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J has just measured the length of a piece
of wood to be 34.4 cm +/- 1 cm -- what does that number mean?
Numbers that have units after them are measurements.
They may have been part of a calculation of some kind, but ultimately they are based on
someone taking the time to use an instrument to measure a characteristic of some object.
All measurements consists of a numeric value (such as length) with a particular precision
(such as one decimal place) and units (such as centimeters).
Measurements may also have associated error (such as plus or minus 1 centimeter).
The precision and unit come from the instrument -- in this case, a ruler.
The numeric value and associated error are decided by the person using the ruler.
For example: I use a thermometer to measure my body’s
temperature.
I can see from the thermometer that the unit is in degrees Fahrenheit (°F), and the smallest
measurement, or precision is 0..That means that I could, theoretically, read a measurement
as 97.6°F which means it is not 97.7°F and not 97.5°F; it is precisely 97.6°F. Let’s
blow up this portion of the thermometer to see that more clearly.
The measurement 97.6°F represents any measurement that lands in the colored zone that marks
the halfway points between 97.6°F and 97.7°F above and 97.5°F below.
If the measurement had landed outside the colored zone up here, it would have rounded
up to 97.7°F. If the measurement had landed below the colored zone down here, it would
have rounded down to 97.5°F. The halfway point here is 97.55°F. The halfway point
here is 97.65°F. So the number 97.6°F really represents all the values from 97.55°F up
to but not including 97.65°F (because 97.65°F would round up to 97.7°F).
Another way to state that is 97.6°F +/-0.05°.
If I think that my measurement was that good, truly in this zone, then I don’t need to
add that error; I just write 97.6°F, and we all know what it means.
However, if I the thermometer numbers were so small that I couldn’t actually be sure
I was reading the measurement correctly, or if the liquid in the thermometer was moving
around a bit, and I might get a different value if I looked at it a second time, I’m
going to add to some error.
For example, this image shows the thermometer with lots of tight little lines separated
by 0.1° -- too tight for me to read easily.
Maybe the value is 97.6°F, but sometimes it looks as high as 97.8°F or as low as 97.4°F,
so I’ll note that error in my final answer by writing 97.6°F +/- 0.2°.
The precision is one decimal place, and that’s set by the instrument limits.
The error is based on how well I can read the instrument and that’s +/-0.2°
What happens when the tool I use to measure has a precision much greater than my error?
For example, when I use Google Earth to measure a distance between two points, I get a precision
of many decimal places, 9.443225 km.
But in this case the two points I’m measuring the distance between are not very precisely
located.
So when I make the measurements multiple times, I get multiple answers.
There seems to be a fair bit of error: plus or minus 0.2 km.
Writing my answer as 9.443225 km +/- 0.2 km is kinda crazy, because while the instrument
has great precision, the error supersedes that and makes all those extra decimal places
meaningless.
So I’m going to round my measurement to one place more precise than my error and the
answer would be 9.44 km +/- 0.2 km.
Now what happens when we perform a mathematical calculation with a measured number?
For example, what if we add or multiply two measured numbers together or double a measured
number?
What do we do about the precision of the final result?
The key here is to ensure that our final result is not implying a greater precision than we
originally measured.
That would be false information.
Let’s look at some examples:
John uses this ruler to measure a piece of wood.
The precision limit of this ruler is 0.1 cm.
His value for the wood is 34.4 cm.
It means it’s not 34.5 cm and not 34.3 cm, so it’s everything between those two numbers
that would round to 34.4 cm -- 34.35 cm all the way up to 34.44 cm -- all these numbers
round to 34.4 cm.
Julia uses a different ruler to measure a similar piece of wood.
The precision limit of her ruler is 1 cm.
She measures the length of the wood to be 34 cm.
It’s not 33 cm and not 35 cm, but anything that would round to 34 cm, so 33.5 through
to 34.4 cm.
John’s value is much more precise than Julia’s.
What do we do when we add their two lengths of wood together?
What’s 34.4 cm + 34 cm?
Just doing the math, we get 68.4 cm, but is it correct to maintain the precision of the
most precise value in the sum or do we have to round to the least precise?
Which one can we stand behind as being correct -- the more precise or least precise value?
If we write 68.4 cm, then that means it’s not 68.5 and not 68.3.
It’s definitively 68.4 cm with an implied error of +/-0.05 cm.
But wait!
Julia’s piece has way more error than that.
She has an error of +/-0.5 cm (ten times the error of John’s piece).
To leave it at 68.4 implies more precision than is correct.
So we round our answers to match the least precise measured number.
Then we can be sure we aren’t implying more precision that we have.
That’s important because a measured number loses its value to us if we can’t trust
its precision.
If we round to the ones place, which is the precision of Julia’s number, we end up with
a final sum of 68 cm, which means it could be anything from 67.5 cm to 68.4 cm, and that
is more reasonably true based on what we know of Julia’s measurement.
What if we take Julia’s wood piece and cut it in thirds?
Does taking a third of Julia’s number change its precision?
Or doubling it?
No.
If there’s no other measured number in the calculation, we simply keep the precision
the same.
34 cm/3 is 11.3 cm which we round to match the original precision of the ones place,
so simply 11 cm.
Again, we can’t increase or decrease the precision just by dividing by 3.
Finally, what do we do when we multiply two measured numbers together?
When we have two measured numbers and we multiply or divide them, we can’t simply round the
answer to match a starting precision.
Let’s see why not and what we do as a result.
What’s the area of a square that has 34.4 cm on one side and 34 cm on the other?
The calculation completed with the original numbers gives us: 34.4 cm x 34 cm = 1,169.6
cm2.
Are all those places and precision correct?
Let’s look at the ranges of the two starting measurements.
We know that 34.4 cm could be as high as 34.44 cm (that’s the error implied by that precision),
and 34 cm could be as high as 34.4 cm (that’s the error implied by that precision).
So we complete the same calculation with the highest error range for each number and again
with the lowest.
Then we can compare those answers to see the error range results and use that to round
our original answer accordingly.
This range of possible answers -- from 1150.725 to 1184.736 cm2 -- starts to vary in the tens
place.
Notice that starting from the left in the thousands place, the numbers are identical.
Similarly in the hundreds place, but once we reach the tens place they diverge.
So there’s no point in keeping precision past the tens place when the tens place is
already showing variation.
We go back to the original answer, 1169.6 cm2 and round it to the tens place but no
further.
Final answer: 1,170 cm2, which has an implied error of +/- 5 cm (or 1165 to 1174 cm2) -- much
closer to the range of possible answers that we calculated above.
Going forward in this lab, it’s important that for all measured numbers you include
the correct precision, units, and if applicable, extra error.
In all calculations with measured numbers, be sure you round the results appropriately
so that your answers imply accurate precision.
The kinds of measuring devices we’l be using in the first few labs include rulers, protractors,
compasses, and latitude and longitude lines.
Anytime you have to measure a number yourself in this class, if you can’t be absolutely
sure that your measurement matches the instrument’s precision, then add some error.
Typically you wll always want to add some error.
A good rule of thumb is to imagine how your answer will match the answer in the key.
If you aren’t absolutely sure you set your ruler or your protractor in the right place
or were able to line everything up perfectly, add a reasonable amount of error, so that
your measurement and the key’s measurement (and any other student’s measurement) would
sit within each other’s error.
How to do this?
Try measuring the value a few times and see what range of answers you get.
If they’re all identical, that’s your number and you need no more error.
If, however, you are getting slightly different answers, add the appropriate +/- error to
be sure that any of the answers you got would be correct.
Pause now. [music]
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