Table of Contents
Significant Figures, Measurement Errors, Accuracy and Precision
1. Introduction
Scientific measurements are never entirely free from uncertainty. Even sophisticated instruments have limitations, and experimental procedures may introduce variations in measured values.
Significant figures, accuracy, precision and measurement errors are important concepts used to evaluate the quality and reliability of scientific measurements.
They help researchers communicate numerical results correctly, assess experimental limitations and determine how much confidence can be placed in measured values.
2. Significant Figures
Significant figures are the digits in a measured or calculated value that convey meaningful information about its precision.
They include all certain digits and one estimated or uncertain digit, depending on the measurement method.
For example:
[
12.45\ \text{cm}
]
This measurement contains four significant figures.
The last digit indicates the level of reported measurement precision.
3. Rules for Counting Significant Figures
3.1 Non-Zero Digits
All non-zero digits are significant.
Examples:
- 25 has 2 significant figures.
- 347 has 3 significant figures.
- 8.642 has 4 significant figures.
3.2 Zeros Between Non-Zero Digits
Zeros between non-zero digits are significant.
Examples:
- 205 has 3 significant figures.
- 1007 has 4 significant figures.
- 4.005 has 4 significant figures.
3.3 Leading Zeros
Zeros before the first non-zero digit are not significant.
Examples:
- 0.005 has 1 significant figure.
- 0.0024 has 2 significant figures.
- 0.00072 has 2 significant figures.
3.4 Trailing Zeros After a Decimal Point
Trailing zeros after a decimal point are significant when they are part of the stated measurement.
Examples:
- 2.50 has 3 significant figures.
- 4.200 has 4 significant figures.
- 0.0800 has 3 significant figures.
3.5 Trailing Zeros in Whole Numbers
Trailing zeros in whole numbers without a decimal point may be ambiguous.
For example, 1500 could represent two, three or four significant figures depending on the measurement context.
Scientific notation removes this ambiguity.
- (1.5 \times 10^3): 2 significant figures.
- (1.50 \times 10^3): 3 significant figures.
- (1.500 \times 10^3): 4 significant figures.
4. Significant Figures in Mathematical Operations
4.1 Multiplication and Division
For multiplication and division, the final result should generally contain the same number of significant figures as the quantity with the fewest significant figures.
Example:
[
2.5 \times 3.42 = 8.55
]
The first value has 2 significant figures, while the second has 3.
Therefore, the result should be reported as:
[
\boxed{8.6}
]
4.2 Addition and Subtraction
For addition and subtraction, the result should generally be rounded to the least precise decimal place among the quantities being combined.
Example:
[
12.11 + 3.4 + 0.256 = 15.766
]
The least precise measurement is 3.4, which is reported to one decimal place.
Therefore:
[
\boxed{15.8}
]
4.3 Exact Numbers
Exact counted quantities and defined conversion factors do not limit the number of significant figures.
For example, if a box contains exactly 12 pencils, the number 12 is an exact count.
Likewise, the conversion:
[
1\ \text{m} = 100\ \text{cm}
]
is exact by definition.
4.4 Rounding Rules
Rounding reduces the number of digits reported.
Common decimal rounding conventions include:
- If the first discarded digit is below 5, round down.
- If it is above 5, round up.
- If it is exactly 5, the result depends on the adopted rounding convention and subsequent digits.
Some scientific and statistical systems use round-half-to-even to reduce cumulative rounding bias.
It is preferable to retain additional digits during intermediate calculations and round only the final result.
5. Accuracy
Accuracy refers to how close a measured value is to the true value or an accepted reference value.
A highly accurate measurement has a small measurement error relative to the reference.
Example:
If the accepted value of a length is 10.00 cm and a measurement gives 9.99 cm, the result is close to the reference and therefore highly accurate.
Accuracy is influenced by:
- Instrument quality.
- Calibration.
- Measurement technique.
- Environmental conditions.
- Systematic errors.
- Reference-standard uncertainty.
6. Precision
Precision refers to the closeness of agreement among repeated measurements made under specified conditions.
A precise measurement process produces closely grouped results.
Example:
Suppose three measurements of a quantity are:
10.21 cm, 10.22 cm and 10.21 cm.
These values show high repeatability and precision.
Precision is influenced by:
- Instrument resolution.
- Random variations.
- Experimental technique.
- Environmental stability.
- Operator consistency.
7. Difference Between Accuracy and Precision
Accuracy and precision are related but distinct concepts.
| Accuracy | Precision |
|---|---|
| Closeness to an accepted reference value | Closeness among repeated measurements |
| Relates to measurement error | Relates to consistency and dispersion |
| Can be affected by systematic errors | Often strongly affected by random variation |
| Requires a reference for evaluation | Can be assessed through repeated measurements |
7.1 Four Measurement Situations
High accuracy and high precision
Measurements are closely grouped around the accepted value.
High precision but low accuracy
Measurements are closely grouped but far from the accepted value, often due to systematic error.
High accuracy on average but low precision
Measurements are widely scattered, but their average may be close to the reference.
Low accuracy and low precision
Measurements are scattered and generally far from the reference value.
8. Measurement Errors
Measurement error is the difference between a measured value and the corresponding reference or true value.
[
\text{Error} = \text{Measured Value} – \text{Reference Value}
]
In practice, the true value may not be known exactly. A calibrated reference value is often used instead.
8.1 Absolute Error
Absolute error is the magnitude of the difference between the measured and reference values.
[
\text{Absolute Error} =
|\text{Measured Value} – \text{Reference Value}|
]
Example:
Reference value = 50.0 cm
Measured value = 49.6 cm
[
\text{Absolute Error} = |49.6-50.0|
]
[
=0.4\ \text{cm}
]
8.2 Relative Error
Relative error is the absolute error divided by the magnitude of the reference value.
[
\text{Relative Error} =
\frac{\text{Absolute Error}}{|\text{Reference Value}|}
]
Using the previous example:
[
\text{Relative Error} = \frac{0.4}{50.0}
]
[
=0.008
]
8.3 Percentage Error
Percentage error is relative error multiplied by 100.
[
\text{Percentage Error} =
\frac{\text{Absolute Error}}{|\text{Reference Value}|}
\times 100
]
Therefore:
[
0.008 \times 100 = 0.8%
]
The percentage error is 0.8%.
9. Types of Measurement Errors
9.1 Systematic Errors
Systematic errors are consistent or predictable measurement deviations caused by identifiable factors.
They may result from:
- Incorrect instrument calibration.
- Zero error.
- Instrument defects.
- Incorrect experimental procedures.
- Environmental influences.
- Faulty measurement assumptions.
Systematic errors primarily affect accuracy.
Examples:
- A weighing scale consistently reads 0.5 kg too high.
- A thermometer has a calibration offset.
- A measuring instrument expands significantly with temperature but no correction is applied.
9.2 Random Errors
Random errors are unpredictable variations that cause repeated measurements to differ.
Possible causes include:
- Small environmental fluctuations.
- Electronic noise.
- Variations in human observation.
- Uncontrolled experimental conditions.
Random errors affect precision.
They can often be reduced by taking repeated measurements and calculating an appropriate average.
9.3 Gross Errors
Gross errors are major mistakes caused by human or procedural failures.
Examples include:
- Reading the wrong scale.
- Recording an incorrect value.
- Using the wrong unit.
- Connecting an instrument incorrectly.
- Miscalculating experimental results.
Careful procedures, verification and independent checks help reduce gross errors.
10. Mean and Standard Deviation
Repeated measurements can be analysed statistically.
10.1 Arithmetic Mean
The arithmetic mean is calculated by adding all observations and dividing by the number of observations.
[
\bar{x} = \frac{\sum x_i}{n}
]
Where:
- (\bar{x}) = arithmetic mean.
- (x_i) = individual observation.
- (n) = number of observations.
Example:
Measurements: 10.1, 10.2, 10.0 and 10.3 cm.
[
\bar{x} = \frac{10.1+10.2+10.0+10.3}{4}
]
[
\bar{x} = 10.15\ \text{cm}
]
10.2 Standard Deviation
Standard deviation measures the spread of observations around their mean.
For a sample:
[
s = \sqrt{\frac{\sum(x_i-\bar{x})^2}{n-1}}
]
A smaller standard deviation generally indicates greater precision when comparing measurements under similar conditions.
A standard deviation does not by itself establish accuracy.
10.3 Range
The range is the difference between the largest and smallest observations.
[
\text{Range} = x_{\max}-x_{\min}
]
It provides a simple indication of the spread of measurements.
11. Measurement Uncertainty
Measurement uncertainty expresses the doubt associated with a measurement result.
It is not identical to measurement error.
Error is the difference from a reference value, while uncertainty characterizes the range or dispersion of values reasonably attributable to the measured quantity.
A measurement may be expressed as:
[
x = 25.4 \pm 0.1\ \text{cm}
]
This indicates a reported value of 25.4 cm with an associated uncertainty of 0.1 cm, subject to the stated interpretation and coverage convention.
11.1 Type A Evaluation
Type A uncertainty is evaluated using statistical analysis of repeated observations.
11.2 Type B Evaluation
Type B uncertainty is evaluated using other information, such as:
- Calibration certificates.
- Instrument specifications.
- Manufacturer information.
- Previous measurement records.
- Scientific reference data.
- Expert judgment.
11.3 Combined Uncertainty
Independent standard uncertainty components are often combined using the root-sum-of-squares method:
[
u_c = \sqrt{u_1^2+u_2^2+\cdots+u_n^2}
]
This formula applies when the components are independent or appropriately treated as uncorrelated.
Correlated uncertainty components require additional treatment.
12. Resolution and Sensitivity
12.1 Resolution
Resolution is the smallest change in a quantity that a measuring instrument can display or distinguish under specified conditions.
For example, a digital scale displaying values to 0.01 kg has a display increment of 0.01 kg.
Resolution is not the same as accuracy.
12.2 Sensitivity
Sensitivity describes how much an instrument’s output changes in response to a change in the measured quantity.
For a linear instrument:
[
\text{Sensitivity} = \frac{\Delta \text{Output}}{\Delta \text{Input}}
]
A highly sensitive instrument produces a relatively large output change for a small input change.
13. Improving Measurement Quality
Measurement quality can be improved through:
- Proper instrument selection.
- Regular calibration.
- Correct measurement techniques.
- Repeated observations.
- Controlled environmental conditions.
- Reduction of parallax error.
- Proper unit conversion.
- Accurate data recording.
- Statistical analysis.
- Reporting measurement uncertainty.
14. Practical Example: Measuring a Table
Suppose a table is measured using a ruler.
The reference length is 120.0 cm, while the observed length is 119.5 cm.
Absolute error:
[
|119.5-120.0|=0.5\ \text{cm}
]
Percentage error:
[
\frac{0.5}{120.0}\times100
=0.4167%
]
Rounded appropriately, the percentage error is approximately 0.42%.
If repeated readings are 119.5, 119.6 and 119.5 cm, the close grouping indicates good precision. However, the readings remain below the reference value, suggesting a possible systematic error.
15. Conclusion
Significant figures communicate the meaningful precision of numerical measurements, while accuracy and precision describe different aspects of measurement quality.
Understanding systematic, random and gross errors, along with uncertainty and statistical analysis, is essential for evaluating scientific experiments and producing reliable results.
36 Key Takeaways for Exams
- Significant figures represent meaningful digits in a measured value.
- All non-zero digits are significant.
- Zeros between non-zero digits are significant.
- Leading zeros are not significant.
- Trailing zeros after a decimal point are significant when reported as part of a measurement.
- Trailing zeros in whole numbers without a decimal point can be ambiguous.
- Scientific notation removes ambiguity about significant figures.
- Multiplication and division generally follow the fewest-significant-figures rule.
- Addition and subtraction generally follow the least precise decimal-place rule.
- Exact counted quantities do not limit significant figures.
- Accuracy refers to closeness to an accepted reference value.
- Precision refers to closeness among repeated measurements.
- High precision does not necessarily mean high accuracy.
- Systematic errors produce consistent or predictable measurement deviations.
- Random errors produce unpredictable measurement variations.
- Gross errors commonly result from major human or procedural mistakes.
- Calibration helps identify and correct systematic measurement deviations.
- Absolute error is the magnitude of the difference between measured and reference values.
- Relative error is absolute error divided by the reference value’s magnitude.
- Percentage error is relative error multiplied by 100.
- Arithmetic mean is the sum of observations divided by their number.
- Standard deviation measures the spread of observations around their mean.
- Range is the difference between the maximum and minimum observations.
- Measurement uncertainty characterizes the doubt associated with a result.
- Measurement error and measurement uncertainty are different concepts.
- Type A uncertainty is evaluated statistically.
- Type B uncertainty is evaluated using other available information.
- Independent uncertainty components may be combined using the root-sum-of-squares method.
- Resolution is the smallest distinguishable or displayable change under specified conditions.
- Sensitivity relates instrument output changes to input changes.
- Parallax error occurs when a scale is viewed from an incorrect angle.
- Repeated measurements can help assess random variation.
- Controlled environmental conditions improve measurement reliability.
- Rounding should generally be postponed until the final calculation.
- A small standard deviation indicates low dispersion but does not prove accuracy.
- Reliable scientific results require appropriate measurement methods, calibration and uncertainty evaluation.