# Tag Info

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The absolute error in any one trial (with name or index $k$) is $$\varepsilon_a^k = \widetilde x_k - x_k,$$ where $x_k$ is the true value of the quantity under consideration in trial $k$, and $\widetilde x_k$ is the value which is inferred of that quantity in trial $k$, with the techniques and observational data available. The average of the absolue ...

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The absolute error can be measured using this formula: $$\varepsilon_a=\frac{x_{max}-x_{min}}{2}$$ That is the difference between the highest value and the lowest value that you get after some measurements. The Relative error is: $$\varepsilon_r=\frac{\varepsilon_a}{\bar{x}}$$ where $\bar{x}$ is the average of all your measurements. There is also there is ...

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P1 - They WERE used on land, extensively. Note on land the horizon is indeterminate as hills and valleys are present. A still pool of water (aka artificial horizon) is known to perfectly parallel to the true horizon and accurate measurements can be made. (Compare the direct image of the Sun to its reflected image and note the angle between the two.) P2 - ...

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Yes, if you measure 10 periods at four digits precision, then after dividing by the 10 (an exact integer) you're still good to four digits. Imagine if instead of a pendulum, you were measuring radio waves at several MHz. You measure let's say exactly one billion cycles using your timer good to 100ths of a second. Divide by one billion. How good should ...

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The standard way to propagate uncertainties is, in this case, $$\delta D = \left|\frac{\partial D}{\partial r}\right|\delta r=\frac{c\, \delta r}{r^2},$$ where $\delta r$ is a positive quantity. Then $\delta D>0$ gives you half the width of your uncertainty interval in $D$.

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Do the calculation at the computer/calculator precision then quote result with the appropriate number of significant figures. Yes that is correct procedure. Otherwise you could introduce rounding error.

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I remember reading somewhere that the problem of exact time-keeping on ships could have been solved a lot earlier than it was if somebody would have had the idea of keeping time with a whole array of imprecise clocks - taking the average of clock-times would have given a precise time. Define precise . Precise for timekeeping in ships would be accuracy ...

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You can get a more accurate measurement from multiple measurements. But each measurement has room to damage the target, and you would not get enough measurements to get to atomic standards. Most likely, you could get to micron measures over 1000000 people, assuming each measurement is to the nearest mill. But as likely, most readers would report 1234.12 ...

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This is highly unlikely. It comes down to bias and variance. Individual people of course will estimate with limited accuracy, whether just guessing, eyeballing, or using latest and greatest measuring technology. By itself that would not be a problem if people were unbiased estimators and their estimates were independent. All errors would then be variance ...

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This will not work. I'm going to use the standard error of the mean as the measure of the precision: $\mathrm{SEM} = \sigma_x / \sqrt{N}$. $N$ is the number of people you have make estimates of he length, and $\sigma_x$ is the standard deviation of the estimates that everyone makes of the length. The standard deviation of the sample is given by the square ...

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I believe the answer is no. Let's simplify the question a little by limiting the number of persons doing the measurements to 1. Of course if you show the same stick over and over again and if the person knows she is shown the same stick, she will be making only one measurement. This can be avoided by sampling different length sticks, but the problem will ...

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No, of course not. Yes, some people will overestimate and others will underestimate. Averaging would cancel out the bias to some extent, but there's no reason to expect it to cancel out the bias perfectly. We all have similar eyes and brains. We are all deceived by the same optical illusions, in the same way. We all have a shared cultural understanding of ...

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The people making measurements don't know what the precise answer is so that they can make 'imprecise' measurements. I think you need to understand what 'error' exactly means. Read this explanation: A measurement may be made of a quantity which has an accepted value which can be looked up in a handbook (e.g.. the density of brass). The difference between ...

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I guess this depends on what exactly we mean by "estimate". If estimate means making up some number that's one thing. But if it implies some kind of [visual] measurement that's another thing. It may be difficult to think about a human being producing any good "measurement" visually, so let's ask a question instead: Can an accurate measurement of an object be ...

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I believe you are thinking of the Central Limit Theorem. The mean and variance of the averages of many measurements are better estimates of the precision of your measuring rule, but don't tell you anything about the accuracy of your measuring rule. Your measuring rule may be biased. The Central Limit Theorem is a part of mathematics. IMO you should also ...

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No because none of them know the actual answer. The averaging process you describe only works if each estimate is of the exact answer plus noise. Otherwise it is known as the "Emperor's nose" problem. Nobody can see the Chinese emperor's face so they ask a million peasants how long his nose is, they average the results, and since they have such a large 'N' ...

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