Chapter 9
Example - a basic calculation on uncertainty
9.1 The measurement - how long is a piece of string?
9.2 Analysis of uncertainty - spreadsheet model
9.1 The measurement - how long is a piece of string?
Suppose you need to make a careful estimate of the length of a piece of string.
Following the steps listed in Section 6.2, the process is as follows.
Step 1. Decide what you need to find out from your measurements. Decide what actual measurements and calculations are needed to produce the final result. You need to make a measurement of the length, using a tape measure. Apart from the actual length reading on the tape measure, you may need to consider:
- Possible errors of the tape measure
- Does it need any correction, or has calibration shown it to read correctly - and what is the uncertainty in the calibration?
- Is the tape prone to stretching?
- Could bending have shortened it? How much could it have changed since it was calibrated?
- What is the resolution, i.e., how small are the divisions on the tape (e.g. millimeters).
- Possible errors due to the item being measured
- Does the string lie straight? Is it under- or over-stretched?
- Does the prevailing temperature or humidity (or anything else) affect its actual length?
- Are the ends of the string well-defined, or are they frayed?
- Possible errors due to the measuring process, and the person making the measurement
- How well can you line up the beginning of the string with the beginning of the tape measure.
- Can the tape be laid properly parallel with the string?
- How repeatable is the measurement?
Can you think of any others?
Step 2. Carry out the measurements needed. You make and record your measurements of length. To be extra thorough, you repeat the measurement a total of 10 times, aligning the tape measure freshly each time (probably not very likely in reality!) Let us suppose you calculate the mean to be 5.017 metres (m), and the estimated standard deviation to be 0.0021 m (i.e., 2.1 millimetres).
For a careful measurement you might also record:
- when you did it
- how you did it, e.g. along the ground or vertically, reversing the tape measure or not, and other details of how you aligned the tape with the string
- which tape measure you used
- environmental conditions (if you think these could affect your results)
- Anything else that could be relevant.
Step 3. Estimate the uncertainty of each input quantity that feeds into the final result. Express all uncertainties in similar terms (standard uncertainty, u). You would look at all the possible sources of uncertainty and estimate the magnitude of each. Let us say that in this case:
- The tape measure has been calibrated. It needs no correction, but the calibration uncertainty is 0.1 percent of reading, at a coverage factor k = 2 (for a normal distribution). In this case, 0.1 percent of 5.017 m is close to 5 mm. Dividing by 2 gives the standard uncertainty (for k = 1) to be u = 2.5 mm.
- The divisions on the tape are millimeters. Reading to the nearest division gives an error of no more than ±0.5 mm. We can take this to be a uniformly distributed uncertainty (the true readings could lie variously anywhere in the 1 mm interval - i.e., ±0.5 mm). To find the standard uncertainty, u, we divide the half-width (0.5 mm) by /3, giving u = 0.3 mm, approximately.
- The tape lies straight, but let us suppose the string unavoidably has a few slight bends in it. Therefore the measurement is likely to underestimate the actual length of the string. Let us guess that the underestimate is about 0.2 percent, and that the uncertainty in this is also 0.2 percent at most. That means we should correct the result by adding 0.2 percent (i.e., 10 mm.) The uncertainty is assumed to be uniformly distributed, in the absence of better information. Dividing the half-width of the uncertainty (10 mm) by /3 gives the standard uncertainty u = 5.8 mm (to the nearest 0.1 mm).
The above are all Type B estimates. Below is a Type A estimate.
- The standard deviation tells us about how repeatable the placement of the tape measure is, and how much this contributes to the uncertainty of the mean value. The estimated standard deviation of the mean of the 10 readings is found using the equation in Section 3.6:
Let us suppose that no other uncertainties need to be counted in this example. (In reality, other things would probably need to be included.)
Step 4. Decide whether the errors of the input quantities are independent of each other. (If you think not, then some extra calculations or information are needed.) In this case, let us say that they are all independent.
Step 5. Calculate the result of your measurement (including any known corrections for things such as calibration). The result comes from the mean reading, together with the correction needed for the string lying slightly crookedly,
i.e., 5.017 m + 0.010 m = 5.027 m.
Step 6. Find the combined standard uncertainty from all the individual aspects. The only calculation used in finding the result was the addition of a correction, so summation in quadrature can be used in its simplest form (using the equation in Section 7.2.1). The standard uncertainties are combined as
Step 7. Express the uncertainty in terms of coverage factor (see Section 7.4 above), together with a size of the uncertainty interval, and state a level of confidence. For a coverage factor k = 2, multiply the combined standard uncertainty by 2, to give an expanded uncertainty of 12.8 mm (i.e., 0.0128 m). This gives a level of confidence of about 95 percent.
Step 8. Write down the measurement result and the uncertainty, and state how you got both of these. You might record:
‘The length of the string was 5.027 m ± 0.013 m. The reported expanded uncertainty is based on a standard uncertainty multiplied by a coverage factor k = 2, providing a level of confidence of approximately 95%.
‘The reported length is the mean of 10 repeated measurements of the string laid horizontally. The result is corrected for the estimated effect of the string not lying completely straight when measured. The uncertainty was estimated according to the method in "A beginner’s guide to uncertainty of measurement".’
9.2 Analysis of uncertainty - spreadsheet model
To help in the process of calculation, it can be useful to summarise the uncertainty analysis or ‘uncertainty budget’ in a spreadsheet as in Table 1 below.
Table 1. Spreadsheet model showing the ‘uncertainty budget’.