Relative measurement error

Absolute error is the actual amount you were off, or mistaken by, when measuring something. Relative error compares the absolute error against the size of the thing you were measuring. In order to calculate relative error, you must...


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Absolute error is the actual amount you were off, or mistaken by, when measuring something. Relative error compares the absolute error against the size of the thing you were measuring. In order to calculate relative error, you must calculate the absolute error as well. If you tried to measure something that was 12 inches long and your measurement was off by 6 inches, the relative error would be very large. But, if you tried to measure something that was 120 feet long and only missed by 6 inches, the relative error would be much smaller — even though the value of the absolute error, 6 inches, has not changed.

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    When given an expected value, subtract the value you got from the expected value to get the Absolute Error. An expected value is usually found on tests and school labs. Basically, this is the most precise, common measurement to come up with, usually for common equations or reactions. You can compare your own results to get Absolute Error, which measures how far off you were from the expected results. To do so, simply subtract the measured value from the expected one. Even if the result is negative, make it positive. This is your absolute error![1]

    • Example: You want to know how accurately you estimate distances by pacing them off. You pace from one tree to another and estimate that they’re 18 feet apart. This is the experimental value. Then you come back with a long measuring tape to measure the exact distance, finding out that the trees are in fact 20 feet (6 meters) apart. That is the «real» value. Your absolute error is 20 — 18 = 2 feet (60.96 centimeters).

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    Alternatively, when measuring something, assume the absolute error to be the smallest unit of measurement at your disposal. For example, if you’re measuring something with a meter stick, the smallest unit marked on the meter stick is 1 millimeter (mm). So you know that your measurement is accurate to within + or — 1 mm; your absolute error is 1 mm.

    • This works for any measurement system. Many scientific tools, like precision droppers and measurement equipment, often has absolute error labeled on the sides as «+/- ____ «

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    Always add the appropriate units. Say your Absolute Error was «2 meters.» This tells your viewers exactly how far off your error was. But if you write that your error was simply «2,» this doesn’t tell your audience anything. Use the same unites as the ones in your measurements.

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    Practice with several examples. The best way to learn how to calculate error is to go ahead and calculate it. Take a stab at the following problems, then highlight the space after the colon (:) to see your answer.

    • Jill is studying chemical reactions. After mixing and matching, her test tube contains 32 grams of substrate. The accepted value for her experiment was 34 grams. Her Absolute Error is: +/- 2 grams
    • Clive is testing reactions in chemistry. It takes 10ml drops of water to cause a reaction, but his dropper claims it is «+/- .5ml.» The Absolute Error in his measurements must be: +/- .5ml

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    Understand what causes error, and how you can work to eliminate it. No scientific study is ever perfectly error free — even Nobel Prize winning papers and discoveries have a margin or error attached. Still, understanding where error comes from is essential to help try and prevent it:[2]

    • Human error is the most common. This is from bad measurements, faulty premises, or mistakes in the lab.
    • Incidental energy/material loss, such as the little fluid left in the beaker after pouring, changes in temperature due to the environment, etc.
    • Imperfect equipment used either for measurement or studies, such as very small, precise measurements or burners that provide uneven heat.[3]

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    Divide the Absolute Error by the Actual Value of the item in question to get Relative Error. The result is the relative error.[4]

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    Multiply the answer by 100 to get an easier to understand percentage. Leave the relative error in fraction form, complete the division to render it in decimal form, or multiply the resulting decimal form by 100 to render your answer as a percentage. This tells you what percentage of the final measurement you messed up by. If you are measuring a 200 foot boat, and miss the measurement by 2 feet, your percentage error will be much lower than missing the 20 foot tree measurement by 2 feet. The error is a smaller percentage of the total measurement.[5]

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    Calculate Relative Error all at once by turning the numerator (top of fraction) into your Absolute Error equation. Once you understand the difference between Absolute and Relative Error, there is really no reason to do everything all by itself. Simply substitute the equation for Absolute Error in for the actual number. Note that the vertical bars are absolute value signs, meaning anything within them must be positive.[6]

    • Relative Error ={frac {|{mathrm {Measured}}-{mathrm {Actual}}|}{{mathrm {Actual}}}}
    • Multiply the whole thing by 100 to get Relative Error Percentage all at once.[7]

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    Always provide units as context. Let the audience know the units you’re using for measurement. However, the relative error does not employ units of measurement. It is expressed as a fraction or a percentage, such as a relative error of 10%.[8]

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Add New Question

  • Question

    What does the +/- sign tell about the relative percentage error?

    Donagan

    It means the reported or estimated amount could be higher or lower than the true amount.

  • Question

    What is the difference between systematic and random errors?

    Community Answer

    Systematic errors are those which occur according to a certain pattern or system; these errors are due to known reasons. Random errors have no set pattern or cause.

  • Question

    If the absolute error was 0.94, then what will the relative error be?

    Aditya Kannan

    Aditya Kannan

    Community Answer

    Relative error, as mentioned in the answer, equals (Absolute Error)/(Actual Value). Hence, it isn’t possible to calculate relative error just by knowing the absolute error.

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  • Make sure that your experimental value and real value are all expressed in the same unit of measurement. For example, if your experimental value is in inches but your real value is in feet, you must convert one of them to the other unit of measurement.

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References

About This Article

Article SummaryX

Before you can calculate relative error, you must calculate the absolute error in your calculations. To do this, subtract your answer from the expected value, or the correct answer. Write the answer as a positive number, even if it’s negative, and add the appropriate units. To get the relative error, divide the absolute error by the actual value of the item in question. If you’d like, you can multiply the answer by 100 to display it as a percentage. To understand when you would need to use relative error, read on!

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Absolute and Relative Error Calculation

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Updated on October 06, 2019

Absolute error and relative error are two types of experimental error. You’ll need to calculate both types of error in science, so it’s good to understand the difference between them and how to calculate them.

Absolute error is a measure of how far ‘off’ a measurement is from a true value or an indication of the uncertainty in a measurement. For example, if you measure the width of a book using a ruler with millimeter marks, the best you can do is measure the width of the book to the nearest millimeter. You measure the book and find it to be 75 mm. You report the absolute error in the measurement as 75 mm +/- 1 mm. The absolute error is 1 mm. Note that absolute error is reported in the same units as the measurement.

Alternatively, you may have a known or calculated value and you want to use absolute error to express how close your measurement is to the ideal value. Here absolute error is expressed as the difference between the expected and actual values.

Absolute Error = Actual Value — Measured Value

For example, if you know a procedure is supposed to yield 1.0 liters of solution and you obtain 0.9 liters of solution, your absolute error is 1.0 — 0.9 = 0.1 liters.

Relative Error

You first need to determine absolute error to calculate relative error. Relative error expresses how large the absolute error is compared with the total size of the object you are measuring. Relative error is expressed as a fraction or is multiplied by 100 and expressed as a percent.

Relative Error = Absolute Error / Known Value

For example, a driver’s speedometer says his car is going 60 miles per hour (mph) when it’s actually going 62 mph. The absolute error of his speedometer is 62 mph — 60 mph = 2 mph. The relative error of the measurement is 2 mph / 60 mph = 0.033 or 3.3%

Sources

  • Hazewinkel, Michiel, ed. (2001). «Theory of Errors.» Encyclopedia of Mathematics. Springer Science+Business Media B.V. / Kluwer Academic Publishers. ISBN 978-1-55608-010-4.
  • Steel, Robert G. D.; Torrie, James H. (1960). Principles and Procedures of Statistics, With Special Reference to Biological Sciences. McGraw-Hill. 
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Absolute, Relative, and Percent Error
Absolute, relative, and percent error are ways to measure the error of a measurement or calculation.

Absolute, relative, and percent error are the most common experimental error calculations in science. Grouped together, they are types of approximation error. Basically, the premise is that no matter how carefully you measure something, you’ll always be off a bit due to the limitations of the measuring instrument. For example, you may be only able to measure to the nearest millimeter on a ruler or the nearest milliliter on a graduated cylinder. Here are the definitions, equations, and examples of how to use these types of error calculations.

Absolute Error

Absolute error is the magnitude (size) of the difference between a measured value and a true or exact value.

Absolute Error = |True Value – Measured Value|

Absolute Error Example:
A measurement is 24.54 mm and the true or known value is 26.00 mm. Find the absolute error.
Absolute Error = |26.00 mm – 25.54 mm|= 0.46 mm
Note absolute error retains its units of measurement.

The vertical bars indicate absolute value. In other words, you drop any negative sign you may get. For this reason, it doesn’t actually matter whether you subtract the measured value from the true value or the other way around. You’ll see the formula written both ways in textbooks and both forms are correct.

What matters is that you interpret the error correctly. If you graph error bars, half of the error is higher than the measured value and half is lower. For example, if your error is 0.2 cm, it is the same as saying ±0.1 cm.

The absolute error tells you how big a difference there is between the measured and true values, but this information isn’t very helpful when you want to know if the measured value is close to the real value or not. For example, an absolute error of 0.1 grams is more significant if the true value is 1.4 grams than if the true value is 114 kilograms! This is where relative error and percent error help.

Relative Error

Relative error puts absolute error into perspective because it compares the size of absolute error to the size of the true value. Note that the units drop off in this calculation, so relative error is dimensionless (unitless).

Relative Error = |True Value – Measured Value| / True Value
Relative Error = Absolute Error / True Value

Relative Error Example:
A measurement is 53 and the true or known value is 55. Find the relative error.
Relative Error = |55 – 53| / 55 = 0.034
Note this value maintains two significant digits.

Note: Relative error is undefined when the true value is zero. Also, relative error only makes sense when a measurement scale starts at a true zero. So, it makes sense for the Kelvin temperature scale, but not for Fahrenheit or Celsius!

Percent Error

Percent error is just relative error multiplied by 100%. It tells what percent of a measurement is questionable.

Percent Error = |True Value – Measured Value| / True Value x 100%
Percent Error = Absolute Error / True Value x 100%
Percent Error = Relative Error x 100%

Percent Error Example:
A speedometer says a car is going 70 mph but its real speed is 72 mph. Find the percent error.
Percent Error = |72 – 70| / 72 x 100% = 2.8%

Mean Absolute Error

Absolute error is fine if you’re only taking one measurement, but what about when you collect more data? Then, mean absolute error is useful. Mean absolute error or MAE is the sum of all the absolute errors divided by the number of errors (data points). In other words, it’s the average of the errors. Mean absolute error, like absolute error, retains its units.

Mean Absolute Error Example:
You weigh yourself three times and get values of 126 lbs, 129 lbs, 127 lbs. Your true weight is 127 lbs. What is the mean absolute error of the measurements.
Mean Absolute Error = [|126-127 lbs|+|129-127 lbs|+|127-127 lbs|]/3 = 1 lb

References

  • Hazewinkel, Michiel, ed. (2001). “Theory of Errors.” Encyclopedia of Mathematics. Springer Science+Business Media B.V. / Kluwer Academic Publishers. ISBN 978-1-55608-010-4.
  • Helfrick, Albert D. (2005). Modern Electronic Instrumentation and Measurement Techniques. ISBN 81-297-0731-4.
  • Steel, Robert G. D.; Torrie, James H. (1960). Principles and Procedures of Statistics, With Special Reference to Biological Sciences. McGraw-Hill. 
Источник

In this explainer, we will learn how to define and calculate the absolute and relative errors of measured values.

When measuring a value, it is important to be able to know how accurate the measurement is. When determining such accuracy,
the value must be compared to some other value that is deemed to be correct, the accepted value.

An accepted value, also called the actual value, is a measured value obtained by an error-free measurement process. It is what all other measured values are compared to. Accepted values are typically constants, such as the gravitational
constant or charge of an electron.

Measurement error is when the measurement of a value differs from the accepted value. If we know that the mass of
a block of cheese is 1 kg, but a scale says it is
1.2 kg, this is an example of measurement error.

Whatever the source of the error is, there are two different ways to quantify it. Let’s first look at
absolute error.

Absolute error is the absolute difference between the accepted value and the measured value. When expressed as an
equation, it looks as follows:
absoluteerroracceptedvaluemeasuredvalue=|−|.

The lines on the right side of the equation indicate that the difference is an absolute value. An absolute value
only cares about the magnitude of the number, meaning it will always be positive, even if the measured value is larger
than the accepted value.

For the cheese, the accepted value is 1 kg, and the measured value
is 1.2 kg. Substituting these values into the equation gives
|1−1.2|=0.2.kgkgkg

So, even though 1−1.2 results in a negative 0.2, because it is an absolute value, it becomes positive. The cheese has an absolute error of 0.2 kg.

Let’s have a look at an example.

Example 1: Calculating the Absolute Error in the Measurement of an Accepted Value

In an experiment, the acceleration due to gravity at the surface of Earth is measured to be
9.90 m/s2. Find the absolute error in the measurement
using an accepted value of 9.81 m/s2.

Answer

To find the absolute error of the measurement value of 9.90 m/s2,
we must find the difference between it and the accepted value of
9.81 m/s2, as shown in the equation for absolute error. Recall
that the equation for absolute error is
absoluteerroracceptedvaluemeasuredvalue=|−|.

The accepted value is 9.81 m/s2, and the measured value is
9.90 m/s2, so substituting these into the equation for absolute
error gives
||9.81/−9.90/||=0.09/.msmsms

Absolute error is an absolute value, and so it will always be positive, even though
9.81−9.90 results in a negative number. The
absolute error is thus 0.09 m/s2.

Absolute error is not always helpful in determining the accuracy of a measurement though. Say that we have a colossal
cheese wheel with an accepted value of mass of 1‎ ‎000 kg. When the
cheese wheel is put on a scale, it has a measured mass of 1‎ ‎000.2 kg.

Using these values, we see that when putting them into the equation for absolute error
|1000−1000.2|=0.2,kgkgkg
we have the same value of absolute error for the colossal 1‎ ‎000 kg
cheese wheel as we had for the considerably smaller 1 kg block of
cheese. The 0.2 kg matters more for smaller masses than larger ones,
and there is a way to express this, relative error.

Relative error is a way of showing the error proportional to the accepted value. It is found by taking the absolute
error and dividing it by the accepted value
𝑟=Δ𝑥𝑥,
where 𝑟 is the relative error, Δ𝑥 is the absolute error, and 𝑥
is the accepted value.

Both the colossal wheel of cheese and the block have the same value of absolute error,
0.2 kg. Since the colossal wheel of cheese has a much larger accepted value,
we should expect the relative error to be smaller than the single block of cheese. The relative error for the wheel is
0.21000=0.0002,kgkg
and the relative error for the block is
0.21=0.2.kgkg

Note that because the units are the same for both the numerator and denominator of the equation, they cancel, making the
relative error unitless.

Let’s have a look at some examples.

Example 2: Calculating an Absolute Error from a Relative Error

If the relative error in measuring an area of 320 m2 was
0.03, calculate the absolute error for that measurement.

Answer

We are given two values initially, the relative error of 0.03 and the accepted value of
320 m2. We need to find the absolute error, which we can do by looking
at the equation for relative error. Recall that the equation for relative error is
𝑟=Δ𝑥𝑥,
where 𝑟 is the relative error, Δ𝑥 is the absolute error, and 𝑥 is the
accepted value.

To isolate the absolute error, Δ𝑥, we need to think algebraically. Let’s multiply both sides
of the equation by the accepted value, 𝑥𝑟×𝑥=Δ𝑥𝑥×𝑥,
which cancels the accepted value on the right side of the equation, giving
𝑟×𝑥=Δ𝑥.

Using this modified equation, we can now substitute in the given values. Relative error is 0.03, and the accepted value is
320 m2:
0.03×320=9.6.mm

Relative error is unitless, so the multiplication inherits the units of
m2. Our value of absolute error is thus
9.6 m2.

Example 3: Identifying the Measurement That Has the Greatest Accuracy

Which of the following measurements of time is the most accurate?

  1. 3.4±0.1 s
  2. 5.2±0.01 s
  3. 7.3±0.2 s
  4. 4.1±0.2 s

Answer

The ± symbol means plus or minus a particular value, with the number following it being the absolute
error. To determine which measurement of time is most accurate, we will need to find the relative error, as the measurement
that has the lowest relative error is the most accurate. Recall that the relative error equation is absolute error
over the accepted value,
𝑟=Δ𝑥𝑥.

In this problem, the absolute error is the number after the ± and the accepted value is before it. Let’s look at each potential answer individually, starting with A:
0.13.4=0.029.ss

Subsequently, the relative error for B is
0.015.2=0.002,ss
the relative error for C is
0.27.3=0.027,ss
and the relative error for D is
0.24.1=0.049.ss

We see from these that answer B has the smallest relative error, of only 0.002. We could also have determined this by looking
at the absolute errors for each option: much smaller absolute errors would also give smaller relative errors.

Relative error is often expressed using a slight modification, making it a percentage.

Percent relative error is relative error expressed as a percentage, which is calculated by multiplying the value by
100%:
𝑟×100%=𝑟,%
where 𝑟% is the percent relative error.

Looking back at the cheese, the smaller block of cheese had a relative error of 0.2. The percent relative error is thus
0.2×100%=20%,
so the block of cheese has a percent relative error of 20%, or the measurement was off by
20%.

The colossal wheel of cheese has a much smaller percent relative error:
0.0002×100%=0.02%.

This larger proportional difference in percentage error for the smaller blocks of cheese means that the errors in
measurement will stack up much faster. If, for instance, you are tasked with measuring out
1‎ ‎000 kg of cheese, choosing the single colossal wheel of
1‎ ‎000 kg will result in an accuracy of 0.02%. If you were to instead choose 1‎ ‎000 of the smaller blocks, the percent relative error would use the much higher 20%.

To get the actual value of how much cheese in kilograms
the percent relative error will result in, divide the percent relative error by 100% to convert back
to the relative error. Comparing the two, the colossal wheel’s is
1000×0.02%100%=0.2,kgkg
while the smaller block of cheese’s is
1000×20%100%=200.kgkg

So, while the colossal wheel’s mass will only vary by 0.2 kg,
choosing to instead use the stack of 1‎ ‎000 smaller cheese blocks will have their mass vary by a full
200 kg. Bringing anywhere between 800 and
1‎ ‎200 kg of cheese when you were supposed to have
1‎ ‎000 kg is a big mistake to make.

Since relative error is based on absolute error and the accepted value, the equation for percent relative error,
𝑟% is written as
𝑟=Δ𝑥𝑥×100%,%
where Δ𝑥 is the absolute error and 𝑥 is the accepted value.

Let’s look at some examples using the percent relative error.

Example 4: Calculating the Relative Error in a Measurement of an Accepted Value

In an experiment, the speed of sound waves on Earth at sea level at a temperature of
21∘C is
333 m/s. Find the percent relative error in the measurement
using an accepted value of 344 m/s. Give your answer to one
decimal place.

Answer

In this problem, the given values are the measured value of 333 m/s
and the accepted value of 344 m/s. Recall the percent relative error
equation
𝑟=Δ𝑥𝑥×100%,%
where Δ𝑥 is the absolute error and 𝑥 is the accepted value.

The absolute error is needed, which is found by taking the difference between the measured and accepted values:
344/−333/=11/.msmsms

The relative error is then calculated by dividing the absolute error,
11 m/s, by the accepted value of
344 m/s:
Δ𝑥𝑥=11/344/11/344/=0.03197…,msmsmsms
making the relative error 0.03197…. The answer should eventually be to one decimal place, but it is
not rounded until the end of the problem for maximum accuracy. To get the percent relative error, this value is then
multiplied by 100%:
0.03197…×100%=3.197…%.

Now that the answer is in its final form, it can be rounded off to one decimal place, making the percent relative error
3.2%.

Example 5: Determining a Value from Its Absolute and Relative Error

The relative and absolute errors in measuring the mass of some box are found to be
1.6% and 0.4 kg respectively. Calculate the actual value of the mass.

Answer

The actual value is the accepted value, and it can be found by using the extended equation for percent relative error
Δ𝑥𝑥×100%=𝑟,%
where Δ𝑥 is the absolute error and 𝑥 is the accepted value.

The accepted value, 𝑥, needs to be isolated, which can be done algebraically. Let’s start by
multiplying both sides by the accepted value:
Δ𝑥𝑥×100%×𝑥=𝑟×𝑥.%

This causes the accepted values on the left to cancel out, leaving behind
Δ𝑥×100%=𝑟×𝑥.%

Both sides can then be divided by the percent relative error to give
Δ𝑥×100%𝑟=𝑟×𝑥𝑟,%%%
making the percent relative error cancel on the right, which forms an equation with an isolated accepted value:
Δ𝑥×100%𝑟=𝑥.%

Now, the values of absolute error, 0.4 kg, and percent relative error
of 1.6% can be substituted in
0.4×100%1.6%=25,kgkg
causing the percentage signs to cancel, leaving behind the accepted value of the mass as
25 kg.

Let’s now summarize what we learned in this explainer.

Key Points

  • The accepted value is the actual value that is considered correct.
  • Measurement error is when the measured value differs from the accepted value.
  • Absolute error is the difference between the accepted value and measured value, and it is in the same units as the values.
  • Relative error is the proportion of absolute error and the accepted value, and it is unitless.
  • Percentage relative error is relative error expressed as a percent.
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HelpYouBetter » Physics » Units and Measurements » Measurement error and types of errors in measurement

I have already written articles about the basics of units and measurement, different conversion charts, dimensions and dimensional analysis etc. In this article, I focus on the error in measurement, different types of errors and the combination of errors which occurs during the measurement of a physical quantity.

So let’s start with the measurement error. While measuring any physical quantity, it is practically impossible to find its true value. The difference between the true value and the measured value of a physical quantity is called the error in its measurement. In other words, we can say, the result of every measurement by any measuring instrument contains some uncertainty and this uncertainty is called the error.

Distinguish between Accuracy, Least Count and Precision.

Before going into the types of errors, let’s distinguish between three terms: accuracy, least count, and precision. The accuracy of a measurement is the relative exemption from errors. That is, accuracy is the measure of how close the measured value is to the actual value of the quantity.

For every instrument, there is a minimum value that can be measured accurately. This is called the least count of that instrument. It is 0.1 cm for an ordinary scale, 0.01 cm for an ordinary vernier calliper and 0.001 cm for an ordinary screw gauge.

Precision describes the limit or resolution of the quantity measured. For example, consider an iron rod of length 12 cm. The scale 1 measures it to be 11.9 cm and scale 2 measures it to be 12.426 cm. Here scale 1 is more accurate but scale 2 is more precise. Now another scale 3 measures it to be 12.0056 cm. We can say scale 3 is both accurate and precise.

Also, learn significant figures and the rules for rounding off the uncertain digits.

What are the different types of errors in measurement?

The errors that may occur in the measurement of a physical quantity can be classified into six types: constant error, systematic error, random error, absolute error, relative error and percentage error. Each type of error in measurement are explained below.

  • Constant error

    Constant errors are those which affect the result by the same amount.
    For eg: If the reading of a thermometer, when placed in melting ice at normal pressure, is 10 C, then the instrument has an error by 10 C.

  • Systematic error

    Systematic errors are due to some known causes according to a definite law and are tend to be in one direction, either positive or negative. We can minimize the systematic errors by selecting better instruments, by improving the experimental techniques or procedures and by removing personal errors as far as possible. For a given experimental set-up, these systematic errors may be calculated to a certain extent and the necessary corrections may be applied to the observed readings.

    Types of systematic error

    There are four sources or types of systematic error: Instrumental error, gross error, error due to external causes and the error due to imperfections.

    1. Instrumental error

      Instrumental errors are errors due to the apparatus or measuring instruments used. It may be errors due to the imperfect design or calibration of the measuring instrument, zero error in the instrument etc. It depends on the limit or resolution of the measuring instrument.

      For eg, using a metre scale with graduations at one mm interval, the accuracy of the reading is limited to one mm. The error in the reading of the metre scale is taken to be of the order of half of the smallest division on the scale; that is, of the order of 0.5 mm. When vernier calliper with least count 0.1 mm is used for the measurement, the error is about 0.05 mm. These errors are called instrumental errors.

    2. Gross error

      The gross error is another type of systematic error which are committed due to the personal peculiarities of the experiment like carelessness in taking observations without observing necessary precautions or lack of proper setting of the measuring instruments, etc. The gross error is also called as the personal error or observation error.

      For example, while taking the reading from the instrument meter observer may read 41 as 47.
      Another example: Parallax error which arises due to the habit of taking measurements by always holding the observer’s head a bit too far to the right or left while reading the position of a needle on the scale.
      We can reduce the gross error by increasing the number of observers who are taking the readings. Also, proper care should be taken while reading and recording the data.

    3. Error due to external causes

      These errors arise due to change of external conditions like temperature, wind velocity, pressure, humidity, electric field or magnetic field etc.
      For eg: During summer, the length of the iron metre scale becomes more than one metre.

    4. Error due to imperfection in experimental technique or procedure

      Some errors occur due to imperfection in the experimental arrangement.
      For example, while determining the human temperature, a thermometer placed under the armpit will always give a temperature lower than the actual value of the body temperature.
      Another example: The loss of heat due to radiation in a calorimeter.

  • Random error

    Random error is the error caused by the individual who measures the quantity. The random error depends on the qualities of the measuring person and the care taken in the measuring process. It is also called as the chance error. In order to minimise random errors, the measurements are repeated several times and the average (arithmetic mean) value is taken as the correct value of the measured quantity. The mean value would be very close to the most accurate reading. When the number of observation is made ‘n’ times, the random error reduces to 1/n times.
    If a1, a2, a3 ….. an are the n different readings of a physical quantity when it is measured, the most accurate value is its arithmetic mean value which is given by large begin{aligned} a_{mean}=frac{a_{1}+a_{2}+.....+a_{n}}{n}=frac{1}{n};sum_{i=1}^{n}a_{i} end{aligned}

  • Absolute error

    The magnitude of the difference between the true value of the quantity and the measured value is called the absolute error in the measurement. Since the true value of the quantity is not known, the arithmetic mean of the measured values may be taken as the true value.
    If a1, a2, ….. are the measured values of a certain quantity, the errors in ∆a1, ∆a2, ……… in the measurements are
    ∆a1 = amean – a1
    ∆a2 = amean – a2
    …………………………………
    The arithematic mean of all the absolute errors is taken as the final absolute error in the measurement and is known as mean absolute error.
    large begin{aligned} Delta a_{mean}=frac{left | Delta a_{1} right |+left | Delta a_{2} right |+....+left | Delta a_{n} right |}{n} = frac{1}{n}sum_{i=1}^{n}left | Delta a_{i} right |end{aligned}
    The value obtained in a single measurement may be in the range
    amean  ±  ∆ amean

  • Relative error

    The ratio of the absolute error to the true value of the measured quantity is called the relative error or fractional error. Since the arithmetic mean value is taken as the true value, the relative error is given by,
    large begin{aligned} Relative;error,delta a = frac{Delta a_{mean}}{a_{mean}}end{aligned}

  • Percentage error

    It is the relative error exprressed in percentage.
    large begin{aligned}Percentage;error = frac{Delta a_{mean}}{a_{mean}}times 100%end{aligned}

Example:

When the diameter of a wire is measured using a screw gauge, the successive readings are found to be 1.10 mm, 1.12 mm, 1.14 mm, 1.08 mm, 1.16mm and 1.17mm. Calculate the absolute errors and the relative error in the measurement.

Ans: Arithmetic mean value of the mesurement is

largebegin{aligned}a_{mean}=frac{1.10+1.12+1.14+1.08+1.16+1.17}{6} = 1.128 mmend{aligned}

Difference between amean and measured value Magnitude of errors
1.128 – 1.10 =    0.028 mm 0.028 mm
1.128 – 1.12 =    0.008 mm 0.008 mm
1.128 – 1.14 = –0.012 mm 0.012 mm
1.128 – 1.08 =   0.048 mm 0.048 mm
1.128 – 1.16 = –0.032 mm 0.032 mm
1.128 – 1.17 = –0.042 mm 0.042 mm

The arithmetic mean of the absolute errors (mean of the magnitudes of the errors),
large begin{aligned}Delta a_{mean}=frac{0.028+0.008+0.012+0.048+.032+.042}{6}=0.028 mmend{aligned}

large begin{aligned} Relative;error,delta a = frac{Delta a_{mean}}{a_{mean}}=frac{0.028}{1.128}=0.0248end{aligned}

Percentage error = largebegin{aligned} = frac{Delta a_{mean}}{a_{mean}}times 100%=frac{0.028times 100}{1.128}=pm 2.48%end{aligned}

More solved problems for the calculation of errors are given in the last section of this article.

Combination of errors

When a quantity is determined by combining several measurements, the errors in the different measurements will combine in some way or other.

Error in the sum of the quantities

Let two quantities, A and B have measured values A ± ∆ A and B ± ∆ B respectively.
∆A and  ∆B are the absolute errors in their measurements.

To find the error ∆Z that may occur in the sum Z = A + B, consider
Z ± ∆Z = (A ± ∆A) + (B ± ∆B)
            = A + B ± ∆A ± ∆B

The maximum possible error in the value of Z is given by ∆Z = ∆A + ∆B.

Thus, when two quantities are added, the absolute error in the result is the sum of the absolute errors in the measured quantities.

Error in the difference of the quantities

Let two quantities, A and B have measured values A ± ∆ A and B ± ∆ B respectively.
∆A and  ∆B are the absolute errors in their measurements.

To find the error ∆Z that may occur in the difference Z = A – B, consider
Z ± ∆Z = (A ± ∆A) – (B ± ∆B)
           = A – B ± ∆A ± ∆B

Similarly, the maximum possible error in the value of Z is given by ∆Z = ∆A + ∆B.

Thus, when two quantities are subtracted, the absolute error in the result is the sum of the absolute errors in the measured quantities.

Error in the product of the quantities

Let two quantities, A and B have measured values A ± ∆ A and B ± ∆ B respectively.
∆A and  ∆B are the absolute errors in their measurements.

To find the error ∆Z that may occur in the product Z = AB, consider
Z ± ∆Z = (A ± ∆A)(B ± ∆B)
           = AB  ± A∆B ± B∆A ± ∆A∆B

Dividing L.H.S by Z and R.H.S by AB we get,
large begin{aligned}1pm frac{Delta Z}{Z}=1pm frac{Delta B}{B}pm frac{Delta A}{A}pm frac{Delta ADelta B}{AB}end{aligned}

Since ∆A and ∆B are small, their products large frac{Delta ADelta B}{AB} is very small and can be neglected. Hence, the maximum fractional error in Z is given by
large begin{aligned}frac{Delta Z}{Z}=frac{Delta A}{A}pm frac{Delta B}{B}end{aligned}

Thus, when two quantities are multiplied, the fractional error in the result is the sum of the fractional errors in the measured quantities.

Error in the quotient of the quantities

Let two quantities, A and B have measured values A ± ∆ A and B ± ∆ B respectively.
∆A and  ∆B are the absolute errors in their measurements.

To find the error ∆Z that may occur in the quotient large Z=frac{A}{B}, consider
large Zpm Delta Z=frac{Apm Delta A}{Bpm Delta B}

Similarly, on solving, we get the maximum fractional error in Z as
large frac{Delta Z}{Z}=frac{Delta A}{A}pm frac{Delta B}{B}

Thus, when two quantities are divided, the fractional error in the result is the sum of the fractional errors in the measured quantities.

Error when a quantity is raised to a power

The error ∆Z that may occur when a quantity is raised to its nth power is n times the fractional error in the quantity itself ie., if Z = An ,
large frac{Delta Z}{Z}=n;frac{Delta A}{A}   ——————- eqn (1)

By this equation large frac{Delta Z}{Z}=frac{Delta A}{A}pm frac{Delta B}{B} it is clear that the maximum percentage error in Z is the sum of the maximum percentage error in A and maximum percentage error in B.

large i.e,;;; frac{Delta Z}{Z}times 100=frac{Delta A}{A}times 100+frac{Delta B}{B}times 100

Similarly from eqn (1), the maximum percentage error in Z is given by
large frac{Delta Z}{Z}times 100=ntimes frac{Delta A}{A}times 100

If large Z = frac{A^{textsc{p}}.B^{textsc{q}}}{C^{textsc{r}}}, then the maximum % error in Z is given by
large frac{Delta Z}{Z}times 100=textsc{p}frac{Delta A}{A}times 100+textsc{q}frac{Delta B}{B}times 100+textsc{r}frac{Delta C}{C}times 100

For example:- The volume V of a cube of side L is given by
V = L3 = L.L.L
Thus as before, large frac{Delta V}{V} = frac{Delta L}{L}+frac{Delta L}{L}+frac{Delta L}{L}
or, the maximum fractional error large frac{Delta V}{V} = 3;frac{Delta L}{L}

Error calculation solved problems

1. A physical quantity p is related to four observations a, b, c and d as follows large p=frac{a^{2}b^{3}}{csqrt{d}} . The percentage error in the measurements in a, b, c and d are 2 %, 3 %, 1 % and 4 % respectively. Calculate the percentage error in p?

Ans:

large p=frac{a^{2}b^{3}}{csqrt{d}}
large therefore frac{Delta p}{p}times 100=2frac{Delta a}{a}times 100+3frac{Delta b}{b}times 100+frac{Delta c}{c}times 100+frac{1}{2}frac{Delta d}{d}times 100

ie, % error in p = 2×(% error in a)+3×(% error in b)+(% error in c)+ ½ ×(% error in d).
= 2 x 2 % + 3 x 3 % + 1 % + ½ x 4 %
= 16 %

2. Period of oscillations of a simple pendulum is measured in an experiment to be 2.01 s, 2.03 s, 1.99 s, 1.98 s, 2.05 s and 2.04 s. Calculate (1) Mean time period of the pendulum, (2) Absolute error in each measurements (3) Average absolute error (4) Relative error and (5) Percentage error.

Ans:

Let T1 = 2.01 s, T2 = 2.03 s, T3 = 1.99 s, T4 = 1.98 s, T5 = 2.05 s, T6 = 2.04 s

(1) Mean time period of the pendulum is
T_{mean}= frac{2.01+2.03+1.99+1.98+2.05+2.04}{6}=2.01667 s
Since there are only three significant figures, it is proper to have only 3 significant figures in the mean also
ie, Tmean = 2.02 s

(2) Absolute errors in each measurement.

Absolute errors = Difference between Tmean and measured value Magnitude of errors
∆T1 = Tmean – T1 = 2.02 – 2.01 =   0.01 s 0.01 s
∆T2 = Tmean – T2 = 2.02 – 2.03 =  -0.01 s 0.01 s
∆T3 = Tmean – T3 = 2.02 – 1.99 =   0.03 s 0.03 s
∆T4 = Tmean – T4 = 2.02 – 1.98 =   0.04 s 0.04 s
∆T5 = Tmean – T5 = 2.02 – 2.05 =  -0.03 s 0.03 s
∆T6 = Tmean – T6 = 2.02 – 2.04 =  -0.02 s 0.02 s

(3) Average absolute error.

Average absolute error = arithmetic mean of the magnitude of the errors

large (Delta T)_{mean}=frac{left | Delta T_{1} right |+left | Delta T_{2} right |+left | Delta T_{3} right |+left | Delta T_{4} right |+left | Delta T_{5} right |+left | Delta T_{6} right |}{6}
large ie, (Delta T)_{mean}= frac{0.01+0.01+0.03+0.04+0.03+0.02}{6}=0.02;s

(4) Relative error.

large Relative;error,delta T = frac{Delta T_{mean}}{T_{mean}}=frac{0.02}{2.02}=0.01

(5) Percentage error.

large Percentage;error = frac{Delta T_{mean}}{T_{mean}}times 100%=frac{0.02times 100}{2.02} approx 1%

3. Calculate the maximum percentage error in P if large p=pi r^{2}frac{X}{L} . Given that r = (0.32 ± 0.03) cm; X = (19 ± 1); L = (72 ± 0.2 )cm; π is a constant.

Ans:

Given large p=pi r^{2}frac{X}{L}
large frac{Delta p}{p}=frac{Delta pi }{pi }+frac{2Delta r}{r}+frac{Delta X}{X}+frac{Delta L}{L} = 0+2times frac{0.03}{0.32}+frac{1}{19}+frac{0.2}{72}=0.244
Maximum percentage of error in p = 0.244 × 100 = 24.4 %

4. Two resistances (80 ± 3)Ω and  (130 ± 4)Ω are connected in series. Calculate the effective resistance with error limit and percentage error?

Ans:

Let R1 = 80 Ω, ∆R1 = 3 Ω, R2 = 130 Ω, ∆R2 = 4 Ω

Effective resistance of this series connection

R = R1 + R2 = 80 + 130 = 210 Ω
∆R = ∆R1 + ∆R2 = 3 + 4 = 7 Ω

∴ Effective resistances with error limit = (210 ± 7)Ω

large Percentage;error = frac{Delta R}{R}times 100 % = frac{7}{210}times 100 = 3.33%

5. The time period of oscillation of a simple pendulum is large T = 2pi sqrt{frac{l}{g}}. The length of the pendulum is measured with a scale of least count 1 mm is 60 cm. If the time for 20 oscillations is measured with a stop watch of resolution 0.1 s is 50 s, what is the percentage error in the determination of g?

Ans:

large \* Given;T = 2pi sqrt{frac{l}{g}}; newline ie,;T^{2}=4pi ^{2}frac{l}{g};newline or;g=4pi ^{2}frac{l}{T^{2}}
Given ∆l = 1 mm = 0.1 cm
l = 60 cm
Time for 20 oscillations = 50 s
∆T = 0.1 s

large \* Percentage;error;in;g=frac{Delta l}{l}times 100% + 2times frac{Delta T}{T}times 100% \ linebreak = frac{0.1}{60}times 100% + 2times frac{0.1}{50}times 100% \ linebreak = 0.167+0.4 \ linebreak =0.567%

I hope the information in this article helps you to get a brief idea about the errors and types of errors in measurement, and if you believe I missed something or if you have any suggestions, do let me know via comments.

Also if you find this article useful, don’t forget to share with your friends and colleagues on Facebook and Twitter.

Источник

Measurement is a major part of scientific calculations. Completely accurate measurement results are absolutely rare. While measuring different parameters, slight errors are common. There are different types of errors, which cause differences in measurement. All the errors can be expressed in mathematical equations. By knowing the errors, we can calculate correctly and find out ways to correct the errors. There are mainly two types of errors – absolute and relative error. In this article, we are going to define absolute error and relative error. Here, we are giving explanations, formulas, and examples of absolute error and relative error along with the definition. The concept of error calculation is essential in measurement. 

Define Absolute Error

Absolute error is defined as the difference between the actual value and the measured value of a quantity. The importance of absolute error depends on the quantity that we are measuring. If the quantity is large such as road distance, a small error in centimetres is negligible. While measuring the length of a machine part an error in centimetre is considerable. Though the errors in both cases are in centimetres, the error in the second case is more important.

Absolute Error Formula

The absolute error is calculated by the subtraction of the actual value and the measured value of a quantity. If the actual value is x₀ and the measured value is x, the absolute error is expressed as,

∆x = x₀- x

Here, ∆x is the absolute error.

Absolute Error Example

Here, we are giving an example of the absolute error in real life. Suppose, we are measuring the length of an eraser. The actual length is 35 mm and the measured length is 34.13 mm. 

So, The Absolute Error = Actual Length — Measured Length

 = (35 — 34.13) mm 

 = 0.87 mm

Classification Of Absolute Error

  1. Absolute accuracy error

Absolute accuracy error is the other name of absolute error. The formula for absolute accuracy error is written as E= E exp – E true, where E is the absolute accuracy error, E exp is the experimental value and E true is the actual value.

  1. Mean absolute error

MAE or the mean absolute error is the mean or average of all absolute errors. The formula for Mean Absolute Error is given as,

MAE = [frac{1}{n}] [sum_{i=1}^{n}mid x_{i} -xmid]

  1. Absolute Precision Error

It is a standard deviation of a group of numbers. Standard deviation helps to know how much data is spread.

Relative Error Definition

The ratio of absolute error of the measurement and the actual value is called relative error. By calculating the relative error, we can have an idea of how good the measurement is compared to the actual size. From the relative error, we can determine the magnitude of absolute error. If the actual value is not available, the relative error can be calculated in terms of the measured value of the quantity. The relative error is dimensionless and it has no unit. It is written in percentage by multiplying it by 100.

Relative Error Formula

The relative error is calculated by the ratio of absolute error and the actual value of the quantity. If the absolute error of the measurement is ∆x, the actual value is x0, the measured value is x, the relative error is expressed as, 

                                                     xᵣ = (x₀ — x)/ x₀ = ∆x/x₀

Here, xr is the relative error.

Relative Error Example

Here, we are giving an example of relative error in real life. Suppose, the actual length of an eraser is 35 mm. Now, the absolute error = (35-34.13) mm = 0.87 mm.

So, the relative error = absolute error/actual length

= 0.87/35

= 0.02485

(Image will be Uploaded Soon)

Relative Error as a Measure of Accuracy

In many cases, relative error is a measure of precision. At the same time, it can also be used as a measure of accuracy. Accuracy is the extent of knowing how accurate the value is as compared to the actual or true value. Students can find the RE accuracy only if they know the true value or measurement. For simplicity, we have the formula for calculating the RE accuracy which is given as

RE[_{accuracy}] = Actual error/ true value * 100%

Absolute Error and Relative Error in Numerical Analysis

Numerical analysis of error calculation is a vital part of the measurement. This analysis finds the actual value and the error quantity. The absolute error determines how good or bad the measurement is. In numerical calculation, the errors are caused by round-off error or truncation error.

Absolute and Relative Error Calculation — Examples

1. Find the absolute and relative errors. The actual value is 125.68 mm and the measured value is 119.66 mm.

Solution:

Absolute Error = |125.68 – 119.66| mm

= 6.02 mm

Relative Error = |125.68 – 119.66| / 125.68

= 0.0478

2. Find out the absolute and relative errors, where the actual and measured values are 252.14 mm and 249.02 mm.

Solution:

Absolute Error = |252.14 – 249.02| mm

                           = 3.12 mm

Relatives Error = 3.12/252.14

= 0.0123

Did You Know?

In different measurements, the quantity is measured more than one time to get an average value of the quantity. Mean absolute error is one of the most important terms in this kind of measurement. The average of all the absolute errors of the collected data is called the MAE ( Mean Absolute Error). It is calculated by dividing the sum of absolute errors by the number of errors. The formula of MAE is –

MAE = [frac{displaystylesumlimits_{i=1}^n mid x_{0}-x mid }{n}]

Here, 

n is the number of errors, 

x₀ is the actual value, 

x is the measured value, and 

|x₀-x| is the absolute error.

Источник

Created by Bogna Szyk

Reviewed by

Dominik Czernia, PhD and Jack Bowater

Last updated:

Nov 10, 2022

If you ever wondered what’s the difference between relative and absolute error, our relative error calculator is right up your alley. In the following text, you’ll discover the absolute and relative error formulas, together with easy-to-follow examples. We also prepared a short section on the differences between the two types of error, as well as one on the reason why the relative error is considered to be more useful.

What is the absolute error?

The absolute error also called the approximation error, is the absolute value of the difference between the actual value and the measured value. The absolute error formula is

absolute error = |actual value - measured value|

The actual value is otherwise known as the real or true value. On the other hand, the measured valueis an approximation.

Very often we talk about absolute error to indicate how inaccurate a measuring device is. For example, imagine that you have a bathroom scale that only displays the result in full pounds — it can’t get more accurate than that. Hence, if you weigh e.g. 140 lbs, you can say that your weight is 140 ± 0.5 lbs, with the measured value equal to 140 lbs and the absolute error equal to 0.5 lb. The actual value will be somewhere between 139.5 and 141.5 lbs.

Remember that the absolute error is expressed in the same unit as the measured and real values. For example, if you measured the height of a tree in feet, the absolute error would also be expressed in feet.

What is the relative error?

Relative error (or percent error), on the other hand, expresses the error in terms of a percentage. You can use the following relative error formula:

relative error = |absolute error / actual value| = |(actual value - measured value) / actual value|

We typically express both the absolute and relative errors as positive values, hence the use of absolute values.

The relative error compares the absolute error to the actual value of the property you’re measuring. For example, let’s say you measure your child’s height in a doctor’s office to the highest degree of accuracy, so the actual value it’s equal to 121.2 cm. When you measure your child at home, you find the measured value to be 120.5 cm.

The relative error is |(121.2 - 120.5) / 121.2| = 0.00578 = 0.578%

As you can see, the relative error is expressed as a percentage and is unitless. Whether you’re analyzing length, weight, or temperature, the unit doesn’t influence the result.

To know more about calculating percentage error, check our percent error calculator. It may also be interesting to visit the error propagation calculator to see how errors in measurements propagate to derived quantities.

How to calculate the absolute error and relative error

You can use our relative error calculator to estimate both the absolute and relative error for any measurement or calculation. Let’s analyse the difference between these two types of error with an example.

Let’s say you want to determine the value of the square root of two. The value you find online is 1.41421356237, but you wonder how accurate it would be to simply write it rounded to two significant figures. Note, that this is different to decimal points — see the significant figures calculator for more information.

  1. To find out the absolute error, subtract the approximated value from the real one:

    |1.41421356237 - 1.41| = 0.00421356237

  2. Divide this value by the real value to obtain the relative error:

    |0.00421356237 / 1.41421356237| = 0.298%

As you can see, the relative error is lower than 1%. In many cases, this is considered a good approximation.

Is my absolute error too high?

The main advantage of the relative error is that, since it can only take values between 0-100%, it’s easy to evaluate whether an error is big or small. It’s much more challenging to determine whether a specific absolute error is of sufficient accuracy. For example, let’s imagine you take measurements of weight with an absolute error of 1 kg:

  • If you are weighing apples in a grocery store, and you are planning on buying 2 kg of apples, an absolute error of 1 kg can result in buying up to 50% more or less than you need. You wouldn’t want to use such scales in a store, would you?
  • When you weigh yourself at home, a 1 kg error makes a substantial difference — after all, you’d like to know whether you weigh 75 or 76 kg. Nevertheless, this error feels more acceptable than in the case of apples.
  • However, if you want to weigh a 20-meter-long steel beam that weighs approximately 2 tonnes, you’re not interested in a difference of one kilogram — it’s a relative error of about 0.05%, which can easily be neglected.

As you can see, the bigger the real value, the higher the accepted absolute error.

FAQ

Is the relative error the same as absolute error?

The absolute error is the discrepancy between your measurement and the true value, while the relative error is the ratio between the absolute error and the absolute value of the true value.

The relative error helps us asses how accurate the measured value is when compared to the true value.

Is there another name for relative error?

Relative error is known under several different names:

  • Relative uncertainty;
  • Approximation error;
  • Fractional error; and
  • Percentage error.

What is the relative error if I measured 42 and the true value is 40?

The answer is 0.05 or 5%. To arrive at this result, we apply the relative error formula:
relative error = |(actual - measured) / actual|.
Plugging in actual = 40 and measured = 42, we obtain
relative error = |(42-40) / 40| = 1/20 = 0.05.

AB testCoefficient of determinationCritical value… 21 more

Источник

To estimate the error in a measurement, we need to know the expected or standard value and compare how far our measured values deviate from the expected value. The absolute error, relative error, and percentage error are different ways to estimate the errors in our measurements.

Error estimation can also use the mean value of all the measurements if there is no expected value or standard value.

The mean value

To calculate the mean, we need to add all measured values of x and divide them by the number of values we took. The formula to calculate the mean is:

Let’s say we have five measurements, with the values 3.4, 3.3, 3.342, 3.56, and 3.28. If we add all these values and divide by the number of measurements (five), we get 3.3764.

As our measurements only have two decimal places, we can round this up to 3.38.

Estimation of errors

Here, we are going to distinguish between estimating the absolute error, the relative error, and the percentage error.

Estimating the absolute error

To estimate the absolute error, we need to calculate the difference between the measured value x0 and the expected value or standard xref:

Imagine you calculate the length of a piece of wood. You know it measures 2.0m with a very high precision of ± 0.00001m. The precision of its length is so high that it is taken as 2.0m. If your instrument reads 2.003m, your absolute error is | 2.003m-2.0m | or 0.003m.

Estimating the relative error

To estimate the relative error, we need to calculate the difference between the measured value x0 and the standard value xref and divide it by the total magnitude of the standard value xref:

Using the figures from the previous example, the relative error in the measurements is | 2.003m-2.0m | / | 2.0m | or 0.0015. As you can see, the relative error is very small and has no units.

Estimating the percentage error

To estimate the percentage error, we need to calculate the relative error and multiply it by one hundred. The percentage error is expressed as ‘error value’%. This error tells us the deviation percentage caused by the error.

Using the figures from the previous example, the percentage error is 0.15%.

What is the line of best fit?

The line of best fit is used when plotting data where one variable depends on another one. By its nature, a variable changes value, and we can measure the changes by plotting them on a graph against another variable such as time. The relationship between two variables will often be linear. The line of best fit is the line that is closest to all the plotted values.

Some values might be far away from the line of best fit. These are called outliers. However, the line of best fit is not a useful method for all data, so we need to know how and when to use it.

Obtaining the line of best fit

To obtain the line of best fit, we need to plot the points as in the example below:

Estimation of Errors, Plotted data with variance, StudySmarterFig. 1 — Data plotted from several measurements showing variation on the y-axis

Here, many of our points are dispersed. However, despite this data dispersion, they appear to follow a linear progression. The line that is closest to all those points is the line of best fit.

When to use the line of best fit

To be able to use the line of best fit, the data need to follow some patterns:

  1. The relationship between the measurements and the data must be linear.
  2. The dispersion of the values can be large, but the trend must be clear.
  3. The line must pass close to all values.

Data outliers

Sometimes in a plot, there are values outside the normal range. These are called outliers. If the outliers are fewer in number than the data points following the line, the outliers can be ignored. However, outliers are often linked to errors in the measurements. In the image below, the red point is an outlier.

Estimation of Errors, Plotted data with outlier, StudySmarterFig. 2 — Data plotted from several measurements showing variation on the y-axis in green and an outlier in pink

Drawing the line of best fit

To draw the line of best fit, we need to draw a line passing through the points of our measurements. If the line intersects with the y-axis before the x-axis, the value of y will be our minimum value when we measure.

The inclination or slope of the line is the direct relationship between x and y, and the larger the slope, the more vertical it will be. A large slope means that the data changes very fast as x increases. A gentle slope indicates a very slow change of the data.

Estimation of Errors, Line of best fit, StudySmarterFigure 3 — The line of best fit is shown in pink, with the slope being shown in light green

Calculating uncertainty in a plot

In a plot or a graph with error bars, there can be many lines passing between the bars. We can calculate the uncertainty of the data using the error bars and the lines passing between them. See the following example of three lines passing between values with error bars:

Estimation of Errors, Plot showing uncertainty approximations, StudySmarter Fig. 4 — Plot showing uncertainty bars and three lines passing between them. The blue and purple lines begin at the extreme values of the uncertainty bars

How to calculate the uncertainty in a plot

To calculate the uncertainty in a plot, we need to know the uncertainty values in the plot.

  • Calculate two lines of best fit.
  • The first line (the green one in the image above) goes from the highest value of the first error bar to the lowest value of the last error bar.
  • The second line (red) goes from the lowest value of the first error bar to the highest value of the last error bar.
  • Calculate the slope m of the lines using the formula below.
  • For the first line, y2 is the value of the point minus its uncertainty, while y1 is the value of the point plus its uncertainty. The values x2 and x1 are the values on the x-axis.
  • For the second line, y2 is the value of the point plus its uncertainty, while y1 is the value of the point minus its uncertainty. The values x2 and x1 are the values on the x-axis.
  • You add both results and divide them by two:

Let’s look at an example of this, using temperature vs time data.

Calculate the uncertainty of the data in the plot below.

Estimation of Errors. Plot showing uncertainty approximations. StudySmarterFigure 6. Plot showing uncertainty bars and three lines passing between them. The red and green lines begin at the extreme values of the uncertainty bars. Source: Manuel R. Camacho, StudySmarter.

The plot is used to approximate the uncertainty and calculate it from the plot.

Time (s) 20 40 60 80
Temperature in Celsius 84.5 ± 1 87 ± 0.9 90.1 ± 0.7 94.9 ± 1

To calculate the uncertainty, you need to draw the line with the highest slope (in red) and the line with the lowest slope (in green).

In order to do this, you need to consider the steeper and the less steep slopes of a line that passes between the points, taking into account the error bars. This method will give you just an approximate result depending on the lines you choose.

You calculate the slope of the red line as below, taking the points from t=80 and t=60.

You now calculate the slope of the green line, taking the points from t=80 and t=20.

Now you subtract the slope of the green one (m2) from the slope of the red one (m1) and divide by 2.

As our temperature measurements take only two significant digits after the decimal point, we round the result to 0.06 Celsius.

Estimation of Errors — Key takeaways

  • You can estimate the errors of a measured value by comparing it to a standard value or reference value.
  • The error can be estimated as an absolute error, a percentage error, or a relative error.
  • The absolute error measures the total difference between the value you expect from a measurement (X0) and the obtained value (Xref), equal to the absolute value difference of both Abs = | Xo-Xref |.
  • The relative and percentage errors measure the fraction of the difference between the expected value and the measured value. In this case, the error is equal to the absolute error divided by the expected value rel = Abs / Xo for the relative error, and divided by the expected value and expressed as a percentage for the percentage error per = (Abs / Xo) * 100. You must add the percentage symbol for percentage errors.
  • You can approximate the relationship between your measured values using a linear function. This approximation can be made simply by drawing a line, which must be the line that passes closest to all values (the line of best fit).
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Absolute error is the difference between measured or inferred value and the actual value of a quantity. The absolute error is inadequate due to the fact that it does not give any details regarding the importance of the error. While measuring distances between cities kilometers apart, an error of a few centimeters is negligible and is irrelevant. Consider another case where an error of centimeters when measuring small machine parts is a very significant error. Both the errors are in the order of centimeters but the second error is more severe than the first one.

Absolute Error Formula

If x is the actual value of a quantity and x0 is the measured value of the quantity, then the absolute error value can be calculated using the formula

Δx = x0-x.

Here, Δx is called an absolute error.

If we consider multiple measurements, then the arithmetic mean of absolute errors of individual measurements should be the final absolute error.

Read more:

  • Arithmetic Mean Statistics
  • Arithmetic Mean and Range
  • Difference Between Average and Mean
  • Arithmetic Progression For Class 10

Absolute Error Example

For example, 24.13 is the actual value of a quantity and 25.09 is the measure or inferred value, then the absolute error will be:

Absolute Error = 25.09 – 24.13

= 0.86

Most of the time it is sufficient to record only two decimal digits of the absolute error. Thus, it is sufficient to state that the absolute error of the approximation 4.55 to the correct value 4.538395 is 0.012.

Relative Error

The relative error is defined as the ratio of the absolute error of the measurement to the actual measurement. Using this method we can determine the magnitude of the absolute error in terms of the actual size of the measurement. If the true measurement of the object is not known, then the relative error can be found using the measured value. The relative error gives an indication of how good measurement is relative to the size of the object being measured.

If x is the actual value of a quantity, x0 is the measured value of the quantity and Δx is the absolute error, then the relative error can be measured using the below formula.

Relative error = (x0-x)/x = (Δx)/x

An important note that relative errors are dimensionless. When writing relative errors it is usual to multiply the fractional error by 100 and express it as a percentage.

Absolute Error and Relative Error in Numerical analysis

Numerical analysis is concerned with the methods of finding the approximate values and the absolute errors in these calculations. The absolute error gives how large the error is, while the relative error gives how large the error is relative to the correct value. In numerical calculation, error may occur due to the following reasons.

  • Round off error
  • Truncation error

Example 1:

Find the absolute and relative errors of the approximation 125.67 to the value 119.66.

Solution:

Absolute error = |125.67-119.66|=6.01

Relative error = |125.67-119.66|/119.66 = 0.05022

Mean Absolute Error

The mean absolute error is the average of all absolute errors of the data collected. It is abbreviated as MAE (Mean Absolute Error). It is obtained by dividing the sum of all the absolute errors with the number of errors. The formula for MAE is:

Mean absolute error

Here,

|xi – x| = absolute errors

n = number of errors

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