Video: Recognizing Measurement Errors Using a Ruler

Scarlett uses a centimeter ruler to measure the length of a straight line, as shown in the diagram. She determines that the length of the line is 10.4 cm. Which of the following statements explains why this answer is incorrect? [A] The maximum resolution of the ruler is 1 cm; thus, the length of the line should be recorded as 10 cm. [B] Measurements using a ruler should always be rounded up; thus, the length of the line should be recorded as 11 cm. [C] The ruler is not parallel to the line; thus, the line is actually shorter than 10.4 cm. [D] The ruler is not parallel to the line; thus, the line is actually longer than 10.4 cm.

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Video Transcript

Scarlett uses a centimeter ruler to measure the length of a straight line as shown in the diagram. She determines that the length of the line is 10.4 centimeters. Which of the following statements explains why this answer is incorrect? A) The maximum resolution of the ruler is one centimeter. Thus, the length of the line should be recorded as 10 centimeters. B) Measurements using a ruler should always be rounded up. Thus, the length of the line should be recorded as 11 centimeters. C) The ruler is not parallel to the line. Thus, the line is actually shorter than 10.4 centimeters. D) The ruler is not parallel to the line. Thus, the line is actually longer than 10.4 centimeters.

Okay, so what Scarlett’s done here is try to measure the length of this line here. The line is labeled 𝑥. And she’s wrote off using the ruler that the length of the line is 10.4 centimeters. Now this is an incorrect measurement of the length of the line. And we need to work out why that is.

So looking at option A as a possible answer, this one says that the maximum resolution of the ruler is one centimeter. Thus, we should record the length of the line as 10 centimeters rather than 10.4. Well, we’ve been told in the initial part of the question that Scarlett uses a centimeter ruler. This means that the large markings on the ruler are every one centimeter. And hence, the smaller markings must be every tenth of a centimeter or every millimeter. This means that the maximum resolution of the ruler is not in fact one centimeter. It’s one millimeter. And hence, option A can be ruled out. Haha, ruled out.

Moving on to option B then, this one says that measurements using a ruler should always be rounded up. Now this is not a very good idea at all because always rounding up a measurement will introduce a systematic error in our whole set of measurements.

Let’s say we’re trying to measure the lengths of different lines all over the place. If we always round up their measurements, then unless one of these lines falls exactly on a perfect centimeter marking, we’re going to approximate them to be longer than they actually are. Because we’ll say that they’re rounded up. Now this is not a good idea in terms of an entire sample. Because the way that rounding normally works is that, on average, 50 percent of the time we will round up. So we will assume that the length of a line is longer than it actually is. And 50 percent of the time we will round down, just as we do with normal rounding. And so any rounding up done in some lines is canceled by the rounding down done in other lines.

However, if we always round up when we make measurements with a ruler, then we’re always overestimating the length of a line, unless, like we said, it falls perfectly on a centimeter marking. So, for example, if a line is measured to be exactly four centimeters, then we wouldn’t need to round up. But overall, we’re still overestimating the lengths of lines in our sample. So this is not a good thing. And hence, option B is out of the question as well.

Let’s then look at option C. This one says that the ruler is not parallel to the line. Thus, the line is actually shorter than 10.4 centimeters. And actually, if we look at option D, it starts off the same way as option C. It says that the ruler is not parallel to the line. Where it differs is in the consequence.

Option D says that the line is actually longer than 10.4 centimeters. So because these are the only two options left, we can safely say that the reason that Scarlett’s measurement is incorrect is because the ruler and the line are not parallel. And that actually is the reason why. Because if we want to measure the length of the line, then we need to be measuring from here to here. And hence, the ruler and the line need to be parallel with each other.

So to work out the consequence of this, let’s imagine that we rotate this line clockwise until it’s parallel with the ruler. Or if we want to, we can imagine rotating the ruler counterclockwise. It doesn’t matter which as long as we rotate one or both until they’re parallel with each other.

So now this is what the line would look like if we rotated it until it was parallel with the ruler. In this case, we can see that the left-hand end of the line is the point about which we rotated it. And the reason we’ve done this is because the left-hand end of the line is aligned perfectly with the zero marking on the ruler. This is a good thing. This is something that Scarlett has done well. Because in order to measure the length of the line once the line is parallel with the ruler, one end needs to be aligned perfectly with the zero marking on the ruler. And the other end is used to take the measurement.

So we rotate about the left-hand end and then take the measurement on this end. And we can see that the measurement is actually roughly 11 centimeters. The exact measurement doesn’t matter. But the point is that the actual measurement of the line when the line is parallel with the ruler is larger than 10.4 centimeters. And hence, we’ve worked out that the line is actually longer than 10.4 centimeters. That’s what option D says. Therefore, option C is incorrect.

And so we have the final answer to our question. The reason that Scarlett’s measurement of 10.4 centimeters is incorrect is because the ruler is not parallel to the line. Therefore, the line is actually longer than 10.4 centimeters.

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