1.3 Accuracy, Precision, and Significant Figures
3 min read•june 18, 2024
Measurements in physics require and . , , and error analysis are crucial tools for ensuring reliable results. These concepts help scientists communicate the level of confidence in their measurements and calculations.
Understanding the difference between accuracy and is key. Accuracy measures how close a value is to the true value, while precision refers to the consistency of repeated measurements. Both are essential for drawing valid conclusions from experiments and advancing our understanding of the physical world.
Measurement and Uncertainty
Significant figures in calculations
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- represent the number of digits in a measured value that are known with certainty plus one estimated digit
- Non-zero digits are always significant (1, 2, 3, etc.)
- Zeros between non-zero digits are significant (101, 1002, etc.)
- Leading zeros are not significant (0.01, 0.0023, etc.)
- Trailing zeros are significant only if the decimal point is present (1.0, 2.300, etc.)
- Addition and subtraction follow the decimal place rule
- The result should have the same number of decimal places as the measurement with the least number of decimal places (1.2 + 3.45 = 4.7)
- Multiplication and division follow the significant figure rule
- The result should have the same number of significant figures as the measurement with the least number of significant figures (2.3 × 1.2345 = 2.8)
- is often used to express very large or very small numbers while maintaining significant figures
Percent uncertainty in experiments
- represents the smallest unit of measurement that can be reliably measured by the measuring instrument (ruler with mm markings has an absolute of 1 mm)
- is the ratio of the absolute uncertainty to the measured value
- Expressed as a fraction or percentage (absolute uncertainty of 1 mm for a measured length of 50 mm gives a relative uncertainty of 1/50 or 0.02)
- Calculate relative uncertainty using the formula:
- is the relative uncertainty expressed as a percentage
- Calculate using the formula: (relative uncertainty of 0.02 gives a percent uncertainty of 2%)
- Apply percent uncertainty to experimental results by expressing the final result with the appropriate number of significant figures based on the percent uncertainty (a measured value of 10.5 cm with a percent uncertainty of 2% should be expressed as 10.5 ± 0.2 cm)
- The of an instrument is the smallest measurement it can reliably make, which contributes to
Accuracy vs precision in measurements
- Accuracy refers to how close a measured value is to the true or accepted value
- High accuracy means the measured value is very close to the true value (measuring a known 100 g mass and obtaining a result of 99.8 g)
- is crucial for maintaining accuracy in measurements
- Precision refers to how close multiple measurements of the same quantity are to each other
- High precision means the measurements are very consistent and have little variation (multiple measurements of the same object yielding 10.1 cm, 10.2 cm, and 10.1 cm)
- (consistency of measurements by the same person) and (consistency of measurements by different people) are aspects of precision
- Importance in physics:
- Accurate measurements are essential for verifying scientific theories and laws (confirming the predicted value of the gravitational acceleration, )
- Precise measurements are necessary for reproducibility and reliability of experimental results (consistent values of the speed of light, , obtained by different researchers)
- Both accuracy and precision are crucial for drawing valid conclusions from scientific experiments and advancing our understanding of the physical world (precise and accurate measurements of atomic spectra leading to the development of quantum mechanics)
Error Analysis and Propagation
- Measurement uncertainty is inherent in all physical measurements and must be accounted for
- involves calculating how uncertainties in individual measurements affect the uncertainty of a final calculated result
- Understanding error propagation is crucial for determining the reliability of experimental conclusions
Key Terms to Review (35)
Absolute Uncertainty: Absolute uncertainty is the smallest possible variation or error in a measurement, representing the limits of the precision of the measuring instrument. It is a fundamental concept in the context of accuracy, precision, and significant figures, as it quantifies the inherent uncertainty associated with any measurement.
Accuracy: Accuracy refers to the closeness of a measurement or calculation to the true or accepted value. It is a measure of how precise and reliable a result is, and it is a critical concept in the fields of physics, engineering, and scientific research.
Error propagation: Error propagation refers to the way in which uncertainties in measured values affect the uncertainty in a calculated result. When combining measurements, the overall uncertainty can be determined through mathematical methods, revealing how the precision of individual measurements impacts the final outcome. Understanding error propagation is crucial for evaluating accuracy and precision, as it highlights the significance of significant figures in reporting results.
Instrument Calibration: Instrument calibration is the process of adjusting the measurements of an instrument to ensure its accuracy and precision, in the context of topics such as accuracy, precision, and significant figures. It involves comparing the instrument's readings to a known standard to identify and correct any deviations or errors.
Kilo-: Kilo- is a metric prefix that denotes a factor of one thousand (1,000 or $10^3$). This prefix is commonly used in various scientific and mathematical contexts to represent large quantities, making it easier to communicate measurements and calculations. Understanding kilo- is crucial in disciplines like physics, where accurate measurements and unit conversions are essential for precision and clarity.
Kilogram: A kilogram is the base unit of mass in the International System of Units (SI). It is defined by taking the fixed numerical value of the Planck constant, $h$, to be $6.62607015 \times 10^{-34}$ when expressed in units of $\text{kg} \cdot \text{m}^2 / \text{s}$.
Kilogram: The kilogram is the base unit of mass in the International System of Units (SI). It is a fundamental physical quantity that is used to measure the amount of matter in an object, and it serves as the foundation for understanding concepts related to physical quantities, accuracy, precision, and Newton's laws of motion.
Least Count: Least count refers to the smallest increment or measurement that can be accurately read on an instrument. This concept is crucial in determining the precision of measurements, as it directly impacts how accurately a value can be represented. The least count is influenced by the instrument's design and can help in evaluating both the accuracy and precision of the measurements taken.
Mean: The mean is the average value of a set of numbers, calculated by adding all the values together and then dividing by the total number of values. It serves as a central tendency measure, offering insight into the overall behavior of a dataset. Understanding the mean is essential for evaluating accuracy and precision, as it helps identify trends and anomalies in measurements.
Measurement Uncertainty: Measurement uncertainty is the range of values within which the true value of a measurement is expected to lie. It represents the degree of reliability or precision associated with a measurement, accounting for factors that can introduce errors or variability in the measurement process.
Median: The median is the middle value in a set of ordered data points. It is a measure of central tendency that divides a distribution into two equal halves, with half the values above the median and half below. The median is particularly useful for describing the typical or central value in a dataset, especially when the distribution is skewed or contains outliers.
Method of adding percents: The method of adding percents involves combining percentage values, ensuring that the base or reference values are consistent. It is a common procedure in scientific measurements to determine cumulative uncertainties or other aggregate percentage-related calculations.
Micrometer: A micrometer is a precise measuring instrument used to measure small distances or thicknesses, typically in the order of one millionth of a meter (1 µm). It is essential for ensuring accuracy and precision in various scientific and engineering applications, as it allows for measurements that require high levels of detail. Understanding micrometers connects to the concepts of accuracy, precision, and significant figures, as it involves careful calibration and the ability to record measurements with a specific number of meaningful digits.
Milli-: The prefix 'milli-' is a metric system prefix that denotes one-thousandth (1/1000) of the base unit. It is commonly used to express very small quantities or measurements in various scientific and engineering contexts.
Nano-: The prefix 'nano-' is derived from the Greek word 'nanos' meaning 'dwarf'. In the context of physics and scientific measurements, the prefix 'nano-' is used to denote one-billionth (1/1,000,000,000) of a particular unit. This prefix is commonly used to describe extremely small quantities or dimensions at the atomic and molecular scale.
Newton’s second law of motion: Newton’s second law of motion states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. Mathematically, it is represented as $F = ma$, where $F$ is the net force, $m$ is the mass, and $a$ is the acceleration.
Percent uncertainty: Percent uncertainty is the ratio of the uncertainty of a measurement to the measured value, multiplied by 100 to express it as a percentage. It provides a way to gauge the relative size of the uncertainty in relation to the measurement itself.
Percent Uncertainty: Percent uncertainty is a measure of the reliability and accuracy of a measurement or calculation. It quantifies the potential error or range of values around a reported result, expressed as a percentage of the measured value.
Precision: Precision is the degree to which repeated measurements under unchanged conditions show the same results. It indicates the consistency and repeatability of measurements.
Precision: Precision refers to the consistency and repeatability of measurements, indicating how closely related a series of measurements are to one another. High precision means that repeated measurements yield similar results, regardless of whether those results are close to the true value. This concept is crucial for evaluating physical quantities and units, understanding significant figures, making approximations, and interpreting null measurements.
Random Error: Random error is the unpredictable variation in a measurement that occurs due to the limitations of the measurement instrument or process. It is a type of measurement error that cannot be eliminated, but can be reduced through repeated measurements and statistical analysis.
Relative Uncertainty: Relative uncertainty is a measure of the precision of a measurement, expressed as a fraction or percentage of the measured value. It provides information about the reliability and variability of a measurement, which is crucial in understanding the accuracy and precision of experimental data.
Repeatability: Repeatability is the ability of a measurement or experiment to be consistently reproduced under the same conditions. It is a crucial aspect of accuracy and precision, ensuring that results can be reliably replicated and trusted.
Reproducibility: Reproducibility is the ability to consistently obtain the same or similar results when an experiment or measurement is repeated under the same conditions. It is a fundamental principle in science, ensuring the reliability and validity of research findings.
Reproducibility is closely tied to the concepts of accuracy, precision, and significant figures, as it speaks to the consistency and reliability of measurements and experimental outcomes.
Rounding Rules: Rounding rules are a set of guidelines used to determine how to properly round numerical values to a specified number of significant figures or decimal places. These rules ensure consistency and accuracy when expressing measurements or calculations in the context of scientific and mathematical applications.
Scientific Notation: Scientific notation is a way of expressing very large or very small numbers in a compact and standardized format. It involves representing a number as a product of a decimal value between 1 and 10, and a power of 10. This method is particularly useful for working with measurements and calculations that involve numbers with many digits.
Second: The second is the base unit of time in the International System of Units (SI). It is the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom. The second is a fundamental physical quantity that is essential for understanding and measuring various physical phenomena across multiple fields, including physics, chemistry, and engineering.
Significant Figure Rules: Significant figure rules are a set of guidelines used to determine the appropriate number of significant figures to report in a measurement or calculation. These rules help ensure the accurate representation of the precision and uncertainty associated with a given value.
Significant figures: Significant figures are the digits in a number that carry meaningful information about its precision. They include all non-zero digits, any zeros between significant digits, and trailing zeros in the decimal portion.
Significant Figures: Significant figures are the digits in a number that carry meaningful information about its precision. They include all non-zero digits, any zeros between significant digits, and trailing zeros in the decimal part. Understanding significant figures is essential when dealing with physical quantities, as it ensures that measurements reflect the accuracy and precision of data collected.
Standard Deviation: Standard deviation is a statistical measure that quantifies the amount of variation or dispersion of a set of data values around the mean or average of the data set. It provides a way to assess the spread or variability of a distribution, which is crucial for understanding the accuracy and precision of measurements or observations.
Systematic Error: Systematic error is a type of measurement error that occurs due to flaws or biases in the measurement process, leading to consistent deviations from the true value. Unlike random errors, systematic errors are reproducible and can be identified and corrected, making them an important consideration in the context of accuracy, precision, and significant figures.
Uncertainty: Uncertainty quantifies the doubt about a measurement's accuracy. It indicates the range within which the true value is expected to lie.
Uncertainty: Uncertainty is the lack of exact knowledge or confidence about a measurement, observation, or outcome. It is a fundamental concept in physics that acknowledges the inherent limitations in our ability to precisely determine or predict physical quantities.
Vernier caliper: A vernier caliper is a precise measuring instrument used to measure internal and external dimensions, as well as depths, with high accuracy. It features a main scale and a sliding vernier scale that enables users to read measurements more finely than standard rulers or tape measures, making it essential for tasks requiring both accuracy and precision. Understanding how to read a vernier caliper connects directly to concepts of measurement reliability, where accuracy relates to how close a measurement is to the true value, while precision refers to the consistency of repeated measurements.