Measurement and Uncertainty Notes


Reporting Measurements and Experimental Results

Best Estimate ± Uncertainty

When scientists make a measurement or calculate some quantity from their data, they generally assume that some exact or "true value" exists based on how they define what is being measured (or calculated). Scientists reporting their results usually specify a range of values that they expect this "true value" to fall within. The most common way to show the range of values is:
measurement = best estimate ± uncertainty
Example: a measurement of 5.07 g ± 0.02 g means that the experimenter is confident that the actual value for the quantity being measured lies between 5.05 g and 5.09 g. The uncertainty is the experimenter's best estimate of how far an experimental quantity might be from the "true value." (The art of estimating this uncertainty is what error analysis is all about).

How many digits should be kept?

Experimental uncertainties should be rounded to one significant figure. Experimental uncertainties are, by nature, inexact. Uncertainties are almost always quoted to one significant digit (example: ±0.05 s). If the uncertainty starts with a one, some scientists quote the uncertainty to two significant digits (example: ±0.0012 kg).

Wrong: 52.3 cm ± 4.1 cm
Correct: 52 cm ± 4 cm

Always round the experimental measurement or result to the same decimal place as the uncertainty. It would be confusing (and perhaps dishonest) to suggest that you knew the digit in the hundredths (or thousandths) place when you admit that you unsure of the tenths place.

Wrong: 1.237 s ± 0.1 s
Correct: 1.2 s ± 0.1 s

Comparing experimentally determined numbers

Uncertainty estimates are crucial for comparing experimental numbers. Are the measurements 0.86 s and 0.98 s the same or different? The answer depends on how exact these two numbers are. If the uncertainty too large, it is impossible to say whether the difference between the two numbers is real or just due to sloppy measurements. That's why estimating uncertainty is so important!

Measurements don't agree 0.86 s ± 0.02 s and 0.98 s ± 0.02 s
Measurements agree 0.86 s ± 0.08 s and 0.98 s ± 0.08 s

If the ranges of two measured values don't overlap, the measurements are discrepant (the two numbers don't agree). If the rangesoverlap, the measurements are said to be consistent.

Estimating uncertainty from a single measurement

In many circumstances, a single measurement of a quantity is often sufficient for the purposes of the measurement being taken. But if you only take one measurement, how can you estimate the uncertainty in that measurement? Estimating the uncertainty in a single measurement requires judgement on the part of the experimenter. The uncertainty of a single measurement is limited by the precision and accuracy of the measuring instrument, along with any other factors that might affect the ability of the experimenter to make the measurement and it is up to the experimenter to estimate the uncertainty (see the examples below).


Try measuring the diameter of a tennis ball using the meter stick. What is the uncertainty in this measurement?

Even though the meterstick can be read to the nearest 0.1 cm, you probably cannot determine the diameter to the nearest 0.1 cm.

Answers: It's hard to line up the edge of the ball with the marks on the ruler and the picture is blurry. Even though there are markings on the ruler for every 0.1 cm, only the markings at each 0.5 cm show up clearly. I figure I can reliably measure where the edge of the tennis ball is to within about half of one of these markings, or about 0.2 cm. The left edge is at about 50.2 cm and the right edge is at about 56.5 cm, so the diameter of the ball is about 6.3 cm ± 0.2 cm.

Another example

Try determining the thickness of a CD case from this picture.

Answers: The best way to do the measurement is to measure the thickness of the stack and divide by the number of cases in the stack. That way, the uncertainty in the measurement is spread out over all 36 CD cases. It's hard to read the ruler in the picture any closer than within about 0.2 cm (see previous example). The stack goes starts at about the 16.5 cm mark and ends at about the 54.5 cm mark, so the stack is about 38.0 cm ± 0.2 cm long. Divide the length of the stack by the number of CD cases in the stack (36) to get the thickness of a single case: 1.056 cm ± 0.006 cm. By "spreading out" the uncertainty over the entire stack of cases, you can get a measurement that is more precise than what can be determined by measuring just one of the cases with the same ruler. We are assuming that all the cases are the same thickness and that there is no space between any of the cases.

Estimating uncertainty from multiple measurements

Increasing precision with multiple measurements

One way to increase your confidence in experimental data is to repeat the same measurement many times. For example, one way to estimate the amount of time it takes something to happen is to simply time it once with a stopwatch. You can decrease the uncertainty in this estimate by making this same measurement multiple times and taking the average. The more measurements you take (provided there is no problem with the clock!), the better your estimate will be.

Taking multiple measurements also allows you to better estimate the uncertainty in your measurements by checking how reproducible the measurements are. How precise your estimate of the time is depends on the spread of the measurements (often measured using a statistic called standard deviation) and the number (N) of repeated measurements you take.

Consider the following example: Maria timed how long it takes for a steel ball to fall from top of a table to the floor using the same stopwatch. She got the following data:

0.32 s, 0.54 s, 0.44 s, 0.29 s, 0.48 s

By taking five measurements, Maria has significantly decreased the uncertainty in the time measurement. Maria also has a crude estimate of the uncertainty in her data; it is very likely that the "true" time it takes the ball to fall is somewhere between 0.29 s and 0.54 s. Statistics is required to get a more sophisticated estimate of the uncertainty.

Some statistical concepts

When dealing with repeated measurements, there are three important statistical quantities: average (or mean), standard deviation, and standard error. These are summarized in the table below:

Statistic What it is Statistical interpretation Symbol
average an estimate of the "true" value of the measurement the central value xave
standard deviation a measure of the "spread" in the data You can be reasonably sure (about 70% sure) that if you repeat the same measurement one more time, that next measurement will be less than one standard deviation away from the average. s
standard error an estimate in the uncertainty in the average of the measurements You can be reasonably sure (about 70% sure) that if you do the entire experiment again with the same number of repetitions, the average value from the new experiment will be less than one standard error away from the average value from this experiment. SE

Maria's data revisited

The statistics for Maria's stopwatch data are given below:

It's pretty clear what the average means, but what do the other statistics say about Maria's data?

Calculating the statistics using Excel

Spreadsheet programs (like Microsoft Excel) can calculate statistics easily. Once you have the data in Excel, you can use the built-in statistics package to calculate the average and the standard deviation.

To calculate the average of cells A4 through A8:
  • Select the cell you want the average to appear in (D1 in this example)
  • Type "=average(a4:a8)"
  • Press the Enter key
To calculate the standard deviation of the five numbers, use Excel's built-in STDEV function.
Excel doesn't have a standard error function, so you need to use the formula for standard error:

where N is the number of observations

Uncertainty in Calculations

What if you want to determine the uncertainty for a quanitity that was calculated from one or more measurements? There are complicated and less complicated methods of doing this. For this course, we will use the simple one. The Upper-Lower Bounds method of uncertainty in calculations is not as formally correct, but will do. The basic idea of this method is to use the uncertainty ranges of each variable to calculate the maximum and minimum values of the function. You can also think of this procedure as exmining the best and worst case scenarios. For exaample, if you want to find the area of a square and measure one side as a length of 1.2 +/- 0.2 m and the other length as 1.3 +/- 0.3 meters, then the area would be:

A = l * w = 1.2 * 1.3 = 1.56 m^2

The minimum area would be using the "minimum" measurements so l = 1.2 - 0.2 = 1.0 and w = 1.3 - 0.3 = 1.0

So the "minimum' area is A-min = 1.0 * 1.0 = 1.0 m^2

Likewise for the maximum area, l = 1.2 + 0.2 = 1.4 and w = 1.3 + 0.3 = 1.6

So the Maximum area is A-max = 1.4 * 1.6 = 2.24 m^2

Thus, we can say the area is A = 1.5 +/- 0.6 m^2