Calculate the average and the uncertainty for each set of data: 15.32, 15.37, 15.33, 25.38, 15.35
Answers
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 meters tick can be read to the nearest 0.1 cm, you probably cannot determine the diameter to the nearest 0.1 cm.
What factors limit your ability to determine the diameter of the ball?
What is a more realistic estimate of the uncertainty in your measurement of the diameter of the ball