37cm, 4.11cm, and 2.56cm, respectively. The nominal volume of the block is therefore, 14.41cm3 ( ).

Of course, each of these measurements has an uncertainty of ± 0.01cm due to imperfections in the caliper. As before we can calculate a lower-limit of the block’s volume:


which is 0.19cm3 lower than the nominal value.

The upper-limit can also be calculated:


which is 0.20cm3 higher than the nominal value. We must choose the highest deviation from the nominal to serve as our uncertainty, so we would report that the volume of the block is 14.14cm3 ± 0.20cm3.

D. Reporting with error bars. When a quantity is graphed, it is common for the uncertainty of that quantity to be represented by error bars. Refer to the Excel tutorial on using error bars to learn how to include this important piece of data into your laboratory reports. You should know that if a quantity has the same uncertainty value for each measurement, for example mass, you can enter a fixed error value as shown in Figure 15. If each data point has a unique uncertainty value, for example volume, you should store those in a column in the spreadsheet and enter the cells of the custom values as shown in Figure 16.

Figure 15.
How to enter fixed uncertainty values.

Figure 16.
How to enter custom uncertainty values.

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