In chemistry laboratory reports, it is important not only to list measured values but also to handle the results with attention to significant figures and errors. Even when the same experiment is performed, measured values always show some degree of variation due to differences in instrument accuracy, reading methods, temperature, experimental procedures, and other factors.
This article clearly explains the concept of significant figures commonly used in chemistry experiments, how to round measured values, how to handle averages and errors, and examples of discussion that can be used in reports. Use it as a reference when writing results and discussions for basic chemistry experiments, analytical chemistry experiments, physical chemistry experiments, and similar laboratory work.
Note: The methods used for handling significant figures and errors may differ depending on the university, course, or instructor. In actual reports, give priority to the laboratory manual distributed in the course and the instructions of your instructor.
- What Are Significant Figures?
- Basic Rules for Significant Figures
- Why Significant Figures Are Important in Chemistry Experiments
- Measuring Instruments and Significant Figures
- Significant Figures in Calculated Results
- What Is Error?
- Types of Error
- Absolute Error and Relative Error
- How to Think About Averages and Variation
- Example of Writing Significant Figures and Errors in the Results
- Common Perspectives for Discussing Causes of Error
- Examples of Discussing Causes of Error
- Difference Between a Superficial Discussion and a Good Discussion
- Common Mistakes with Significant Figures and Errors
- Example Expressions That Can Be Used in Reports
- Checklist for Significant Figures and Errors
- Summary
What Are Significant Figures?
Significant figures are the digits in a measured value that carry meaningful information. The number of reliable digits in an experimental measurement is determined by the precision of the measuring instrument. Therefore, writing a calculated result to an unnecessarily large number of decimal places may make the experimental value appear more precise than it actually is.
For example, the number of reliable digits differs between a volume measured with a graduated cylinder and a volume measured with a volumetric pipette. Graduated cylinders are often used for approximate volume measurements, while volumetric pipettes and burettes are used for more precise volume measurements.
In a laboratory report, values must be recorded with a number of digits appropriate to the precision of the measuring instrument, and calculated results must also be expressed using appropriate significant figures.
Basic Rules for Significant Figures
When determining significant figures, it is useful to understand the following basic rules.
All Nonzero Digits Are Significant
For example, 12.3 has three significant figures. 4.56 also has three significant figures.
Zeros Between Nonzero Digits Are Significant
For example, 10.5 has three significant figures. 1002 has four significant figures.
Trailing Zeros After a Decimal Point Are Significant
For example, 1.20 has three significant figures. 1.200 has four significant figures. Numerically, both represent the same value as 1.2, but as measured values they have different meanings.
Leading Zeros Before the First Nonzero Digit Are Not Significant
For example, 0.0123 has three significant figures. The leading zeros only indicate the position of the decimal point and do not represent measurement precision.
| Value | Significant Figures | Explanation |
|---|---|---|
| 12.3 | 3 | 1, 2, and 3 are significant |
| 1.20 | 3 | The final 0 also indicates measurement precision |
| 0.0123 | 3 | Leading zeros are not included as significant figures |
| 1002 | 4 | Zeros between digits are significant |
| 0.0100 | 3 | 1, 0, and 0 are significant |
Why Significant Figures Are Important in Chemistry Experiments
Chemistry experiments involve many types of measured values, including mass, volume, concentration, pH, absorbance, and temperature. These values contain limitations arising from instrument precision and reading errors.
For example, a sample measured as 1.234 g with an electronic balance and a sample measured as 1.2 g with a top-pan balance may appear to have similar masses, but the number of reliable digits is different. If this difference is ignored in calculations, the precision of the experimental values cannot be represented correctly.
By paying attention to significant figures, results can be presented with a precision appropriate to the reliability of the measurements. This also affects the persuasiveness of the results and discussion in a laboratory report.
Measuring Instruments and Significant Figures
The number of digits that can be read differs depending on the measuring instrument. When writing a report, significant figures must be determined by considering the precision of the instrument used.
| Instrument / Measurement Method | Characteristics | Points to Consider for Significant Figures |
|---|---|---|
| Electronic balance | Can measure mass precisely | Record according to the number of displayed digits |
| Graduated cylinder | Measures approximate volume | Do not record more digits than justified by the graduation spacing |
| Volumetric pipette | Measures a fixed volume precisely | Consider the volume accuracy indicated for the instrument |
| Burette | Measures titration volume precisely | Pay attention to the smallest graduation and visual reading error |
| pH meter | Measures pH | Consider the displayed digits and calibration condition |
| Thermometer | Measures temperature | Pay attention to the smallest graduation and temperature changes |
For example, writing a volume measured with a graduated cylinder as “10.000 mL” is not appropriate. The instrument cannot measure the volume with that degree of precision.
Significant Figures in Calculated Results
In laboratory reports, measured values are used to calculate concentrations, yields, averages, error percentages, and other quantities. It is important not to report calculated results with an unnaturally large number of digits compared with the original measured values.
Multiplication and Division
For multiplication and division, the result is generally expressed using the same number of significant figures as the value with the fewest significant figures.
Example:
1.23 × 4.5 = 5.535
1.23 has three significant figures and 4.5 has two significant figures, so the answer is expressed with two significant figures as 5.5.
Addition and Subtraction
For addition and subtraction, the result is generally expressed to the same number of decimal places as the value with the fewest decimal places.
Example:
12.34 + 1.2 = 13.54
Because 1.2 is given to one decimal place, the answer is rounded to one decimal place as 13.5.
Do Not Round Too Much During Intermediate Calculations
Repeatedly rounding during intermediate calculations can increase the error in the final result. It is useful to retain slightly more digits during the calculation and round to the appropriate number of significant figures at the end.
What Is Error?
Error is the difference between a measured value and the true value, or between a measured value and a theoretical or literature value. In chemistry experiments, errors cannot be completely eliminated no matter how carefully the experiment is performed.
The presence of error itself is not necessarily a problem. What is important is to consider how large the error is and what caused it.
In a laboratory report, simply writing that “an error occurred” is insufficient. It is necessary to explain specifically “which operation,” “which measured value,” and “in which direction” the error affected the result.
Types of Error
Errors that occur in experiments can broadly be classified into random errors and systematic errors.
Random Error
Random error is an error that occurs irregularly each time a measurement is performed. Examples include slight differences in reading graduations, small changes in temperature, and variation in determining the titration endpoint.
The effect of random error can be reduced by performing multiple measurements and calculating the average.
Systematic Error
Systematic error is an error that shifts measured values consistently in one direction. Examples include improper instrument calibration, a standard solution with an incorrect concentration, a measuring instrument with an incorrect zero point, and an insufficiently dried sample.
Because systematic errors shift measurements in the same direction each time, simply calculating an average does not easily eliminate them. Instrument calibration and confirmation of experimental conditions are necessary.
| Type of Error | Characteristics | Examples |
|---|---|---|
| Random error | Changes irregularly between measurements | Differences in reading graduations, variation in endpoint determination |
| Systematic error | Shifts in a consistent direction | Improper instrument calibration, incorrect standard solution concentration |
Absolute Error and Relative Error
Absolute error and relative error are methods used to express the magnitude of an error.
Absolute Error
Absolute error is the difference between a measured value and a theoretical or literature value.
Absolute error = |Measured value − Theoretical value|
For example, if the theoretical value is 10.00 g and the measured value is 9.80 g, the absolute error is 0.20 g.
Relative Error
Relative error is the absolute error divided by the theoretical value. It is often expressed as a percentage and makes it easier to compare how much an experimental value differs from the theoretical value.
Relative error (%) = |Measured value − Theoretical value| ÷ Theoretical value × 100
Example:
Theoretical value: 10.00 g
Measured value: 9.80 g
Absolute error: |9.80 − 10.00| = 0.20 g
Relative error: 0.20 ÷ 10.00 × 100 = 2.0%
How to Think About Averages and Variation
When the same measurement is performed multiple times, the average of the measured values may be calculated. Using an average can reduce the effect of random error.
Average = Sum of measured values ÷ Number of measurements
Example of Titration Volumes:
1st measurement: 8.72 mL
2nd measurement: 8.70 mL
3rd measurement: 8.74 mL
Average = (8.72 + 8.70 + 8.74) ÷ 3 = 8.72 mL
However, care is needed when deciding whether to include a measurement that clearly overshot the endpoint or involved an operational error in the average. If a value is excluded, the reason for excluding it should be explained in the report.
Discussion Example:
In the first titration, the endpoint was exceeded and the solution became a dark red color, so the titration volume may have been overestimated. Therefore, the average titration volume was calculated using the second and third measurements, in which the endpoint was judged appropriately.
Example of Writing Significant Figures and Errors in the Results
In the results section of a report, measured values, averages, and calculated results should be presented with attention to significant figures. When necessary, organizing the values in a table can make them easier to read.
| Measurement | Titration Volume / mL |
|---|---|
| 1st | 8.72 |
| 2nd | 8.70 |
| 3rd | 8.74 |
| Average | 8.72 |
In a table, including the unit in the heading makes the values easy to understand without writing the unit after every number. However, when presenting a value in the text, include the unit, as in “8.72 mL.”
Common Perspectives for Discussing Causes of Error
When considering causes of error, think specifically about where in the experimental procedure the value may have shifted. In chemistry experiments, the following perspectives are commonly used.
Errors Caused by Instruments
- Errors in reading graduations
- Calibration errors of burettes or pipettes
- Zero-point shift of an electronic balance
- Improper calibration of a pH meter
- Errors in reading a thermometer
Errors Caused by Experimental Procedures
- Part of the sample was spilled
- Precipitate or crystals remained on equipment during filtration
- Washing was insufficient
- Drying was insufficient
- The endpoint was exceeded during titration
- The heating time or temperature was inappropriate
Errors Caused by Reagents or Samples
- The reagent concentration was not accurate
- The sample contained impurities
- The sample absorbed moisture
- The reaction did not proceed to completion
- A side reaction occurred
Errors Caused by Environmental Conditions
- Temperature changes
- Effects of humidity
- Evaporation
- Effects of carbon dioxide or moisture in the air
In the discussion, rather than simply listing these causes, it is important to select those that match your own experimental results and explain how they affected the results.
Examples of Discussing Causes of Error
This section presents examples of discussing causes of error that are easy to use in laboratory reports. Adjust the expressions to match your own experimental results.
Discussion Example for a Titration Experiment
One possible reason for the variation in titration volume is a difference in endpoint determination. Because the color change of the indicator is judged visually, the titration volume changes depending on the point at which a faint color change is considered the endpoint. Particularly when solution is added beyond the endpoint, the titration volume becomes larger than the actual value, causing an error in the calculated concentration.
Discussion Example for Mass Measurement
One possible cause of error in mass measurement is insufficient drying of the sample. If moisture remained in the sample, the measured mass would be larger than the actual sample mass. As a result, the yield may have been overestimated relative to the theoretical value.
Discussion Example for Volume Measurement
One possible cause of error in volume measurement is a difference in the position used to read the meniscus. If the eye level was not at the same height as the graduation, the measured volume may have been read as larger or smaller than the actual volume. Such reading errors also affect the results of concentration calculations.
Discussion Example for Yield Calculation
Possible reasons why the yield was lower than the theoretical value include incomplete reaction and loss of part of the product during filtration or transfer. In addition, if recrystallization was performed, part of the desired substance may have remained dissolved in the solvent, reducing the amount recovered.
Difference Between a Superficial Discussion and a Good Discussion
In discussing errors, simply writing that “there was human error” or “there was a measurement error” is insufficient. It is necessary to explain what type of error occurred, which value it affected, and how it affected that value.
| Superficial Discussion | Good Discussion |
|---|---|
| An error occurred because of a measurement mistake. | If the eye level was not aligned with the graduations when reading the burette, the titration volume may have been read as larger or smaller than the actual value, causing an error in the calculated concentration. |
| I think the sample was insufficiently dried. | If drying was insufficient and moisture remained in the sample, the measured mass would increase and the yield may have been calculated as higher than the actual value. |
| The experiment did not go well. | Because part of the product adhered to the filter paper or beaker during filtration and could not be recovered, the obtained mass decreased and the yield is considered to have become lower. |
A good discussion clearly describes the connection between cause and result. The key is to explain not only “what happened” but also “how the measured value changed as a result.”
Common Mistakes with Significant Figures and Errors
The following mistakes are commonly seen in chemistry laboratory reports when handling significant figures and errors.
Writing Too Many Digits in Calculated Results
If the value displayed by a calculator is copied directly, there may be too many significant figures. For example, instead of writing a yield as 66.666666%, express it as 66.7% or 67% according to the precision of the measured values.
Forgetting Units
Always include units with numerical values. If only “8.72” is written, it is unclear whether the unit is mL or g.
Writing Only General Causes of Error
Writing only “human error” or “instrument error” lacks specificity. It is necessary to explain which operation in your own experiment was related to the error.
Not Explaining the Difference from the Theoretical Value
When comparing with a theoretical or literature value, do not stop after stating whether the values agreed or differed. Consider the reason for the difference. Also pay attention to the direction of the deviation and explain whether the measured value became larger or smaller.
Example Expressions That Can Be Used in Reports
The following expressions can be used when discussing significant figures and errors.
- The measured values were generally consistent, and the reproducibility is considered to have been relatively high.
- One possible reason for the variation in measured values is an error in reading the graduations.
- Because the endpoint was determined visually, individual differences may have occurred in the titration volume.
- If drying was insufficient, moisture may have remained in the sample, causing the measured mass to become larger than the actual value.
- Loss of part of the sample during filtration or transfer is considered to have reduced the amount recovered.
- The calculated result was expressed using significant figures appropriate to the measured values used.
- The difference from the theoretical value is considered to be attributable to the measurement procedure or sample purity.
- Because the relative error was small, the measured value can be considered close to the theoretical value.
Checklist for Significant Figures and Errors
Before submitting a report, it is useful to check the following points.
- Are units included with measured values?
- Are values recorded with a number of digits appropriate to the precision of the measuring instrument?
- Have calculated results been written without simply copying all digits displayed by the calculator?
- For multiplication and division, have you considered the number of significant figures?
- For addition and subtraction, have you considered the number of decimal places?
- When calculating an average, have you avoided including inappropriate values?
- If a value was excluded, have you explained the reason?
- Have you explained the causes of error specifically?
- Have you considered whether the error makes the measured value larger or smaller?
- Have you compared the result with a theoretical or literature value?
Summary
In chemistry laboratory reports, proper handling of significant figures and errors is very important. Every measured value has limitations, and the number of reliable digits is determined by the instruments and procedures used. Therefore, calculated results should not be written with unnecessarily many digits but should instead be expressed using significant figures appropriate to the precision of the measurements.
In addition, when discussing errors, it is important not simply to write that “an error occurred,” but to explain specifically which operation or condition affected which measured value and how it affected it. Understanding the differences between random and systematic errors and the concepts of absolute and relative error makes it easier to evaluate results and write discussions.
Correctly handling significant figures and errors is fundamental to accurately communicating experimental results. When writing a report, check the number of digits in measured values, units, calculated results, and causes of error one by one as you organize the report.
