Measurement accuracy and error

Some of the topics which have been explained are as follows:

• Definitions of accuracy

• Accuracy and Precision With Respect to Physics

• Accuracy and Precision in the Context of Errors

• Accuracy

• Precision

• Error

• Uncertainty

Precision is a depiction of irregular errors, a proportion of statistical changeability.

Accuracy, However, Has Two Definitions

Usually, it is used to explain «it is a depiction of systematic errors, a proportion of statistical bias (on the researcher’s or the experimenter’s part); as these are causative of a distinction between an outcome and an «actual or true» value, ISO calls this trueness. On the other hand, ISO characterizes accuracy as portraying a blend of the two kinds of observational error above (irregular and methodical), so high accuracy requires both high precision and high certainty. 

In the least difficult terms, given a lot of information focuses on repetitive measurements of a similar quantity or result, the set can be said to be exact if the values are near one another, while the set can be said to be accurate if their average is near the true value of the quantity being valued. In the first, common definition given above, the two ideas are autonomous of one another, so a specific arrangement of information can be said to be either accurate, or precise, or both, or neither of the two. 

In the domains of science and engineering, the accuracy of an estimation framework is the degree of closeness of a quantity to that quantity’s actual or true value. The accuracy of an estimation framework, identified with reproducibility and repeatability, is the degree up to which repeated estimations under unaltered conditions demonstrate the equivalent results. 

Although the two words accuracy and precision can be synonymous in everyday use, they are intentionally differentiated when used in the context of science and engineering. As mentioned earlier, an estimation framework can be accurate yet not precise, precise yet not accurate, neither or the other, or both. For instance, on the off chance that an investigation contains a systematic error, at that point expanding the sample size, for the most part, builds precision yet does not necessarily improve accuracy. 

The outcome would be a consistent yet inaccurate series of results from the defective examination. Taking out the systematic error improves accuracy however does not change precision. An estimation framework is viewed as valid on the off chance that it is both accurate and precise. Related terms include bias (either non-arbitrary and coordinated impacts brought about by a factor or factors disconnected to the independent variable) and error (random fluctuation). 

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Accuracy and Precision With Respect to Physics

Accuracy can be described as the measure of uncertainty in an experiment concerning an absolute standard. Accuracy particulars, more often than not, contain the impact of errors because of gain and counterbalance parameters. Offset errors can be given as a unit of estimation, for example, volts or ohms, and are free of the size of the input signal being estimated. 

A model may be given as ±1.0 millivolt (mV) offset error, paying little mind to the range of gain settings. Interestingly, gain errors do rely upon the extent of the input signal and are communicated as a percentage of the reading, for example, ±0.1%. All out or total accuracy is hence equivalent to the aggregate of the two: ± (0.1% of information +1.0 mV). 

Precision depicts the reproducibility of the estimation. For instance, measure a steady state’s signal repeatedly. For this situation in the event that the values are near one another, at that point, it has a high level of precision or repeatability. The values don’t need to be the true values simply assembled together. Take the average of the estimations and the difference that is between it and the true value is accuracy. If upon repeated trials the same values or nearest possible values are observed then that becomes precision.

Accuracy and precision are also predominantly affected by another factor called resolution in Physics.

Resolution May Be Expressed in Two Ways

1. It is the proportion between the highest magnitudes of signal that is measured to the smallest part that can be settled — ordinarily with an analog to-computerized (A/D) converter.

2. It is the level of change that can be hypothetically recognized, usually communicated as a certain number of bits. This relates the number of bits of resolution to the true voltage estimations.

So as to decide the goals of a framework regarding voltage, we need to make a couple of estimations. In the first place, assume an estimation framework equipped for making estimations over a ±10V territory (20V range) utilizing a 16-bits A/D converter. Next, decide the smallest conceivable addition we can distinguish at 16 bits. That is, 216 = 65,536, or 1 section (part) in 65,536, so 20V÷65536 = 305 microvolt (uV) per A/D tally. Along these lines, the smallest hypothetical change we can recognize is 305 uV.

However, unfortunately, different elements enter the condition to reduce the hypothetical number of bits that can 3-Jan-2022be utilized, for example, noise (basically any form of disruption that may cause an imbalance in an environment to disrupt the experiment). An information procurement framework determined to have a 16-bit goal may likewise contain 16 counts of noise. Thinking about this noise, the 16 counts only equals 4 bits (24 = 16); along these lines, the 16 bits of resolution determined for the estimation framework is reduced by four bits, so the A/D converter really resolves just 12 bits, not 16 bits. 

A procedure called averaging can improve the goals, yet it brings down the speed. Averaging lessens the noise by the square root of the number of samples, consequently, it requires various readings to be included and afterward divided by the total number of tests. For instance, in a framework with three bits of noise, 23 = 8, that is, eight tallies of noise averaging 64 tests would lessen the noise commitment to one tally, √64 = 8: 8÷8 = 1. Be that as it may, this system can’t decrease the effects of non-linearity, and the noise must have a Gaussian dispersion. 

Sensitivity is an absolute amount, the smallest total measure of progress that can be identified by an estimation. Consider an estimation gadget that has a ±1.0 volt input extend and ±4 checks of noise if the A/D converter resolution is 212 the crest to-top affectability will be ±4 tallies x (2 ÷ 4096) or ±1.9mV p-p. This will direct how the sensor reacts. For instance, take a sensor that is evaluated for 1000 units with a yield voltage of 0-1 volts (V). This implies at 1 volt the equal estimation is 1000 units or 1mV equals one unit. However, the affectability is 1.9mV p-p so it will take two units before the input distinguishes a change.

Accuracy and Precision in the Context of Errors

When managing error analysis, it is a smart thought to recognize what we truly mean by error. To start with, we should discuss what error isn’t. An error isn’t a silly mistake, for example, neglecting to put the decimal point in a perfect spot, utilizing the wrong units, transposing numbers, etc. The error isn’t your lab accomplice breaking your hardware. The error isn’t even the distinction between your very own estimation and some commonly accepted value. (That is a disparity.) 

Accepted values additionally have errors related to them; they are simply better estimations over what you will almost certainly make in a three-hour undergraduate material science lab. What we truly mean by error has to do with uncertainty in estimations. Not every person in the lab will have the same estimations that you do and yet (with some undeniable special cases because of errors) we may not give preference to one individual’s outcomes over another’s. Therefore, we have to group sorts of errors.

By and large, there are two sorts of errors: 1) systematic errors and 2) arbitrary or random errors. Systematic errors are errors that are steady and dependable of a similar sign and in this way may not be diminished by averaging over a lot of information. Instances of systematic errors would be time estimations by a clock that runs excessively quick or slow, removing estimations by an inaccurately stamped meter stick, current estimations by wrongly aligned ammeters, and so forth. Usually, systematic errors are difficult to relate to a solitary analysis. In situations where it is critical, systematic errors might be segregated by performing tests utilizing unique strategies and looking at results. 

On the off chance that the methodology is genuinely unique, the systematic errors ought to likewise be unique and ideally effectively distinguished. An investigation that has few systematic errors is said to have a high level of accuracy. Random errors are an entire diverse sack. These errors are delivered by any of various unusual and obscure varieties in the examination. 

Some of the examples may include changes in room temperature, changes in line voltage, mechanical vibrations, cosmic rays, and so forth. Trials with little random errors are said to have a high level of precision. Since arbitrary errors produce varieties both above and beneath some average value, we may, by and large, evaluate their importance utilizing statistical methods.

Accuracy

Accuracy alludes to the understanding between estimation and the genuine or right value. In the event that a clock strikes twelve when the sun is actually overhead, the clock is said to be accurate. The estimation of the clock (twelve) and the phenomena that it is intended to gauge (The sun situated at apex) are in the ascension. Accuracy can’t be talked about meaningfully unless the genuine value is known or is understandable. (Note: The true value of measurement can never be precisely known.)

Precision

Precision alludes to the repeatability of estimation. It doesn’t exactly require us to know the right or genuine value. In the event that every day for quite a while a clock peruses precisely 10:17 AM the point at which the sun is at the apex, this clock is exact. Since there are above thirty million seconds in a year this gadget is more precise than one part in one million! That is a fine clock in reality! You should observe here that we don’t have to consider the complicated calculations of edges of time zones to choose this is a good clock. The genuine significance of early afternoon isn’t critical in light of the fact that we just consider that the clock is giving an exact repeatable outcome.

Error

Error alludes to the contradiction between estimation and the genuine or accepted value. You might be shocked to find that error isn’t that vital in the discourse of experimental outcomes.

Uncertainty

The uncertainty of an estimated value is an interim around that value to such an extent that any reiteration of the estimation will create another outcome that exists in this interim. This uncertainty interim is assigned by the experimenter following built-up principles that estimate the probable uncertainty in the outcome of the experiment.

From Wikipedia, the free encyclopedia

Accuracy and precision are two measures of observational error.
Accuracy is how close a given set of measurements (observations or readings) are to their true value, while precision is how close the measurements are to each other.

In other words, precision is a description of random errors, a measure of statistical variability. Accuracy has two definitions:

1. More commonly, it is a description of only systematic errors, a measure of statistical bias of a given measure of central tendency; low accuracy causes a difference between a result and a true value; ISO calls this trueness.
2. Alternatively, ISO defines^[1] accuracy as describing a combination of both types of observational error (random and systematic), so high accuracy requires both high precision and high trueness.

In the first, more common definition of «accuracy» above, the concept is independent of «precision», so a particular set of data can be said to be accurate, precise, both, or neither.

In simpler terms, given a statistical sample or set of data points from repeated measurements of the same quantity, the sample or set can be said to be accurate if their average is close to the true value of the quantity being measured, while the set can be said to be precise if their standard deviation is relatively small.

Common technical definition[edit]

In the fields of science and engineering, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to that quantity’s true value.^[2] The precision of a measurement system, related to reproducibility and repeatability, is the degree to which repeated measurements under unchanged conditions show the same results.^[2][3] Although the two words precision and accuracy can be synonymous in colloquial use, they are deliberately contrasted in the context of the scientific method.

The field of statistics, where the interpretation of measurements plays a central role, prefers to use the terms bias and variability instead of accuracy and precision: bias is the amount of inaccuracy and variability is the amount of imprecision.

A measurement system can be accurate but not precise, precise but not accurate, neither, or both. For example, if an experiment contains a systematic error, then increasing the sample size generally increases precision but does not improve accuracy. The result would be a consistent yet inaccurate string of results from the flawed experiment. Eliminating the systematic error improves accuracy but does not change precision.

A measurement system is considered valid if it is both accurate and precise. Related terms include bias (non-random or directed effects caused by a factor or factors unrelated to the independent variable) and error (random variability).

The terminology is also applied to indirect measurements—that is, values obtained by a computational procedure from observed data.

In addition to accuracy and precision, measurements may also have a measurement resolution, which is the smallest change in the underlying physical quantity that produces a response in the measurement.

In numerical analysis, accuracy is also the nearness of a calculation to the true value; while precision is the resolution of the representation, typically defined by the number of decimal or binary digits.

In military terms, accuracy refers primarily to the accuracy of fire (justesse de tir), the precision of fire expressed by the closeness of a grouping of shots at and around the centre of the target.^[4]

Quantification[edit]

In industrial instrumentation, accuracy is the measurement tolerance, or transmission of the instrument and defines the limits of the errors made when the instrument is used in normal operating conditions.^[5]

Ideally a measurement device is both accurate and precise, with measurements all close to and tightly clustered around the true value. The accuracy and precision of a measurement process is usually established by repeatedly measuring some traceable reference standard. Such standards are defined in the International System of Units (abbreviated SI from French: Système international d’unités) and maintained by national standards organizations such as the National Institute of Standards and Technology in the United States.

This also applies when measurements are repeated and averaged. In that case, the term standard error is properly applied: the precision of the average is equal to the known standard deviation of the process divided by the square root of the number of measurements averaged. Further, the central limit theorem shows that the probability distribution of the averaged measurements will be closer to a normal distribution than that of individual measurements.

With regard to accuracy we can distinguish:

• the difference between the mean of the measurements and the reference value, the bias. Establishing and correcting for bias is necessary for calibration.
• the combined effect of that and precision.

A common convention in science and engineering is to express accuracy and/or precision implicitly by means of significant figures. Where not explicitly stated, the margin of error is understood to be one-half the value of the last significant place. For instance, a recording of 843.6 m, or 843.0 m, or 800.0 m would imply a margin of 0.05 m (the last significant place is the tenths place), while a recording of 843 m would imply a margin of error of 0.5 m (the last significant digits are the units).

A reading of 8,000 m, with trailing zeros and no decimal point, is ambiguous; the trailing zeros may or may not be intended as significant figures. To avoid this ambiguity, the number could be represented in scientific notation: 8.0 × 10³ m indicates that the first zero is significant (hence a margin of 50 m) while 8.000 × 10³ m indicates that all three zeros are significant, giving a margin of 0.5 m. Similarly, one can use a multiple of the basic measurement unit: 8.0 km is equivalent to 8.0 × 10³ m. It indicates a margin of 0.05 km (50 m). However, reliance on this convention can lead to false precision errors when accepting data from sources that do not obey it. For example, a source reporting a number like 153,753 with precision +/- 5,000 looks like it has precision +/- 0.5. Under the convention it would have been rounded to 154,000.

Alternatively, in a scientific context, if it is desired to indicate the margin of error with more precision, one can use a notation such as 7.54398(23) × 10⁻¹⁰ m, meaning a range of between 7.54375 and 7.54421 × 10⁻¹⁰ m.

Precision includes:

• repeatability — the variation arising when all efforts are made to keep conditions constant by using the same instrument and operator, and repeating during a short time period; and
• reproducibility — the variation arising using the same measurement process among different instruments and operators, and over longer time periods.

In engineering, precision is often taken as three times Standard Deviation of measurements taken, representing the range that 99.73% of measurements can occur within.^[6] For example, an ergonomist measuring the human body can be confident that 99.73% of their extracted measurements fall within ± 0.7 cm — if using the GRYPHON processing system — or ± 13 cm — if using unprocessed data.^[7]

ISO definition (ISO 5725)[edit]

A shift in the meaning of these terms appeared with the publication of the ISO 5725 series of standards in 1994, which is also reflected in the 2008 issue of the «BIPM International Vocabulary of Metrology» (VIM), items 2.13 and 2.14.^[2]

According to ISO 5725-1,^[1] the general term «accuracy» is used to describe the closeness of a measurement to the true value. When the term is applied to sets of measurements of the same measurand, it involves a component of random error and a component of systematic error. In this case trueness is the closeness of the mean of a set of measurement results to the actual (true) value and precision is the closeness of agreement among a set of results.

ISO 5725-1 and VIM also avoid the use of the term «bias», previously specified in BS 5497-1,^[8] because it has different connotations outside the fields of science and engineering, as in medicine and law.

In classification[edit]

In binary classification[edit]

Accuracy is also used as a statistical measure of how well a binary classification test correctly identifies or excludes a condition. That is, the accuracy is the proportion of correct predictions (both true positives and true negatives) among the total number of cases examined.^[9] As such, it compares estimates of pre- and post-test probability. To make the context clear by the semantics, it is often referred to as the «Rand accuracy» or «Rand index».^[10][11][12] It is a parameter of the test.
The formula for quantifying binary accuracy is:

where TP = True positive; FP = False positive; TN = True negative; FN = False negative

Note that, in this context, the concepts of trueness and precision as defined by ISO 5725-1 are not applicable. One reason is that there is not a single “true value” of a quantity, but rather two possible true values for every case, while accuracy is an average across all cases and therefore takes into account both values. However, the term precision is used in this context to mean a different metric originating from the field of information retrieval (see below).

In multiclass classification[edit]

When computing accuracy in multiclass classification, accuracy is simply the fraction of correct classifications:^[13]

This is usually expressed as a percentage. For example, if a classifier makes ten predictions and nine of them are correct, the accuracy is 90%.

Accuracy is also called top-1 accuracy to distinguish it from top-5 accuracy, common in convolutional neural network evaluation. To evaluate top-5 accuracy, the classifier must provide relative likelihoods for each class. When these are sorted, a classification is considered correct if the correct classification falls anywhere within the top 5 predictions made by the network. Top-5 accuracy was popularized by the ImageNet challenge. It is usually higher than top-1 accuracy, as any correct predictions in the 2nd through 5th positions will not improve the top-1 score, but do improve the top-5 score.

In psychometrics and psychophysics[edit]

In psychometrics and psychophysics, the term accuracy is interchangeably used with validity and constant error. Precision is a synonym for reliability and variable error. The validity of a measurement instrument or psychological test is established through experiment or correlation with behavior. Reliability is established with a variety of statistical techniques, classically through an internal consistency test like Cronbach’s alpha to ensure sets of related questions have related responses, and then comparison of those related question between reference and target population.^{[citation needed]}

In logic simulation[edit]

In logic simulation, a common mistake in evaluation of accurate models is to compare a logic simulation model to a transistor circuit simulation model. This is a comparison of differences in precision, not accuracy. Precision is measured with respect to detail and accuracy is measured with respect to reality.^[14][15]

In information systems[edit]

Information retrieval systems, such as databases and web search engines, are evaluated by many different metrics, some of which are derived from the confusion matrix, which divides results into true positives (documents correctly retrieved), true negatives (documents correctly not retrieved), false positives (documents incorrectly retrieved), and false negatives (documents incorrectly not retrieved). Commonly used metrics include the notions of precision and recall. In this context, precision is defined as the fraction of retrieved documents which are relevant to the query (true positives divided by true+false positives), using a set of ground truth relevant results selected by humans. Recall is defined as the fraction of relevant documents retrieved compared to the total number of relevant documents (true positives divided by true positives+false negatives). Less commonly, the metric of accuracy is used, is defined as the total number of correct classifications (true positives plus true negatives) divided by the total number of documents.

None of these metrics take into account the ranking of results. Ranking is very important for web search engines because readers seldom go past the first page of results, and there are too many documents on the web to manually classify all of them as to whether they should be included or excluded from a given search. Adding a cutoff at a particular number of results takes ranking into account to some degree. The measure precision at k, for example, is a measure of precision looking only at the top ten (k=10) search results. More sophisticated metrics, such as discounted cumulative gain, take into account each individual ranking, and are more commonly used where this is important.

In cognitive systems[edit]

In cognitive systems, accuracy and precision is used to characterize and measure results of a cognitive process performed by biological or artificial entities where a cognitive process is a transformation of data, information, knowledge, or wisdom to a higher-valued form. (DIKW Pyramid) Sometimes, a cognitive process produces exactly the intended or desired output but sometimes produces output far from the intended or desired. Furthermore, repetitions of a cognitive process do not always produce the same output. Cognitive accuracy (C_A) is the propensity of a cognitive process to produce the intended or desired output. Cognitive precision (C_P) is the propensity of a cognitive process to produce only the intended or desired output.^[16][17][18] To measure augmented cognition in human/cog ensembles, where one or more humans work collaboratively with one or more cognitive systems (cogs), increases in cognitive accuracy and cognitive precision assist in measuring the degree of cognitive augmentation.

See also[edit]

References[edit]

External links[edit]

• BIPM — Guides in metrology, Guide to the Expression of Uncertainty in Measurement (GUM) and International Vocabulary of Metrology (VIM)
• «Beyond NIST Traceability: What really creates accuracy», Controlled Environments magazine
• Precision and Accuracy with Three Psychophysical Methods
• Appendix D.1: Terminology, Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results
• Accuracy and Precision
• Accuracy vs Precision — a brief video by Matt Parker
• What’s the difference between accuracy and precision? by Matt Anticole at TED-Ed

Hello, and welcome to this video about precision, accuracy, and error! Today we’ll learn about the difference between precision and accuracy and when it’s used. We’ll also discuss different types of errors.

Before we get started, let’s review a few things. First, it’s important to remember that many math problems involve measurements of real-world quantities. These measurements include distance, weight, area, volume, temperature, and time. Whenever a quantity is being measured, there is some approximation occurring.

For instance, let’s say you need to measure a length of a wall with a tape measure before hanging a picture frame. To make sure you have the exact length, it’s best to measure the space more than once. When measuring something more than once, it’s possible to have different results. Even though you used the same tool (a tape measure) and are measuring the same space (the wall), differences in measurement can occur.

These differences are called variations, or errors. In this context, the term error does not mean a mistake. Instead, it refers to the difference between a measurement and its actual value, which is also called the known value.

Mathematicians use certain words to talk about differences between a measurement and its known value.

Precision refers to how close repeated measurements are to one another. In other words, it’s how often we get the same result, regardless of if it’s correct or not. If the measurement is consistent, it is considered to be more reliable.

Think of the measurements as points on a target.

The target on the left shows precision because all measurements are approximately the same. The target on the right does not show precision because these measurements are not approximately the same.

Accuracy refers to how close a measurement comes to the true or actual measurement of the object. In other words, it’s used to describe how close the data is to the correct data.

Consider these two targets:

Assuming that the true or actual measurement is the target’s bullseye, the target on the left shows accuracy and precision. All measurements are located at the bullseye. The target on the right does not show accuracy, even though it is precise, because the measurements are not located at or near the bullseye.

Since accuracy and precision mean two different things, measurements can be both accurate and precise. Measuring systems are considered valid if they are both accurate and precise. On the other hand, measurements can be precise without being accurate, and they can be accurate without being precise. Consider these targets:

As you can see, the upper left target shows measurements that are accurate and precise. All measurements are consistent and at the bullseye. In the upper-right target, we can see measurements that are accurate but imprecise. These measurements are not consistent but they are all close to the bullseye. In the lower-left target, the measurements are precise but inaccurate. Although they’re consistent, the measurements are not near the bullseye. Finally, in the lower right target, the measurements are neither precise nor accurate. They are inconsistent and not near the bullseye.

Now that we’ve covered accuracy and precision, let’s talk about errors.

Approximate error refers to the amount of error in a physical measurement. Approximate error is often reported as the measurement, followed by the plus or minus sign ((pm )) and the amount of the approximate error. For instance, consider an approximate error of (6.8pm 0.05) inches. This gives us a range of (6.8-0.05) to (6.8+0.05), which equals 6.75 to 6.85 inches. This range is known as a tolerance interval. It is the range in which measurements are tolerated or considered acceptable.

The greatest possible error (also called the maximum possible error) is calculated by adding or subtracting half of the measuring unit to the measurement. For instance, let’s say you measure the length of a wall to be 90 inches. The unit of measurement is 1 inch, so the greatest possible error is one-half of 1 inch, or 0.5 inches. In other words, any measurements in the range of 89.5 inches to 90.5 inches are considered valid. The tolerance interval is written as (90pm 0.5) inches.

Let’s look at an example of using the greatest possible error together. One gallon of water weighs about 8.3 pounds. What is the greatest possible error and the tolerance interval created by that error?

First, we need to identify the unit of measurement. In this case, the unit of measurement is one-tenth, or 0.1. Therefore, the greatest possible error is one-half of one-tenth, or 0.05, and the tolerance interval would be (8.3pm 0.05) pounds.

Now it’s your turn.

A furniture delivery box weighs about 22 pounds. What is the greatest possible error and the tolerance interval?

Pause the video and see if you can do this one by yourself. When you’re ready, press play and we’ll go over everything together.

Okay, let’s work through this.

First, we need to identify the unit of measurement, which is 1 pound. The greatest possible error is one-half of the unit of measurement, which is 0.5 pounds. Therefore, the tolerance interval is (22pm 0.5) pounds.

Great job!

Now onto some different methods to show errors in measurement. One of these methods is called absolute error. The absolute error is the absolute value of the difference between a measurement’s measured value and its known value.

(text {Absolute error}=left | text{measured value-known value} right |)

The difference is always reported as a positive number, which is why there are absolute value bars shown in the formula. Let’s look at an example.

Say you know a chemistry experiment will yield 10 milliliters but your solution is 9.5 milliliters. Since (10-9.5=0.5), the absolute error in this scenario is 0.5 milliliters.

When the known value is not known or is not given, then the absolute error is equal to the greatest possible error. For example, in the measurement (18pm 0.5) centimeters, no known value is given. In this case, the absolute error is 0.5 centimeters.

Another way to represent errors in measurement is with relative error. Relative error shows the significance of an error by comparing it to the original measurement. To find the relative error, find the difference between the measured value and the known value, and divide its absolute value by the known value.

(text {Relative error}=frac{left | text{measured value-known value} right |}{text{known value}})

Let’s find the relative error of the chemistry experiment example we just discussed. Recall that the known value was 10 milliliters and the measured value was 9.5 milliliters. To find the relative error, substitute these values into the formula and solve.

Since our measured value is 9.5, and our known value is 10, take the absolute value of (9.5-10) and divide it by 10.

If there is no known value given, then the relative error is found by dividing the greatest possible error by the measured value.

(text {Relative error}=frac{text{greatest possible error}}{text{measured value}})

In the measurement (18pm 0.5) centimeters, no known value is given. Recall that in this case, the greatest possible error is 0.5 centimeters. To find the relative error, substitute these values into the formula and solve:

(text {Relative error}=frac{text{greatest possible error}}{text{measured value}}=frac{0.5}{18}approx 0.027text{ centimeters})

Compared to absolute error, relative error is often seen as more useful because absolute error doesn’t show the significance of the error in context. For example, a soccer field is 2,700 inches long and your measurement is off by 0.5 inches, 2,699.5 inches. In this case, that 0.5 inches won’t throw off your calculations too much. However, if a post-it note is 3 inches long and your measured value is off by the same 0.5 inches, and you get 2.5 inches, that’s a much more significant difference due to the size of the post-it note.

In our relative error formula, dividing by the known value gives us that context of if our object is off by 0.5 inches compared to a really large number like with our soccer field, or off by 0.5 inches compared to a really small number like with our post-it note. Absolute error only tells us that we are off by 0.5 inches.

A third way to represent errors in measurement is by calculating the percent of error. To find the percent of error, multiply the relative error by 100 to make it a percent.

In the chemistry example, we found that the relative error was 0.05 milliliters. 0.05 times 100 equals 5, so the percent of error in this experiment is 5%.

Now that we know how to show different errors in measurement, let’s take a look at an example together.

Consider a bedroom with a known length of 16 feet. The bedroom is measured to be 16.5 feet long. Let’s find the absolute error, relative error, and percent of error.

(0.03125times 100=3.125%)

Therefore, the difference in measurement of the bedroom was 0.5 feet, which is just over a 3% error.

Let’s take a look at another example. Consider the measurement of (2.5pm 0.5) inches. Unlike our last example, there is no known value given.

(text{Relative error}=frac{text{greatest possible error}}{text{measured value}}=frac{0.5}{2.5}=0.2text{ inches})

(0.2times 100=20%)

Therefore, the 0.5-inch difference in measurement is a 20% error.

Now it’s your turn. I’m going to give you a set of measurements, and you’re going to find the absolute error, relative error, and percent of error. This one is a little more challenging, but I know you can do it.

Micah is using a ruler to find the radius of a circle. He measures the radius to be 5.5 centimeters, but the actual length of the radius is 5.8 centimeters. Find Micah’s absolute error, relative error, and percent of error.

Pause the video and see if you can do this one by yourself. When you’re ready, press play and we’ll go over it together.

Now that you’ve tried this problem by yourself, let’s go over it together.

(0.05172414times 100=5.17241379% approx 5.17%)

Therefore, the difference in measurement of the circle is 0.3 centimeters, which is just over a 5% error.

I hope this video about precision, accuracy, and error was helpful. Thanks for watching, and happy studying!

Practice Questions