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Error of measurement is defined as the result of a measurement minus a true value of the measurand (Source: Page no. 36)

Uncertainty (of measurement) is a parameter, associated with the result of a measurement, characterizes the dispersion of the values that could reasonably be attributed to the measurand. (Source: Page no. 2)

The definition of uncertainty of measurement given in above is an operational one that focuses on the measurement result and its evaluated uncertainty. However, it is not inconsistent with other concepts of uncertainty of measurement, such as

⎯ a measure of the possible error in the estimated value of the measurand as provided by the result of a measurement;

⎯ an estimate characterizing the range of values within which the true value of a measurand lies (VIM:1984, definition 3.09).

Although these two traditional concepts are valid as ideals, they focus on unknowable quantities: the “error” of the result of a measurement and the “true value” of the measurand (in contrast to its estimated value), respectively. Nevertheless, whichever concept of uncertainty is adopted, an uncertainty component is always evaluated using the same data and related information.

(Source: Page no. 2,3)

Above, the uncertainty is accepted as a measure of possible error in the estimated value of the measurand as provided by the result of a measurement then, But the following resource claims that error and uncertainty are distinct concepts:

‘Error’ and ‘uncertainty’ are two complementary, but distinct, aspects of the characterization of measurements. (Source: Abstract)

Question: Are the two concepts are the same thing, or is there a subtle difference between them?

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    $\begingroup$ @Altair25 I have since retracted my post on the grounds that I answered you in haste and now believe that I had something a little different in mind, the other answers given here are superior to my own, thus please disregard my earlier input. $\endgroup$ Jan 5 at 16:42
  • $\begingroup$ The quality of 11th standard student has really increased from the time I was there $\endgroup$ Jan 7 at 6:29

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The 'error' refers to the specific unknowable difference between the measured value and the unknowable true value, while 'uncertainty' refers to the likely range of possible values of the error of the measurement.

  • An error can be positive or negative, since the measured value can be more or less than the true value.
  • An uncertainty is always positive since it refers to the width of the distribution of possible values of the error, and a width is always positive.
  • The error has a specific single value, even though this value is unknowable.
  • The uncertainty always refers to a distribution and its value depends on the conceptual framework (e.g. frequentist vs bayesian) and how the measurement system is modelled.
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In practice, working physicists will use the words "error" and "uncertainty" interchangeably. Often, precisely what is meant is glossed over (I have found many more people use the concept of error bars than can give an accurate description of what a frequentist confidence interval really means rigorously). But, often, this rough and ready understanding that any measurement has some degree of "fuzziness" is "good enough."

However, at the level of precision of your question, using the two definitions you provided, there is a difference between error and uncertainty. Now, I don't think every physicist you meet will use the same concept of error that is stated here -- I think more often than not real life physicists are likely to (perhaps sloppily) use the word error to mean something closer to what you've defined as uncertainty -- but for the purposes of this question, let's take your definitions as given.

The concept of error, as you've defined it, relies on the idea of a "true value" for some physical quantity. This concept implicitly assumes that there is some true (but unknowable) number in Nature that represents, say, the mass of an electron. We will never know this number exactly, but we hope to get closer and closer to it with better and better measurements. Then the "error" is the difference between whatever number our measurement process comes up with, and this true value. Because "error" in this sense uses the concept of an unknowable true value, we cannot actually know what the error is in our measurement. (If we did know the error exactly, we would be able to solve for the true value!). Instead, one must estimate the error, based on an understanding of statistics and of the physics of the detector.

The concept of uncertainty feels more natural to me. I would classify this as a degree of (dis)belief that the outcome of a measurement that you took gave you the answer you expected to get. From a Bayesian point of view, we can think of uncertainty as a probability distribution (representing our degree of belief) over possible values of some parameter after taking a measurement. There are several reasons your measurement will be uncertain (in other words that the distribution won't be completely sharp and focused on the "right" value), and then it is your job as an experimenter to understand and quantify them, and to minimize as much as possible the dominant sources of uncertainty:

  • "Statistical uncertainty." This represents fluctuations in the measurement output that are complicated to model, so we treat them as random, but are be uncorrelated with what you are trying to measure and should therefore tend to cancel out if you take many measurements. As a toy example, say you want to measure the length of an iron bar. Small variations in temperature in the room can cause the bar to grow or shrink by a small amount. However, by doing multiple measurements of the bar, you expect to get as many hot fluctuations as cold fluctuations relative to room temperature, and so these differences will tend to average out.
  • "Systematic uncertainty." These represent differences in the measurement output relative to what you expect, that are correlated with what you are trying to measure. These are much nastier and don't automatically "cancel out" by taking more measurements. You need to understand possible sources of systematic uncertainty, and try to eliminate them as much as possible. Some common examples are:
    • "Calibration uncertainty." The measuring apparatus itself will not be perfectly calibrated, so two apparently identical devices will give different answers when measuring the same physical object.
    • Your measurement depends on some assumed value or relationship that is not correct. For example, in gravitational wave experiments there is a degeneracy between the angle at which we observe a source, and the distance to that source. If one isn't careful about this, and (say) assumes that they are viewing a source face on when it is really edge on, then one will end up inferring a biased value of the distance.
    • "Confirmation bias." There are several fascinating examples in history, such as the Millikan oil drop experiment, where people will systematically fail to publish results that disagree with the current consensus, and publish ones that do agree with it, so that "the literature" contains a biased measurement of some value.
    • The quantity you are trying to measure might not be completely well defined past a certain point of precision. For example, your height is stable and makes sense to, say, two or three significant figures. But if you try to measure your height to atomic precision, you will find that it is constantly changing since atoms are bonding and unbonding from your hair, it matters which "vertical slice" of you is being measured, and things like that.
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Whether a "true" value even exists is a philosophical question, It is a useful tool when teaching new students, where we have widely accepted information which the students confirm to test their laboratory skills. But every widely-accepted measurement was once an unknown measurement. You can characterize the uncertainty associated with a measurement whether a "true" value is "known" or not. If all of the high-precision measurements agree with each other but disagree with the accepted value, generally the accepted value changes to some average of the trustworthy measurements.

I have a scale in my bathroom that tells me my body mass. It has a digital display with four significant figures. Do I therefore know my body weight to a precision of 0.1 pounds? I do not, for a number of reasons. I generally don't take off all of my clothes before standing on the scale. (At the doctor's office, they sometimes don't even have me empty my pockets.) The scale counts whatever food and drink I happen to be carrying in my belly, whether I'm going to incorporate that into my body or whether it will be disposed of otherwise. Does the air in my lungs contribute to my body mass? It doesn't contribute to my weight, which is what the scale actually measures, because my bathroom usually contains an air atmosphere which exerts a buoyant force on my entire body, including the lungs. A high-precision scale might show this effective weight changing if I compress my lungs by squeezing my torso muscles. For that matter, the scale really measures the compression force applied to it, and so its calibration with mass assumes my floor is level.

A lungful of air has a mass of about a gram. At precisions much better than this, "body mass" stops being meaningful quantity, because such high-precision measurements would capture mass transfer processes in the lungs, as I expel combustion products and replace them with fresh oxygen. But I can always define the uncertainty associated with a mass measurement.

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  • $\begingroup$ Thank you for your explanation 🙂 $\endgroup$
    – Altair25
    Jan 6 at 11:05
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    $\begingroup$ You're welcome! Note that comments which say only "thank you" are generally discouraged here. Instead, please pay your gratitude forward by upvoting helpful answers to your own questions, and by participating in our community in other constructive ways. $\endgroup$
    – rob
    Jan 6 at 16:33
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Typically, we try to estimate a single true value $x$ (which is usually unknown) by sampling a set of measurements $\{\hat{x}_1, \hat{x}_2, \ldots, \hat{x}_n\}$ from some random variable $\hat{X}$. From these measurements, we can construct a distribution $p(x)$ describing the likelihood of various possible values we believe the true value $x$ takes. In the figure below, the pink dots denote sample measurements, and the purple line represents one reasonable choice1 of a distribution $p(x)$.

samples

From this distribution, we can pick a single $\hat{x}$. Some common choices include:

  • The "mean" of the samples: $\hat{x} = \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i$.
  • The "mean" of the distribution: $\hat{x} = \mu = \int_{-\infty}^{\infty} y \, p(y) \, dy$.
  • The "most likely" value of the distribution: $\hat{x} = \operatorname{arg\,max}_{y} \, p(y)$.

Sometimes, these coincide (e.g., when the constructed distribution is a Gaussian with the sample mean).

The error between the final predicted value $\hat{x}$ and true value $x$ is then $\hat{e} = \hat{x} - x$, as visualized below.

error

There is some uncertainty within this prediction. This is sometimes expressed as $\sigma$, where $\sigma^2 = \int \, (y - \mu)^2 \, p(y) \, dy$ is the expected squared error in the prediction. The "$3 \sigma$" uncertainties are visualized below, for two different distributions.

uncertainty

The uncertainty of the prediction determined from the green distribution is much smaller than the uncertainty of the purple distribution. The more measurements we make of a given random variable, the less uncertain we become about the prediction. So, to improve the certainty (i.e., purple $\to$ green), we can simply make more measurements.


1 One frequent assumption is that the independent sample measurements are normally distributed around the true value $x$. In this case, the best predictor of the true value is simply the sample mean, i.e., $\hat{x} = \bar{x}$. Then, we know $p(x)$ exactly to be $p(x) = {t_{n - 1}}\left(\frac{x - \bar{x}}{s / \sqrt{n}}\right)$, where $s^2$ is the sample variance and $t_{n - 1}$ denotes the t-distribution density function for $n - 1$ degrees of freedom. See here for more.

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  • $\begingroup$ For this answer you might even say that the "Expected Value of the square of the error" IS equal to the "Variance of the underlying distribution" . IF we take Uncertainty to mean square root of variance (ie standard deviation) then we say that the uncertainty is the expected value of the magnitude of error $\endgroup$ Jan 6 at 18:05

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