Relative error is an indicator that expresses, as a proportion, how far an experimentally obtained measured value deviates from a theoretical value, literature value, or true value.
In chemistry experiments, it is used when comparing values such as concentration, yield, melting point, density, molar mass, values obtained from absorbance, and values obtained by titration with theoretical values.
It is important because it allows evaluation not only of the absolute magnitude of the deviation but also of how large that deviation is relative to the reference value.
In a discussion of relative error, it is not sufficient to write only that “the value differed from the theoretical value” or “the error was large.”
It is necessary to explain whether the measured value was higher or lower than the theoretical value, which operation caused the deviation, whether the error was systematic or random, and how significant the error was relative to the magnitude of the measured value.
Even when the relative error is small, it is also important to consider whether the measurement was truly accurate or merely happened to be close to the theoretical value by chance.
This article clearly explains, as examples of discussions that can be used in laboratory reports on relative error, the difference from absolute error, calculation of relative error, explanation of differences from theoretical values, causes of measured values becoming larger, causes of measured values becoming smaller, systematic error, random error, measurement accuracy, and points for improvement.
Note:
This article is a reference intended to assist with discussions of differences between measured values and theoretical values obtained in basic chemistry experiments, analytical chemistry experiments, and physical chemistry experiments at universities and similar institutions.
For actual error calculations, significant figures, handling of theoretical values, methods for comparison with literature values, and handling of outliers, always follow the instructions in your university’s laboratory manual and those given by your instructor or TA.
- What Is Relative Error?
- Main Items to Include in the Results
- Difference Between Absolute Error and Relative Error
- Discussion When the Relative Error Is Small
- Discussion When the Relative Error Is Large
- When the Measured Value Is Larger Than the Theoretical Value
- When the Measured Value Is Smaller Than the Theoretical Value
- Check the Assumptions Underlying the Theoretical Value
- Relative Error Caused by Systematic Error
- Relative Error Caused by Random Error
- Relative Error in Titration Experiments
- Relative Error in Mass Measurements
- Relative Error in Absorbance Measurements
- Relative Error in Yield Calculations
- Difference Between Relative Error and Standard Deviation
- Cases Requiring Caution Even When Relative Error Is Small
- Conditions Under Which Relative Error Tends to Become Large
- Significant Figures and Relative Error
- Causes of Relative Error
- When the Results Can Be Considered Good
- Example Discussions When the Experiment Did Not Go Well
- How to Write Points for Improvement
- Difference Between a Superficial Discussion and a Good Discussion
- Examples of Expressions That Can Be Used in Reports
- Points to Check When Discussing Relative Error
- Summary
What Is Relative Error?
Relative error expresses the difference between a measured value and a theoretical value as a proportion of the theoretical value.
Simply saying that a value “differs by 0.1” does not indicate whether 0.1 is a large or small error, because this depends on the magnitude of the quantity being measured.
Using relative error makes it easier to compare how far a measured value deviates from the theoretical value.
Relative error is often expressed as a percentage and is also called percent error.
For example, if the theoretical value is 100 and the measured value is 98, the difference is 2 and the relative error is 2%.
On the other hand, if the theoretical value is 10 and the measured value is 8, the difference is also 2, but the relative error is 20%, so it is evaluated as a larger deviation.
Relative error (%) = |Measured value – Theoretical value| / Theoretical value × 100
Example Discussion:
Relative error is an indicator that shows, as a proportion of the theoretical value, how far the measured value deviates from the theoretical value.
In this experiment, the difference between the measured value and theoretical value was divided by the theoretical value to evaluate the proportion of the error relative to the magnitude of the measured value.
Therefore, it is possible to evaluate the validity of the experimental result relatively, rather than considering only the absolute magnitude of the difference.
Main Items to Include in the Results
To discuss relative error, organize the measured value, theoretical value, absolute error, relative error, measurement conditions, formula used for calculation, units, and significant figures.
It is important not only to show the numerical value of the error but also to explain which operation or measurement condition caused that error.
Main Items to Include in the Results
- Theoretical value or literature value
- Measured value
- Whether the measured value is larger or smaller than the theoretical value
- Absolute error
- Relative error
- Unit or percentage notation of relative error
- Number of measurements
- Mean
- Standard deviation
- Formula used
- Measurement conditions
- Assumptions underlying the theoretical value
- Possibility of systematic error
- Possibility of random error
- Points for improvement
Example of How to Write the Results:
The measured value was smaller than the theoretical value, and the relative error was several percent.
This difference may have been caused by loss of the sample during recovery, reading errors during measurement, or incomplete reaction.
Although the relative error was not large, because the difference from the theoretical value appeared in one direction, it is necessary to consider not only random error but also the effect of systematic error.
Difference Between Absolute Error and Relative Error
Absolute error represents the difference itself between the measured value and the theoretical value.
For example, if the theoretical value is 50.0 and the measured value is 48.0, the absolute error is 2.0.
Relative error, on the other hand, expresses what proportion of the theoretical value this difference represents.
Even with the same absolute error, the relative error differs depending on whether the theoretical value is large or small.
In chemistry experiments, data with different magnitudes may be compared.
In such cases, absolute error alone makes comparison difficult, so relative error is used.
Because relative error indicates the proportion of error relative to the magnitude of the measured quantity, it is a useful indicator for comparing errors among different experimental results.
| Item | Meaning | Feature |
|---|---|---|
| Absolute error | Difference between measured value and theoretical value | Directly expresses the magnitude of the deviation |
| Relative error | Proportion of the deviation relative to the theoretical value | Allows comparison of error relative to the magnitude of the measured value |
Example Discussion:
Absolute error directly shows the difference between the measured value and theoretical value, whereas relative error shows what proportion of the theoretical value that difference represents.
Therefore, when the magnitudes of the quantities being measured differ, relative error makes it easier to compare the size of the errors.
In this experiment, relative error was used to evaluate the proportion of deviation from the theoretical value and discuss the validity of the results.
Discussion When the Relative Error Is Small
When the relative error is small, the measured value is close to the theoretical value, and the experimental result can be considered to agree well with the theoretical prediction.
The measurement procedure, sample preparation, reaction conditions, and calculation method may have been relatively appropriate.
However, a small relative error does not mean that there were no errors in any of the procedures.
Multiple errors may accidentally cancel one another, making the result close to the theoretical value.
Also, if the standard deviation is large, reproducibility may be low even if the mean is close to the theoretical value.
Even when the relative error is small, it is important to check the variation in measured values and the validity of the procedure as well.
Example Discussion:
Because the relative error was small, the measured value was close to the theoretical value, and the experimental result was considered generally valid.
This indicates that sample preparation and measurement operations were performed relatively accurately.
However, because there is also a possibility that errors happened to cancel one another, the standard deviation and reproducibility of the measurements must also be evaluated.
Discussion When the Relative Error Is Large
When the relative error is large, the measured value deviates greatly from the theoretical value, suggesting that there may have been problems with the experimental procedure or measurement conditions.
Possible causes include errors in measuring the sample amount, errors in preparing concentrations, incomplete reaction, loss of product, insufficient drying, contamination with impurities, insufficient instrument calibration, and reading errors.
When the relative error is large, first check whether the measured value is larger or smaller than the theoretical value.
The possible causes differ depending on whether the measured value is larger or smaller.
In addition, whether a large error occurred in only one measurement or whether repeated measurements deviated in the same direction changes the judgment of whether the error was random or systematic.
Example Discussion:
Because the relative error was large, the measured value deviated greatly from the theoretical value, suggesting that some error was included in the measurement or procedure.
Possible causes include concentration errors during sample preparation, insufficient instrument calibration, incomplete reaction, and loss of product.
To identify the cause of the error, it is necessary to check whether the measured value was higher or lower than the theoretical value and discuss it in relation to each stage of the procedure.
When the Measured Value Is Larger Than the Theoretical Value
When the measured value is larger than the theoretical value, components other than the target substance may be included in the measured value.
For example, in mass measurements of a product, residual solvent, moisture, impurities, or contamination with drying agents may increase the measured value.
In titration or concentration measurements, possible causes include adding titrant beyond the endpoint, insufficient blank correction, or an incorrect standard-solution concentration.
If the measured value exceeds the theoretical value, it should not simply be judged as a “good experiment.”
If a value exceeds a theoretical limit that should not be exceeded, the measured value may contain extra components or a systematic shift.
The same applies when the yield exceeds 100%.
Example Discussion:
Possible reasons why the measured value was larger than the theoretical value include residual solvent, moisture, or impurities remaining in the sample.
If these are added to the mass of the target substance, the measured value will be larger than the actual value.
Therefore, when the measured value exceeds the theoretical value, it is necessary to check whether the measurement includes components other than the target substance.
When the Measured Value Is Smaller Than the Theoretical Value
When the measured value is smaller than the theoretical value, possible causes include loss of the target substance, incomplete reaction, or insufficient measurement.
In synthesis experiments, possible causes include incomplete reaction, loss of the target substance during filtration or extraction, target substance remaining in the washing solution or mother liquor, or scattering during concentration.
In quantitative experiments, possible causes include collecting too little sample, judging the titration endpoint too early, or insufficient detection sensitivity.
When the measured value is small, it is useful to consider at which stage the target substance was lost.
Dividing the discussion into the reaction stage, separation stage, washing stage, drying stage, weighing stage, and measurement stage makes it easier to explain the cause specifically.
Example Discussion:
Possible reasons why the measured value was smaller than the theoretical value include unreacted material remaining because the reaction did not proceed completely and loss of the target substance during post-treatment.
Particularly during extraction, washing, filtration, and recrystallization, part of the target substance may remain in the aqueous phase, mother liquor, or on the filter paper.
Therefore, a decrease in the measured value must be discussed not only in terms of the reaction itself but also in terms of losses during separation and purification.
Check the Assumptions Underlying the Theoretical Value
Theoretical values are calculated based on specific assumptions.
For example, the theoretical yield is calculated by assuming that the limiting reagent is completely converted into the target product and that no side reactions or losses occur.
However, in actual experiments, incomplete reactions, side reactions, product loss, and contamination with impurities occur, so the theoretical value and measured value do not completely agree.
When discussing the difference from a theoretical value, confirm the assumptions under which the theoretical value was obtained.
If the theoretical value is a literature value, it is also important to determine whether the measurement conditions, sample purity, temperature, pressure, and measurement method are the same as in your own experiment.
If the assumptions differ, it may be natural for a difference from the measured value to occur.
Example Discussion:
The theoretical value is calculated based on the assumption that the reaction proceeds completely and that there is no loss of the target substance or side reaction.
In the actual experiment, incomplete reaction and losses during post-treatment cannot be completely avoided, so the measured value was considered not to have matched the theoretical value exactly.
Therefore, when discussing relative error, it is necessary to confirm differences between the assumptions underlying the theoretical value and the actual experimental conditions.
Relative Error Caused by Systematic Error
Systematic error is an error that causes measured values to shift consistently in the same direction.
For example, if the concentration of a standard solution is actually higher than assumed, the balance is improperly calibrated, blank correction is insufficient, or temperature correction is not performed, measured values may become biased in one direction.
When systematic error is present, relative error may also become large in a consistent direction.
Systematic error is difficult to eliminate by averaging even if the measurement is repeated.
Therefore, if the relative error appears repeatedly in the same direction, systematic error should be suspected.
To reduce systematic error, calibration of equipment, preparation of fresh standard solutions, blank correction, and measurement of standard samples are necessary.
Example Discussion:
If the measured value was consistently larger than the theoretical value in repeated measurements, systematic error rather than random error may have been present.
For example, if blank correction was insufficient and the measurement signal was consistently too large, the calculated concentration would also be overestimated.
Therefore, when the relative error is biased in one direction, instrument calibration, standard-solution concentration, and blank measurements must be reviewed.
Relative Error Caused by Random Error
Random error is a small error that occurs irregularly from one measurement to another.
Possible causes include slight differences in pipetting, judgment of the titration endpoint, balance readings, instrument noise, and small temperature changes.
Random errors may cause measured values to become either larger or smaller than the theoretical value.
The effect of random error can be reduced by performing multiple measurements and averaging the results.
However, if random error is large, the standard deviation also becomes large, and the relative error itself may vary widely.
When discussing relative error, check not only the mean but also the variation in the measured values.
Example Discussion:
Possible causes of the relative error include random errors such as pipetting differences and reading of the titration endpoint.
Because random errors occur irregularly in each measurement, measured values may be either larger or smaller than the theoretical value.
Therefore, increasing the number of measurements and calculating the mean can reduce the effect of random error.
Relative Error in Titration Experiments
In titration experiments, relative error may be calculated by comparing the titration volume or calculated concentration with a theoretical value.
Causes of relative error include endpoint judgment, burette reading, errors in the concentration of the standard solution, errors in sample volume, and the color-change range of the indicator.
If titrant is added past the endpoint, the measured value may become larger, while stopping before the endpoint may make it smaller.
In titration, even a difference of one drop near the endpoint can affect the relative error.
Particularly in small-volume titrations, the volume of one drop represents a large proportion of the total, so the relative error tends to become large.
It is important to add titrant slowly near the endpoint and judge the endpoint using the same color-change criterion each time.
Example Discussion:
A possible reason why the concentration obtained by titration was larger than the theoretical value is that titrant was added beyond the endpoint.
If the endpoint is exceeded, the amount of titrant used becomes larger than the actual amount, and the calculated concentration may also be overestimated.
Therefore, to reduce the relative error, the addition rate should be slowed near the endpoint and the color change should be judged using a consistent criterion.
Relative Error in Mass Measurements
In mass measurements, relative error may be considered from the difference between the measured mass and theoretical value.
Causes of an excessively large measured mass include insufficient drying, residual moisture or solvent, contamination with impurities or drying agents, and errors in subtracting the tare mass of the container.
Causes of an excessively small measured mass include spilling the sample, adhesion to filter paper or containers, scattering during drying, and loss of volatile components.
With small-mass samples, even slight adhesion or spillage produces a large relative error.
Therefore, when weighing small samples, attention must be paid to adhesion to equipment, static electricity, air currents, moisture absorption, and the drying condition.
Sufficient drying and careful transfer are important for reducing relative error.
Example Discussion:
A possible reason why the measured mass was larger than the theoretical value is that residual solvent or moisture remained in the sample.
On the other hand, if the measured value was smaller, adhesion during transfer or loss during filtration may have occurred.
To reduce relative error in mass measurements, the sample must be dried sufficiently and adhesion to containers and filter paper should be minimized as much as possible.
Relative Error in Absorbance Measurements
In absorbance measurements, relative error may be calculated by comparing the concentration obtained from a calibration curve with a theoretical or known concentration.
Causes of error include errors in preparing standard solutions, insufficient blank correction, contamination of the cell, bubbles, incorrect wavelength settings, sample turbidity, and measurement outside the calibration-curve range.
If absorbance is too high, the measurement may deviate from the Beer-Lambert law, and the linear relationship between concentration and absorbance may break down.
At low concentrations, the effect of noise becomes large.
It is important to dilute or concentrate unknown samples so that measurements fall within the linear range of the calibration curve.
Example Discussion:
Possible causes of relative error in the concentration obtained from absorbance include errors in preparing the standard solutions and insufficient blank correction.
In addition, contamination or bubbles in the cell can change the absorbance and affect the calculated concentration.
Therefore, to reduce relative error, measurements must be performed within the linear range of the calibration curve, and blank correction and the condition of the cell must be checked.
Relative Error in Yield Calculations
In synthesis experiments, the difference between theoretical yield and actual yield may be considered as relative error.
If the actual yield is smaller than the theoretical yield, possible causes include incomplete reaction, side reactions, losses during extraction or filtration, dissolution of the target substance in the recrystallization mother liquor, and scattering during drying.
If the actual yield is larger than the theoretical yield, possible causes include residual solvent, impurities, moisture, or contamination with drying agents.
In theory, yield should not exceed 100%.
If it exceeds 100%, it is highly likely that mass other than that of the product is included.
When discussing relative error in yield, consider not only the efficiency of the reaction itself but also the effects of post-treatment, drying, and weighing.
Example Discussion:
Possible reasons why the actual yield was smaller than the theoretical yield and relative error occurred include incomplete reaction and loss of the target substance during extraction, washing, and recrystallization.
On the other hand, if the actual yield exceeded the theoretical yield, residual solvent or impurities may have been included in the product mass.
Therefore, relative error related to yield must be discussed from both the reaction stage and the post-treatment stage.
Difference Between Relative Error and Standard Deviation
Relative error and standard deviation are both indicators used to evaluate the reliability of experimental results, but they have different meanings.
Relative error indicates how far the measured value deviates from the theoretical value.
Standard deviation, on the other hand, indicates how widely multiple measured values are distributed around the mean.
Relative error is related to accuracy, while standard deviation is related to precision and reproducibility.
Even if the relative error is small, a large standard deviation may indicate that the measured values are close to the theoretical value but have low reproducibility.
Conversely, even if the standard deviation is small, a large relative error may indicate that the measurements are stable but deviate from the theoretical value.
| Indicator | What It Represents | Main Evaluation Target |
|---|---|---|
| Relative error | Deviation from the theoretical value | Accuracy |
| Standard deviation | Variation in measured values | Precision and reproducibility |
Example Discussion:
Relative error indicates how far the measured value deviates from the theoretical value, while standard deviation indicates the variation among values obtained from repeated measurements.
If the standard deviation is small but the relative error is large in this experiment, the reproducibility of the measurement may be high, but the mean may have deviated from the theoretical value because of systematic error.
Therefore, the reliability of experimental results must be evaluated using both relative error and standard deviation.
Cases Requiring Caution Even When Relative Error Is Small
Even when the relative error is small, caution may still be necessary.
For example, if the variation in measured values is large but only the mean happens to be close to the theoretical value, the relative error may appear small.
Also, if multiple errors cancel one another out, the result may appear close to the theoretical value.
Therefore, even when relative error is small, it is necessary to check the number of measurements, standard deviation, procedural records, and assumptions underlying the theoretical value.
Relative error is an important indicator, but it is important not to judge the quality of an experiment based on relative error alone.
Example Discussion:
Even if the relative error was small, if the variation in the measured values was large, the mean may have happened to be close to the theoretical value by chance.
It is also possible that multiple errors canceled one another out, producing an apparently good result.
Therefore, even when relative error is small, it is important to confirm the standard deviation and operating conditions as well.
Conditions Under Which Relative Error Tends to Become Large
Relative error tends to become larger as the quantity being measured becomes smaller.
For example, when handling a small amount of sample, even slight spillage or adhesion represents a large proportion of the theoretical value.
In trace analysis, small-scale synthesis, and low-concentration measurements, the same absolute error tends to produce a larger relative error.
In addition, at low concentrations, the effects of instrument noise and blank values become relatively large.
With small samples, adhesion to equipment and reading errors cannot be ignored.
When considering relative error, attention must also be paid to the magnitude of the measured value itself.
Example Discussion:
Because the amount of sample handled in this experiment was small, even slight adhesion or loss during transfer was considered likely to produce a large relative error.
Because relative error is the proportion of error relative to the theoretical value, even a small absolute error appears as a large proportion when the quantity being measured is small.
Therefore, particularly with small samples, careful transfer and highly sensitive measurement are necessary.
Significant Figures and Relative Error
When calculating relative error, attention must also be paid to significant figures.
Showing relative error to more decimal places than justified by the significant figures of the measured and theoretical values may have no experimental meaning.
For example, if the measured value is known only to three significant figures, it is not appropriate to report the relative error to many decimal places.
The notation used for relative error should be consistent with the precision of the measured value and the standard deviation.
In a report, the calculation result should not simply be written with many digits but should be presented using significant figures appropriate to the measurement precision.
Using consistent significant figures correctly expresses the reliability of the results.
Example Discussion:
Because relative error is calculated from the measured value and theoretical value, the significant figures of the calculated result must be consistent with the precision of the original measured values.
Showing relative error to more decimal places than justified by the precision of the measurements may give the impression that the measurement was more precise than it actually was.
Therefore, relative error must be reported while considering the significant figures and standard deviation of the measured values.
Causes of Relative Error
Causes of relative error include errors in sample preparation, concentration calculation errors, insufficient calibration of equipment, reading errors, incomplete reaction, side reactions, loss of the target substance, contamination with impurities, insufficient drying, insufficient blank correction, and differences in temperature or pH.
These factors may cause the measured value to become either larger or smaller than the theoretical value.
When discussing causes of error, use the direction of the deviation in the measured value as a clue.
If the measured value is too large, suspect excess components or overmeasurement.
If the measured value is too small, suspect loss of the target substance or incomplete reaction.
Explaining the direction of the relative error in relation to the experimental procedure produces a more persuasive discussion.
Example Discussion:
Possible causes of relative error include concentration errors during sample preparation, reading errors of the measuring instrument, incomplete reaction, and loss of the target substance during post-treatment.
Because the measured value was smaller than the theoretical value, incomplete reaction or partial loss of the target substance was particularly likely to have been a major cause.
In this way, it is necessary to organize the causes by focusing on whether the measured value was larger or smaller than the theoretical value.
When the Results Can Be Considered Good
From the perspective of relative error, results can be considered good when the measured value is close to the theoretical or literature value and the relative error is small.
Furthermore, if the variation among repeated measurements is small and the standard deviation is also small, reproducibility can also be considered good.
Results with both a small relative error and high reproducibility are easier to judge as reliable.
However, even if the measured values appear to agree with the theoretical value, it is necessary to check the variation in the measurements and the possibility of systematic error.
Good results are not merely values close to the theoretical value, but results for which the operating conditions are appropriate, the measured values are stable, and the causes of error can be explained as small.
Example Discussion:
In this experiment, the relative error was small, and the measured value was close to the theoretical value.
In addition, because the variation among repeated measurements was small, the reproducibility of the measurements was also considered relatively high.
Therefore, the experimental conditions and measurement procedures used in this experiment were generally appropriate, and the obtained results were judged to be valid.
Example Discussions When the Experiment Did Not Go Well
When the relative error is large, first confirm whether the measured value is larger or smaller than the theoretical value.
Then consider at which stage of sample preparation, reaction, separation, drying, measurement, or calculation the deviation may have occurred.
Writing the causes of error separately for each operation makes the discussion more specific.
Example Discussion:
A possible reason why the measured value was smaller than the theoretical value and the relative error became large is that the reaction did not proceed completely.
If part of the reactants remained unreacted, the amount obtained as the target product would be smaller than the theoretical value.
In addition, loss of the target substance during extraction, filtration, or washing may also have caused the measured value to become smaller.
Another Example Discussion:
A possible reason why the measured value was larger than the theoretical value is that moisture or residual solvent remained in the sample.
If weighing is performed before sufficient drying, the mass of substances other than the target product is also included in the measured value, causing it to be overestimated.
Therefore, the sample must be dried sufficiently and weighed only after confirming that its mass has become constant.
Another Example Discussion:
If the relative error was large despite a small standard deviation, the measured values may have been stable but deviated from the theoretical value because of systematic error.
For example, if the concentration of the standard solution was incorrect, repeated measurements would produce values shifted in the same direction.
In this case, simply increasing the number of measurements would not improve the result, so the preparation of the standard solution and instrument calibration must be reviewed.
How to Write Points for Improvement
In a discussion of relative error, it is important to write not only about the causes of the error but also about how the difference from the theoretical value can be reduced in future experiments.
Points for improvement are easier to organize when divided into sample preparation, reaction conditions, post-treatment, drying, measurement, and calculation checks.
Improvements to Sample Preparation and Measurement
- Prepare standard solutions accurately
- Use pipettes and volumetric flasks correctly
- Mix the sample thoroughly
- Perform blank correction before measurement
- Check the zero point and calibration of the instrument
- Confirm that the measurement range is appropriate
- Perform multiple measurements and use the mean
- Handle significant figures appropriately
Improvements to Reaction and Post-Treatment
- Confirm that the reaction has been completed using TLC or a confirmation reaction
- Control reaction temperature and reaction time appropriately
- Select conditions that suppress side reactions
- Use an appropriate number of extraction steps
- Reduce loss of the target substance into washing solutions
- Reduce adhesion during filtration and transfer
- Dry the product sufficiently
- Confirm purity after purification
Example of How to Write Points for Improvement:
To reduce relative error, it is necessary to take separate measures for causes that make the measured value larger than the theoretical value and causes that make it smaller.
If the measured value is too large, sufficient drying and purification should be performed to remove residual solvent and impurities, while if the measured value is too small, it is important to confirm completion of the reaction and reduce loss of the target substance during post-treatment.
It is also necessary to check the preparation of standard solutions and instrument calibration to reduce systematic error.
Difference Between a Superficial Discussion and a Good Discussion
In a discussion of relative error, simply writing “it differed from the theoretical value” or “there was an error” results in a superficial discussion.
A good discussion explains in which direction the measured value deviated, which operation caused the deviation, and whether the error was systematic or random.
| Superficial Discussion | Good Discussion |
|---|---|
| There was relative error. | Because the measured value was smaller than the theoretical value, incomplete reaction or loss of the target substance during post-treatment may have been the main causes of the relative error. |
| It was larger than the theoretical value. | The measured value may have exceeded the theoretical value because residual solvent, moisture, or impurities were included in the measurement, causing the measured value to be overestimated. |
| The error was small. | Because the relative error was small, the measured value was close to the theoretical value, but the standard deviation and number of measurements must also be checked to evaluate reproducibility. |
| I think it was a measurement error. | Specific operations that may have shifted the measured value in one direction, such as pipetting, titration endpoint judgment, instrument calibration, and blank correction, must be checked. |
| I will be more careful next time. | Next time, relative error can be reduced by accurately preparing the standard solution, confirming completion of the reaction, and weighing only after the mass has become constant following drying. |
Examples of Expressions That Can Be Used in Reports
The following expressions can be used when writing the results and discussion of relative error.
Adjust the necessary parts according to your own experimental results.
- Relative error is an indicator that expresses the difference between the measured value and theoretical value as a proportion of the theoretical value.
- Because the relative error was small, the measured value was considered relatively close to the theoretical value.
- Possible reasons why the measured value was smaller than the theoretical value include incomplete reaction and loss of the target substance.
- Possible reasons why the measured value was larger than the theoretical value include residual solvent or contamination with impurities.
- If the relative error is large, the measurement procedure or sample preparation may have included a large error.
- If deviations in the same direction are repeatedly observed, systematic error may have affected the result.
- If the values vary between measurements, random error may have had a large effect.
- Relative error is related to accuracy, while reproducibility must be evaluated using the standard deviation.
- With small samples, even a slight loss tends to produce a large relative error.
- To reduce relative error, sample preparation, measurement, drying, and post-treatment must be performed accurately.
Points to Check When Discussing Relative Error
Checking the following points before writing the report makes the discussion easier to write.
- Are the theoretical value and measured value clearly shown?
- Is the formula for calculating relative error written?
- Are absolute error and relative error distinguished?
- Has it been checked whether the measured value is larger or smaller than the theoretical value?
- Are the assumptions underlying the theoretical value explained?
- Has the possibility of systematic error been considered?
- Has the possibility of random error been considered?
- Has the result been evaluated together with the standard deviation and reproducibility?
- Has it been considered whether the relative error is large because the measured value itself is small?
- Are significant figures handled appropriately?
- Are the causes of error specified for each operation?
- Do the points for improvement correspond to the causes of error?
Summary
Relative error is an indicator that expresses, as a proportion of the theoretical value, how far a measured value deviates from a theoretical or literature value.
While absolute error represents the magnitude of the deviation itself, relative error expresses the proportion of that deviation relative to the magnitude of the quantity being measured.
Therefore, it is useful when comparing errors in measured values or experimental results of different magnitudes.
When discussing relative error, it is important to check whether the measured value is larger or smaller than the theoretical value.
If the measured value is large, possible causes include residual solvent, moisture, contamination with impurities, and overmeasurement.
If the measured value is small, possible causes include incomplete reaction, loss of the target substance, insufficient extraction, and insufficient measurement.
In addition, deviations in the same direction can be organized as systematic error, while variation among individual measurements can be organized as random error.
In a report, rather than simply writing that “the value differed from the theoretical value,” organize and discuss the calculation of relative error, the direction of the difference from the theoretical value, systematic error, random error, measurement accuracy, standard deviation, sample preparation, post-treatment, drying, instrument calibration, and points for improvement.
Discussion of relative error is important for determining how reliable the experimental results are and which procedures should be improved.
