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Uncertainty Tracking
What is uncertainty?
Uncertainty is a part of all measurements and calculations that are based on those measurements. Uncertainty is the level of precision you have due to the tool, your observation, and the situation - think of it as wiggle room for your measurement.
56.09 +/- 0.02 cm
The above measurement has an uncertainty of 0.02 cm, which is close to the precision of a standard meter stick. The measurement has the +/- symbol which can be interpreted as "give or take." You would read the above measurement as "Fifty-six point zero nine, give or take two hundredths of a centimeter." The meaning of the statement is part of the reliability of the measurement. The person making the measurement is sure that if someone else with the same tool and situation would be able to measure the same thing and get an answer within 0.02 cm of the measurement.
Anatomy of a measurement
There are three parts to a measurement: the measurement, the uncertainty, and the units.
Where it gets tricky is when you are making calculations with measurements. The brute-force method, which works all the time, takes a while and requires you to make at least two sets of calculations. For example, using the measurement above, you would use 56.09 cm, 56.11 cm, and 56.07 cm in three separate calculations. Let's say that you were going to find the area of a rectangle that you measured.
Length: 56.09 +/- 0.02 cm
Width: 3.12 +/- 0.02 cm
1. First, do your normal calculation:
56.09 cm × 3.12 cm = 175.0008 cm^2
2. Next, you determine the largest and smallest possible answers:
56.11 cm × 3.14 cm = 176.1854 cm^2
56.09 cm × 3.12 cm = 173.817 cm^2
3. Now you just see how far, on average, the values are from the normal result:
176.1854 cm^2 - 175.0008 cm^2 = 1.1846 cm^2
175.0008 cm^2 - 173.817 cm^2 = 1.1838 cm^2
Average = +/- 1.1842 cm^2
4. Round the result down to 1 significant digit. Some researches use 2 significant digits, but for high school, 1 is sufficient.
+/- 1 cm^2
5. Put it all together, rounding your result to the tens place that you have in the uncertainty (in this case, ones).
175 +/- 1 cm^2
If you are just looking for an answer and don't need the uncertainty, the cool thing about this is that you get exactly where you can round your result (or when you have zeros at the end, you can tell which place is significant).
If we follow the significant figures rules, since the least precise measurement had three significant digits, then our result must be rounded to three significant digits. Neat how the uncertainty calculation gives us the same result as the more rough estimate given by the significant figure rules!
Now, wouldn't it be neat to be able to have Google Sheets calculate the values for you and round to the correct place?