Standard deviation is an indicator that shows how widely values obtained from multiple measurements are distributed around the mean.
In chemistry experiments, titration values, absorbance, mass, concentration, pH, reaction time, melting point, conductivity, and other quantities are measured multiple times, and standard deviation is used to evaluate the reliability of the measured values.
Because the mean alone does not indicate the stability of the measurement results, it is important to show the standard deviation together with the mean.
In a discussion of standard deviation, it is not sufficient to write only that “the standard deviation was small” or “the variation was large.”
It is necessary to explain what the standard deviation represents, whether small variation means that reproducibility is high, which operations may have caused a large standard deviation, and how outliers and the number of measurements affected the results.
It is also important to note that standard deviation is an indicator of measurement precision and does not directly indicate whether the measured value is close to the true value.
This article clearly explains, as examples of discussions that can be used in laboratory reports on standard deviation, the mean, standard deviation, variation, reproducibility, precision, accuracy, relative standard deviation, outliers, number of measurements, random error, systematic error, and points for improvement.
Note:
This article is a reference intended to assist with discussions of standard deviation and measurement precision in basic chemistry experiments, analytical chemistry experiments, and physical chemistry experiments at universities and similar institutions.
For the actual method of calculating standard deviation, handling of outliers, significant figures, statistical processing, and notation in reports, always follow the instructions in your university’s laboratory manual and those given by your instructor or TA.
- What Is Standard Deviation?
- Main Items to Include in the Results
- Relationship Between the Mean and Standard Deviation
- Discussion When the Standard Deviation Is Small
- Discussion When the Standard Deviation Is Large
- What Is Variation?
- What Is Reproducibility?
- Difference Between Precision and Accuracy
- What Is Relative Standard Deviation?
- Relationship Between the Number of Measurements and Standard Deviation
- Difference Between Standard Deviation and Standard Error
- Outliers and Standard Deviation
- Random Error and Standard Deviation
- Systematic Error and Standard Deviation
- Discussion of Standard Deviation in Titration Experiments
- Discussion of Standard Deviation in Absorbance Measurements
- Discussion of Standard Deviation in Mass Measurements
- Standard Deviation and Significant Figures
- Meaning of Showing Standard Deviation on a Graph
- Causes of Error That Increase Standard Deviation
- 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 Standard Deviation
- Summary
What Is Standard Deviation?
Standard deviation is a value that indicates how far multiple measured values are from the mean.
The smaller the standard deviation, the more closely the measured values are clustered around the mean, indicating less variation.
The larger the standard deviation, the more widely the measured values are scattered from the mean, indicating greater variation.
In chemistry experiments, even when the same sample is measured multiple times under the same conditions, exactly the same value is often not obtained.
This is because random errors arise from instrument readings, pipetting, temperature changes, instrument noise, sample homogeneity, and other factors.
Standard deviation is used to express this type of variation in measured values numerically.
Example Discussion:
Standard deviation is an indicator that shows how widely values obtained from multiple measurements are distributed around the mean.
Because the standard deviation in this experiment was small, each measured value was close to the mean, and the reproducibility of the measurement was considered relatively high.
However, standard deviation indicates the magnitude of variation and does not directly indicate that the mean is close to the true value.
Main Items to Include in the Results
To discuss standard deviation, organize the measured values, mean, standard deviation, number of measurements, units, relative standard deviation, and presence or absence of outliers.
Rather than showing standard deviation alone, presenting it together with the mean makes it easier to judge the magnitude of variation relative to the magnitude of the measured values.
Main Items to Include in the Results
- Each measured value
- Number of measurements
- Mean
- Standard deviation
- Relative standard deviation
- Units
- Maximum and minimum values
- Range of measured values
- Presence or absence of outliers
- Whether outliers were excluded
- Comparison with theoretical or literature values
- Evaluation of reproducibility
- Causes of error
- Points for improvement
Example of How to Write the Results:
The same sample was measured three times, and the mean and standard deviation were calculated.
Because the measured values were clustered around the mean and the standard deviation was also small, the variation in the measurements was judged to be relatively small.
On the other hand, because one measured value deviated slightly from the others, there may have been an error in reading the titration endpoint or in the amount of sample collected during that measurement.
Relationship Between the Mean and Standard Deviation
The mean is a representative value of multiple measurements.
Standard deviation, on the other hand, indicates how widely those measured values are distributed around the mean.
In other words, the mean represents the center of the measurement results, while standard deviation represents the spread of the measurement results.
If only the mean is shown, it is impossible to tell whether the measured values were stable or widely scattered.
For example, even when the mean is the same, the reliability of the results differs between a case in which all measured values are close together and a case in which they are widely scattered.
Therefore, the mean and standard deviation must be considered together.
Example notation for measurement results: Mean ± Standard deviation
Example: 10.25 ± 0.08 mL
Example Discussion:
The mean represents the representative value of the measurements, while presenting the standard deviation together with it makes it possible to evaluate how widely the measured values are distributed.
In this experiment, because the standard deviation was small relative to the mean, the measured values were considered to be clustered around the mean.
Therefore, by showing not only the mean but also the standard deviation, the reproducibility of the measurement results can be evaluated more appropriately.
Discussion When the Standard Deviation Is Small
When the standard deviation is small, the measured values are clustered around the mean and the variation can be judged to be small.
This may indicate that the measurement procedure was stable, the instrument response was consistent, the sample was homogeneous, and the measurements were performed under the same conditions.
The smaller the standard deviation, the higher the reproducibility of the measurement can be considered.
However, a small standard deviation does not necessarily mean that the measured values are close to the true value.
For example, if the equipment is improperly calibrated, all measured values may be shifted in the same direction while the standard deviation remains small.
Standard deviation is an indicator of precision and must be distinguished from accuracy.
Example Discussion:
Because the standard deviation was small, the values obtained from multiple measurements were clustered around the mean, and the reproducibility of the measurement was considered high.
This indicates that sample collection and measurement procedures were performed relatively consistently.
However, a small standard deviation only indicates that the variation is small and does not directly guarantee that the measured values are close to the theoretical value.
Discussion When the Standard Deviation Is Large
A large standard deviation indicates that the variation in the measured values is large.
Possible causes include differences in the amount of sample collected, errors in reading the titration endpoint, variation in pipetting, temperature changes, instrument noise, and insufficient sample mixing.
It is necessary to consider the possibility that the measurement conditions were not constant.
When the standard deviation is large, the reliability of the mean also decreases.
If the variation is large, improvements such as increasing the number of measurements, standardizing procedures, using equipment correctly, and homogenizing the sample are necessary.
In addition, if only one specific point deviates greatly, the possibility of an outlier should also be considered.
Example Discussion:
Because the standard deviation was large, the variation in the measured values was large, and reproducibility was considered insufficient.
Possible causes include variation in pipetting, insufficient mixing of the sample, and errors in reading the titration endpoint.
In such a case, judging the result based only on the mean is insufficient, and it is desirable to standardize the operating conditions and repeat the measurement.
What Is Variation?
Variation means that multiple measured values do not become exactly the same value and differ from one another.
In chemistry experiments, repeating the same procedure often does not produce exactly the same result.
This is because of random errors and slight differences in operation.
The smaller the variation, the higher the reproducibility of the measurement procedure is considered to be.
If the variation is large, there may have been unstable factors in the experimental procedure or measurement conditions.
Standard deviation is a numerical value used to objectively express this variation.
Example Discussion:
The variation in the measured values was considered to have arisen from slight differences in the amount of sample collected, instrument readings, and instrument response.
By calculating the standard deviation, this variation can be evaluated numerically.
In this experiment, the reproducibility of the measurement procedure could be judged from the magnitude of the variation.
What Is Reproducibility?
Reproducibility is a property that indicates how closely the results agree when an experiment or measurement is repeated under the same conditions.
If nearly the same measured value is obtained each time, reproducibility is high, whereas if the measured values vary greatly, reproducibility is low.
Standard deviation is commonly used to evaluate reproducibility.
Results with high reproducibility indicate that the procedure was stable and random error was small.
However, high reproducibility does not necessarily mean high accuracy.
For example, if an uncalibrated balance produces the same shifted value every time, the standard deviation may be small even though the values differ from the true value.
Reproducibility and accuracy must be distinguished.
Example Discussion:
Because the standard deviation of the measured values was small, the measurement results under the same conditions agreed well, and reproducibility was considered high.
This indicates that the measurement procedure and instrument conditions were stable.
However, because high reproducibility does not necessarily mean that the values are close to the true value, comparison with a theoretical value or standard sample is also necessary.
Difference Between Precision and Accuracy
Precision indicates how close measured values are to one another.
The smaller the standard deviation, the more precise the measurement can be considered.
Accuracy, on the other hand, indicates how close the measured values are to the true or theoretical value.
Precision and accuracy are similar concepts, but they have different meanings.
Standard deviation is mainly an indicator used to evaluate precision.
Even if the measured values are close to one another, if they are all shifted from the true value, the result is precise but not accurate.
Conversely, even if the mean is close to the theoretical value, if the measured values are widely scattered, the result may appear accurate but have low precision.
| Item | Meaning | Relationship with Standard Deviation |
|---|---|---|
| Precision | Whether measured values are close to one another | Higher when the standard deviation is smaller |
| Accuracy | Whether the values are close to the true or theoretical value | Cannot be judged from standard deviation alone |
| Reproducibility | Whether repeated measurements produce the same result | Higher when the standard deviation is smaller |
Example Discussion:
Because the standard deviation was small, the variation among the measured values was small, and the precision can be judged to have been high.
However, if the mean deviates greatly from the theoretical value, the measurement may be precise but not accurate.
Therefore, when evaluating measurement results, precision based on standard deviation and accuracy based on the difference from the theoretical value must be considered separately.
What Is Relative Standard Deviation?
Relative standard deviation is obtained by dividing the standard deviation by the mean and expressing the magnitude of variation as a proportion of the mean.
It is often expressed as a percentage and is also called RSD.
It is useful when comparing variation among data sets with different magnitudes.
For example, even if the standard deviation is the same value of 0.1, the meaning of the variation differs greatly when the mean is 1.0 compared with when the mean is 100.0.
Relative standard deviation makes it possible to evaluate the magnitude of variation relative to the mean.
In analytical chemistry, it may be used to compare measurement precision.
Relative standard deviation RSD(%) = Standard deviation / Mean × 100
Example Discussion:
Relative standard deviation is an indicator that expresses the standard deviation as a proportion of the mean.
Because the RSD in this experiment was small, the variation relative to the mean was small, and the measurement precision was considered relatively good.
By using RSD in addition to the absolute value of the standard deviation, variation can be compared more easily among results with different magnitudes.
Relationship Between the Number of Measurements and Standard Deviation
When the number of measurements is small, standard deviation is more easily affected by chance.
For example, with only two or three measurements, one measurement error can have a large effect on the standard deviation.
Increasing the number of measurements makes it easier to evaluate the tendency of variation more stably.
However, even if the number of measurements is increased, accuracy will not improve if there is systematic error in the procedure or instrument.
Increasing the number of measurements is effective for evaluating random error, but separate measures are required for systematic errors such as calibration errors or errors in the concentration of a standard solution.
In a report, whether the number of measurements was sufficient can also be discussed.
Example Discussion:
When the number of measurements is small, a deviation in a single measured value has a large effect on the standard deviation, leaving uncertainty in the evaluation of variation.
Increasing the number of measurements makes it possible to evaluate the effect of random error more appropriately.
However, because systematic error cannot be removed simply by increasing the number of measurements, calibration of equipment and accuracy of sample preparation are also important.
Difference Between Standard Deviation and Standard Error
Standard deviation and standard error are easily confused, but they have different meanings.
Standard deviation indicates how widely individual measured values are distributed around the mean.
Standard error, on the other hand, is an indicator of the uncertainty of the mean itself.
Standard error is calculated by dividing the standard deviation by the square root of the number of measurements.
Standard deviation indicates the variation in measured values, while standard error indicates the reliability of the mean.
Standard deviation is important when discussing the reproducibility of measured values, while standard error may be used when discussing the uncertainty of the mean.
In a report, it is necessary to clearly state which one is being used.
Standard error = Standard deviation / √n
n: Number of measurements
Example Discussion:
Standard deviation represents the variation in individual measured values, whereas standard error represents the uncertainty of the mean.
Standard deviation is used when evaluating the reproducibility of measured values, while standard error is used when evaluating how reliable the mean is.
Therefore, in a report, standard deviation and standard error must not be confused and should be used according to the purpose.
Outliers and Standard Deviation
An outlier is a value that is far from the other measured values.
When an outlier is included, the standard deviation becomes larger.
Particularly when the number of measurements is small, a single outlier can have a large effect on the standard deviation.
Possible causes of outliers include operational errors, reading errors, insufficient sample mixing, and contaminated equipment.
However, outliers must not be excluded without careful consideration.
Excluding an outlier requires evidence such as a clear operational error or measurement abnormality.
If the cause is unknown, it is more appropriate to discuss the increase in standard deviation with the outlier included and state the need for remeasurement.
Example Discussion:
Because one measured value deviated greatly from the other values, the standard deviation was considered to have increased.
Possible causes of this outlier include an error in reading the titration endpoint or an error in the amount of sample collected.
However, if no clear operational error can be confirmed, the value should not be readily excluded and should instead be discussed as a cause of large variation in the measured values.
Random Error and Standard Deviation
Random error is a small error that occurs irregularly with each measurement.
Slight differences in pipetting, differences in readings, instrument noise, and small changes in temperature are included among random errors.
Because random errors cause measured values to vary around the mean, they are reflected in the standard deviation.
Methods for reducing random error include standardizing procedures, using the same equipment, keeping the measurement environment constant, and increasing the number of measurements and averaging them.
When the standard deviation is large, the possibility that random error was large can be discussed.
Example Discussion:
Possible causes of the variation in the measured values include random errors caused by slight differences in pipetting and reading operations.
Because random errors occur in different directions in each measurement, they appear as variation in the measured values and are reflected in the standard deviation.
Increasing the number of measurements and standardizing the operating conditions can reduce the effect of random error.
Systematic Error and Standard Deviation
Systematic error is an error that causes measured values to consistently deviate in the same direction.
Calibration errors in balances or pipettes, errors in the concentration of standard solutions, insufficient blank correction, and zero-point drift of instruments are examples of systematic error.
Because systematic error shifts all measured values in the same direction, it may not appear clearly in the standard deviation.
In other words, even if the standard deviation is small, the mean may deviate from the true value if systematic error is present.
The accuracy of an experiment must not be judged from standard deviation alone.
To check for systematic error, standard samples, blank measurements, equipment calibration, and comparison with theoretical values are necessary.
Example Discussion:
Even if the standard deviation was small, if the concentration of the standard solution was incorrect, all measured values may have shifted in the same direction.
Because this type of systematic error does not greatly increase the variation in measured values, it is difficult to detect using standard deviation alone.
Therefore, in addition to evaluating precision using standard deviation, accuracy must be checked by comparison with theoretical values or standard samples.
Discussion of Standard Deviation in Titration Experiments
In titration experiments, the mean and standard deviation may be calculated from multiple titration values.
If the standard deviation is small, the endpoint judgment and burette readings can be considered stable.
If the standard deviation is large, there may have been variation in judging the endpoint color change, dropping rate, burette reading, or amount of sample collected.
In titration, a difference of one drop may affect the measured value.
It is important to add titrant slowly near the endpoint and judge the color change using the same criterion.
If the standard deviation is large, the reproducibility of the titration procedure must be improved.
Example Discussion:
One possible reason why the standard deviation of the titration values was large is that there was variation in judging the color change at the endpoint.
In measurements in which titration was stopped only after the endpoint had been exceeded, the titration volume became larger, increasing the variation in the measured values.
Therefore, it is important to slow the addition rate near the endpoint and terminate the titration using the same color change as the criterion.
Discussion of Standard Deviation in Absorbance Measurements
In absorbance measurements, the same sample may be measured multiple times and the standard deviation calculated.
If the standard deviation is small, the condition of the cell and the response of the instrument can be considered stable.
If the standard deviation is large, possible causes include contamination of the cell, bubbles, sample turbidity, insufficient blank correction, incorrect wavelength settings, and insufficient sample mixing.
Absorbance may also be affected by the orientation of the cell and contamination on its surface.
Cleaning the cell before measurement, removing bubbles, and measuring with the cell in the same orientation help improve reproducibility.
Blank measurement is also important.
Example Discussion:
Possible causes of the large standard deviation in absorbance include contamination on the surface of the cell, bubbles, and insufficient sample mixing.
These factors affect the amount of transmitted light and cause variation in the measured values.
To improve measurement reproducibility, the cell must be kept clean, bubbles must be removed, blank correction must be performed, and measurements must be carried out under the same conditions.
Discussion of Standard Deviation in Mass Measurements
In mass measurements, the sensitivity of the balance, drying condition of the sample, moisture absorption, static electricity, air currents, and contamination of the sample pan affect the measured value.
If the standard deviation is large, the sample may not have been sufficiently dried, may have absorbed moisture during measurement, or the balance may not have been stable.
In mass measurements, it is important to allow the sample to return to room temperature before measurement, store it in a desiccator, close the draft shield of the balance, and use the same container.
The smaller the standard deviation, the higher the reproducibility of the weighing procedure can be considered.
Example Discussion:
Possible causes of the large standard deviation in the mass measurements include insufficient drying of the sample and moisture absorption.
If moisture in the sample evaporates or is absorbed during measurement, the mass changes and the measured values vary.
Therefore, in mass measurements, it is important to dry the sample sufficiently, cool and store it in a desiccator, and then weigh it.
Standard Deviation and Significant Figures
When reporting standard deviation, attention must also be paid to significant figures.
Because standard deviation represents the uncertainty of measured values, it is not appropriate to report the mean to unnecessarily fine decimal places.
In general, the decimal place of the mean is matched to that of the standard deviation.
For example, if the standard deviation is approximately 0.1, writing the mean as 10.123456 gives no meaningful information in the lower digits.
In a report, the number of digits in the mean and standard deviation should be made consistent and expressed using significant figures appropriate to the measurement precision.
Example Discussion:
Because standard deviation indicates the variation in measured values, the significant figures of the mean must be reported according to the decimal place of the standard deviation.
Writing the mean to finer decimal places than the standard deviation may appear to provide information beyond the actual measurement precision.
Therefore, it is important to present the mean and standard deviation using a number of digits appropriate to the measurement precision.
Meaning of Showing Standard Deviation on a Graph
When measured values are plotted on a graph, standard deviation may be shown as error bars.
Error bars visually represent the magnitude of variation at each measurement point.
Smaller error bars indicate higher reproducibility of the measured values, while larger error bars indicate greater variation.
Showing error bars makes it easier to judge whether differences between measurement points are truly meaningful or fall within the range of variation.
However, it is necessary to clearly state whether the error bars represent standard deviation, standard error, or a confidence interval.
Example Discussion:
By showing standard deviation as error bars on the graph, the magnitude of variation at each measurement point can be visually compared.
At measurement points with large error bars, the reproducibility of the measured values is low, and discussion based on those points contains greater uncertainty.
Therefore, it is important to show not only the mean but also information about variation on graphs.
Causes of Error That Increase Standard Deviation
Major causes of a large standard deviation include variation in sample preparation, reading errors of measuring equipment, instrument noise, changes in temperature and humidity, sample inhomogeneity, differences in endpoint judgment, insufficient cleaning of equipment, and differences in operating technique among operators.
It is important to specifically consider which operations are likely to affect variation.
For example, endpoint judgment is important in titration, cell condition in absorbance measurements, drying condition in mass measurements, and electrode stability in pH measurements.
A good discussion specifies the causes of a large standard deviation according to the details of the experiment.
Example Discussion:
Possible causes of the large standard deviation include variation in the amount of sample collected, reading errors of measuring equipment, and insufficient sample mixing.
These random errors caused individual measured values to deviate from the mean and increased the variation in the measurement results.
To improve measurement precision, it is effective to standardize sample preparation and measurement procedures and, when necessary, increase the number of measurements.
When the Results Can Be Considered Good
From the perspective of standard deviation, results can be considered good when the variation in the measured values is small and the standard deviation or relative standard deviation is small relative to the mean.
In this case, the reproducibility of the measurement procedure is high, and the effect of random error can be considered relatively small.
However, if the mean deviates greatly from the theoretical or literature value, the result cannot be considered good even if the standard deviation is small.
Precision based on standard deviation and accuracy based on agreement with the theoretical value must be judged together.
Good results are those in which the variation is small and reasonable values are obtained.
Example Discussion:
In this experiment, the standard deviation was small and the relative standard deviation was also low, so the variation in the measured values was small and reproducibility was considered good.
In addition, if the mean was close to the theoretical value, the results could be judged good in terms of both precision and accuracy.
Therefore, the measurement conditions used in this experiment were stable, and the measurement precision was considered relatively high.
Example Discussions When the Experiment Did Not Go Well
If the standard deviation is large, the measured values do not agree, one point deviates greatly, the relative standard deviation is high, or the mean deviates from the theoretical value, consider the causes separately.
Distinguishing whether the variation was caused by random error or whether the shift in the mean was caused by systematic error makes the discussion easier to write.
Example Discussion:
One possible reason why the standard deviation was large is that the amount of sample collected differed slightly among measurements.
Because changes in the sample amount also change the measured value, variation occurred among the results of repeated measurements.
To improve this, pipettes and volumetric flasks must be used correctly, and a fixed amount of sample must be collected after thorough mixing.
Another Example Discussion:
Because one measured value deviated greatly from the other measured values, the standard deviation increased.
During this measurement, there may have been a delay in judging the endpoint or an error in reading the equipment.
If an outlier is excluded, a clear operational basis is required, and if the basis is insufficient, remeasurement is desirable.
Another Example Discussion:
If the standard deviation was small but the mean deviated greatly from the theoretical value, the reproducibility of the measurement procedure may have been high, but systematic error may have been present.
For example, an error in the concentration of the standard solution or zero-point drift of the instrument would shift all measured values in the same direction.
Therefore, it is necessary to check not only the standard deviation but also the difference from the theoretical value.
How to Write Points for Improvement
In a discussion of standard deviation, writing not only about the causes of large variation but also about how the variation can be reduced makes the report easier to organize.
Points for improvement can be organized into sample preparation, measurement procedures, equipment, instruments, number of measurements, and handling of outliers.
Improvements to Sample Preparation and Measurement Procedures
- Mix the sample thoroughly before measurement
- Use pipettes and volumetric flasks correctly
- Repeat measurements using the same operating procedure
- During titration, add titrant slowly near the endpoint
- In absorbance measurements, remove contamination and bubbles from the cell
- In mass measurements, dry the sample sufficiently
- Keep the measurement temperature and pH constant
- Allow the instrument to stabilize before measurement
Improvements to Data Processing and Confirmation
- Increase the number of measurements
- Present the mean and standard deviation together
- Compare variation using relative standard deviation
- Check the cause of outliers
- If an outlier is excluded, clearly state the basis
- Compare with theoretical values or standard samples
- Do not confuse standard deviation with standard error
- Match significant figures to the standard deviation
Example of How to Write Points for Improvement:
To reduce variation in measured values, it is important to standardize sample preparation and measurement procedures and perform multiple measurements under the same conditions.
In titration, endpoint judgment must be kept consistent; in absorbance measurements, contamination and bubbles must be removed from the cell; and in mass measurements, the drying condition of the sample must be standardized.
In addition, if an outlier is present, checking its cause and performing remeasurement when necessary can make the standard deviation a more reliable value.
Difference Between a Superficial Discussion and a Good Discussion
In a discussion of standard deviation, simply writing that it “was small” or “was large” results in a superficial discussion.
A good discussion explains what the standard deviation indicates, which operations caused the variation, and how it affected the reproducibility and measurement precision of the results.
| Superficial Discussion | Good Discussion |
|---|---|
| The standard deviation was small. | Because the standard deviation was small, the measured values were clustered around the mean, and the reproducibility of measurements under the same conditions was considered high. |
| The standard deviation was large. | Possible causes of the large standard deviation include random errors in sample collection, endpoint judgment, and instrument noise that differed among measurements. |
| Reproducibility was good. | Because the variation among repeated measurements was small and the relative standard deviation was also low, the reproducibility of the measurement procedure was judged to be good. |
| The measurement was accurate. | Standard deviation is an indicator of precision, and to judge accuracy, it is necessary to separately confirm whether the mean is close to the theoretical or standard value. |
| The outlier was excluded. | Excluding an outlier requires clear evidence such as a reading error or operational error, and if there is no such basis, remeasurement is desirable. |
Examples of Expressions That Can Be Used in Reports
The following expressions can be used when writing the results and discussion of standard deviation and variation.
Adjust the necessary parts according to your own experimental results.
- Standard deviation indicates how widely values obtained from multiple measurements are distributed around the mean.
- Because the standard deviation was small, the variation in the measured values was small and reproducibility was considered high.
- Because the standard deviation was large, there may have been variation in the measurement procedure or sample preparation.
- Because the relative standard deviation was small, the variation relative to the mean was judged to be small.
- Standard deviation indicates precision but does not indicate accuracy.
- If the mean deviates from the theoretical value, the effect of systematic error must be considered.
- If an outlier is included, the standard deviation increases and affects the evaluation of reproducibility.
- If an outlier is excluded, a clear operational basis is required.
- If the number of measurements is small, uncertainty remains in the evaluation of standard deviation.
- To reduce the standard deviation, it is important to standardize the measurement procedure and repeat measurements under the same conditions.
Points to Check When Discussing Standard Deviation
Checking the following points before writing the report makes the discussion easier to write.
- Is the number of measurements stated?
- Are the mean and standard deviation shown together?
- Has the unit of the standard deviation been checked?
- Is relative standard deviation used when necessary?
- Is the reason why the standard deviation is small explained?
- Have the causes of a large standard deviation been considered specifically?
- Are reproducibility and standard deviation related in the discussion?
- Are precision and accuracy distinguished?
- Has the presence or absence of outliers been checked?
- Are random error and systematic error considered separately?
- Are the significant figures appropriate for the standard deviation?
- Do the points for improvement correspond to the causes of error?
Summary
Standard deviation is an indicator that shows how widely values obtained from multiple measurements are distributed around the mean.
The smaller the standard deviation, the more closely the measured values are clustered around the mean, and the higher the reproducibility and precision can be judged to be.
On the other hand, if the standard deviation is large, there may have been causes of variation in sample preparation, measurement procedures, instruments, or environmental conditions.
However, standard deviation is an indicator of measurement precision and does not directly indicate whether the measured values are close to the true value.
Even if the standard deviation is small, if there is systematic error such as an error in the concentration of a standard solution or zero-point drift of an instrument, the mean may deviate from the theoretical value.
Therefore, evaluation of reproducibility using standard deviation and evaluation of accuracy by comparison with theoretical or standard values must be considered separately.
In a report, rather than simply writing that “the standard deviation was small” or “the variation was large,” organize and discuss the mean, standard deviation, relative standard deviation, number of measurements, outliers, random error, systematic error, reproducibility, precision, accuracy, and points for improvement.
Discussion of standard deviation is important for evaluating the reliability of measured values and making experimental procedures more stable.
