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00:00 Uncertainty
No measurement is perfect. It’s not possible to measure anything to infinite precision. That means there is always some level of uncertainty associated with a measurement. When doing experimental science it is a good practice to explicitly notate how much uncertainty is associated with a measurement. There are several ways to do this:
Absolute Uncertainty
In these examples, the exact quantity of uncertainty is explicitly written down:
2315 ± 2 mm
5.125 ± 0.015 m
12 ± 1 kg
This is an alternate notation that can be used that omits the ± symbol:
2315(2) mm
5.125(15) m
12(1) kg
Percent or Fractional Uncertainty
In this case the uncertainty is written as a percentage:
2315 mm ± 0.1%
5.125 m ± 0.3%
12 kg ± 10%
07:24 Significant Figures
Significant figures are a way of expressing uncertainty without the need to explicitly write down the uncertainty. Instead the number of digits in a number implies the level of uncertainty in the measurement. For example:
2315 mm
This measurement has no digits to the right of the 5. No tenths of a mm, no hundredths of a mm.
2315 mm ≠ 2315.00000000
This implies that the measurement was not precise enough to measure anything smaller than a single mm. In this example then, there are only 4 significant digits that carry information. We don’t write down extra digits if we can’t measure them.
The only exception is that sometimes zeroes (0) are used as placeholders and this can make things confusing. For example, we can convert
2,315 mm → 2,315,000 μm
In this case the three trailing zeroes are not significant and do not communicate any knowledge or measurement information. Instead they are used as placeholders. Unfortunately this makes zeroes ambiguous and confusing when discussing significant figures. It would be better if we could write numbers down with an explicit placeholder symbol, like 2,315,### μm. In this case it would be obvious that the # symbols are not significant and do not confer information, however this is not a standard practice.
15:07 Scientific Notation
Instead we can rely on scientific notation to help us out. In scientific notation, all digits are significant. Numbers written in scientific notation look like this:
6.22 × 103 kg
8.194 × 1021 nm
Numbers always begin with a non-zero digit, followed by a decimal point, then some number of digits, and end with a power of 10. All digits are significant and it’s relatively easy to multiply and divide numbers in scientific notation because we can add or subtract the powers of 10.
18:34 Precision vs Accuracy
While these words are used interchangeably in everyday English, in science they have distinct meanings.
When we speak of Precision we are talking about how finely we are able to make a measurement. A measurement with high precision has many significant figures.
Accuracy is more a measure of how close to the true value we get. If the true length of a wall is 4.124 m and we measure it to be 4.119 m, while our friend measures it to be 4.117 m, then our measurement is more accurate because it is closer to the true value. You can imagine a third person could measure the wall using a different technique and come up with a value of 4.11543 m. In this case his answer would be more precise, but also less accurate.
21:34 Arithmetic with Significant Figures
When multiplying and dividing numbers, the # significant figures in the result is equal to the smallest # of significant figures in the input terms. For example:
1.251 x 21.04 ÷ 9.8 = 2.685820408163265 → 2.7
We have to round our answer to 2 significant figures because one of the inputs, 9.8, only has 2 significant figures. The higher levels of precision in the other inputs essentially get “washed away” by the poorer precision in 9.8.
When Adding and Subtracting numbers, we don’t care about counting the # of significant figures. Instead we care about place value and our result can only be as precise as our least precise input term. For example:
1,203.5 + 0.52 = 1,204.02 → 1,204.0
Our first number, 1,203.5, is only measured up until the tenths place. That means any digits more precise than the tenths place are completely uncertain to us. So our final result cannot contain digits more precise than the tenths place. Our uncertainty beyond the tenths place essentially “washes away” any digits there, so they get rounded off in the final answer.
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