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Discussion
We saw in the rounding numbers section that the level of precision available when measuring values can impact the final answer of a calculation. Finding the volume of water in a graduated cylinder with markings every 1 mL, for example, could result in a different answer than finding the volume of the same amount of water in a different graduated cylinder that has markings every 0.25 mL.
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Both of these cylinders have the same amount of water in them, but if we took the measurement using the first cylinder we'd get 25 mL, and if we took it using the second, we'd get 25.25 mL.
What range of numbers do you think the "actual" volume of the water might fall in, based off of the measurement given by first cylinder? Based off of the second?
When you read 25mL on a scale marked every 1mL, the water level must be closer to the 25mL mark than to either the 24mL or the 26mL marks, but you do not know exactly where between them. The point where it would instead look closer to 24mL is halfway between 24 and 25, which is
and the point where it would look closer to 26mL is halfway between 25 and 26, which is
So any true volume between 24.5mL and 25.5mL would still be read as 25mL. Therefore,
On the 0.25mL-graduation cylinder, a reading of 25.25mL means the water level is nearer the 25.25 mark than to 25.00 or 25.50. The halfway points are
So any true volume between 25.125mL and 25.375mL would register as 25.25mL, giving
Approximation error bounds
A measurement is always accurate to ±21 ("plus or minus one half") of the smallest division on the scale. The same idea applies to rounding numbers: the rounded value is accurate to ±21 the value of the digit "place" (tens, ones, tenths, hundredths, etc.) it is rounded to.
Typically, we express the error bounds of a number x using inequalities:
where x is the measured value, u is the smallest unit on the scale we used. Note that some people equivalently write
The upper bound, x+21u, is not included since it would be rounded up to the next increment.
This is because real-world measurements are found using some kind of measuring tool. Think about using a ruler with centimeter markings to find the length of some object. Even if the object's length were, say, 27.657 centimeters, we couldn't use this ruler to find that value since it's only marked off every whole centimeter. We would instead report the length as 28 centimeters in order to be honest about the degree of precision.
So when we say the object's length is 28 centimeters based off of a measurement taken with that ruler, what we're really saying is that the length is closer to 28 than any other whole centimeter value. It's this idea of "closeness" that is captured by the error bounds. The object's length would be "closest" to 28 if its exact value lay anywhere from 28cm−21(1cm)=27.5cm to just below28+21(1cm)=28.5cm, since any value in this range has 28 as the closest tick.
Hence we report error bounds using inequality notation. They're basically a way of being honest about how accurate our real-world tools allow us to be.
Checkpoint
A ruler in millimeter increments is used to measure an object. The object's length is found to be 105 millimeters. What are the bounds of the object's actual length?
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Discussion
It's useful to be able to describe a range of potential error values, but merely stating a range of numbers doesn't give us much information about how impactful the error could be on our calculation. Imagine that the same graduated cylinder from earlier, with markings every milliliter, is used again, this time to measure out 5 mL of water instead of 25 mL.
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Do you think this measurement will be more or less accurate than the one for 25 mL? Why?
Can you give any numerical value for the accuracy of this measurement? Specifically, how "wrong" do you expect it to be in comparison to the total volume?
When you read a scale marked in 1mL steps, your eye can land up to half a division above or below the true level. Since each division is 1mL, your reading can be off by at most 0.5mL. In other words, the 5mL you pour could in fact be anywhere between 4.5mL and 5.5mL.
To see how significant that possible 0.5mL mismatch is compared with the amount you’ve poured, we take the ratio of the error to the total volume. This “relative” measure tells us how impactful the reading uncertainty is for different volumes.
As a fraction of the 5mL poured, that is
By contrast, when you pour 25mL the same ±0.5mL reading-error represents
Because 10% ≫ 2%, the 5mL measurement is less accurate in relative terms than the 25mL measurement.
This discussion captures the idea of absolute versus percentage error. Even though the error bounds might be the same for two different measurements, an error of ±21 mL represents a far larger percentage of 5 mL than 25 mL.
We can describe these errors precisely if we know the difference between approximated and exact values.
Absolute and percentage error
The actual size of an error is the difference between the approximated value (VA) and the exact value (VE). We call this the absolute error, and can calculate it with the equation
Expressing the error as a percentage of the exact value allows us to see how much the error "matters" in a specific scenario. This is the percentage error, which we calculate with the equation
Checkpoint
Find the absolute and percentage error of an approximation of 12.5 m when the actual value is 10.8 m.
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Discussion
If we're using reported measurements to calculate another value, the error bounds of those original measurements will impact the error bounds of our calculation.
To see this, consider taking the measurements of a rug to find its area. The width is measured at w=3.8 ft and the length is measured at l=4.2 ft, based on a ruler with increments of 0.1 ft.
What are the smallest possible values for length and width? How about the largest possible values?
Can you use these to find the smallest and largest possible area for the rug?
The ruler’s smallest division is 0.1ft, so each measurement is accurate to within ±0.05ft. Hence
Because both width w and length l are positive, the area
increases whenever either w or l increases. Thus, the smallest area occurs at the smallest w and smallest l, and the largest area at the largest w and largest l:
(uncertainty picture)
Bounds of calculation based on other approximations
The error bounds of values like length and width impact the error bounds of any secondary calculation, like area, that we use them to find. For example, if an object of length l and width w has error bounds 0.9≤l≤1.1 and 2.6≤w≤2.9, then the object's minimum and maximum area are
i.e., the error bounds are 2.34≤A<3.19.
Checkpoint
An object measured to the nearest tenth of a foot is found to have length l=23.5 ft and width w=17 ft.
Find the error bounds of the object's area.
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Exercise
Martha wants to buy a new tank for her tropical fish. She measures the dimensions of the current tank as w=52 cm in width, l=176 cm in length, and h=48 cm in height, rounded to the nearest centimeter.
Based on the error bounds of her measuring tool, Martha estimates that the actual volume V of the tank is between 4.29×105 cm3 and a×10k cm3, where 1≤a<10 and k∈Z.
Use Martha's width, length, and height measurements to find a and k.
Martha calculates the volume of the tank to be 4.39×105 cm3.
Based on your answer to part (a), find the maximum percentage error in Martha's calculation.
The store only sells tanks in increments of 0.01 m3.
State the volume, in cubic meters, of the tank Martha should purchase to be as close as possible to the volume of the old tank.
Martha decides to purchase a 4.4×105 cm3 tank. Each of her fish needs, at minimum, 3.8×103 cm3 of tank space.
Based on these values for tank size and fish space, how many fish, at most, can Martha fit in her new tank?
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