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Questions and Answers
When values are assigned to the nearest of a set of quantization values, separated by $ \Delta V $, the errors have this form of ______.
When values are assigned to the nearest of a set of quantization values, separated by $ \Delta V $, the errors have this form of ______.
distribution
If a number of values, such as voltage and current, that each have their own degree of uncertainty, are combined together, one must estimate the ______ of the final, calculated value.
If a number of values, such as voltage and current, that each have their own degree of uncertainty, are combined together, one must estimate the ______ of the final, calculated value.
uncertainty
For continuous variables, the most commonly used Probability Density Function form, also called a PDF, in engineering is the ______.
For continuous variables, the most commonly used Probability Density Function form, also called a PDF, in engineering is the ______.
gaussian
When there are many possible events that could occur but each one is very unlikely, a ______ type of distribution can be used to describe it.
When there are many possible events that could occur but each one is very unlikely, a ______ type of distribution can be used to describe it.
A measure of uncertainty used to quantify the 'width' of a Probability Distribution Function (PDF) is known as the ______.
A measure of uncertainty used to quantify the 'width' of a Probability Distribution Function (PDF) is known as the ______.
Arising due to the fact that a resistor is not at absolute zero, ______ causes free electrons to be in constant, random motion, even when zero current is flowing.
Arising due to the fact that a resistor is not at absolute zero, ______ causes free electrons to be in constant, random motion, even when zero current is flowing.
In a measurement, uncertainty related to how much variation there is when a series of measurements are taken indicates the likely ______.
In a measurement, uncertainty related to how much variation there is when a series of measurements are taken indicates the likely ______.
The difference between the analogue value and the digital value chosen is referred to as ______ or quantisation noise.
The difference between the analogue value and the digital value chosen is referred to as ______ or quantisation noise.
In the method of 'Least Squares' fitting the line found represents the most ______ set of parameters to match to the data.
In the method of 'Least Squares' fitting the line found represents the most ______ set of parameters to match to the data.
If using a photo-detector to measure a light level of photons, for the lowest fractional uncertainty, it is better to count ______ electrons as compared to only counting 10 or 100.
If using a photo-detector to measure a light level of photons, for the lowest fractional uncertainty, it is better to count ______ electrons as compared to only counting 10 or 100.
Flashcards
Quantum Noise
Quantum Noise
Error due to the fundamental, quantum nature of current flow. Arises because charge is carried by discrete electrons.
Thermal Noise
Thermal Noise
Noise arising from the thermal energy of a resistor, causing constant random motion of free electrons, even without applied voltage.
Electromagnetic Interference
Electromagnetic Interference
Errors caused by stray electromagnetic fields inducing currents in the measurement circuit, corrupting the measured value.
Quantization Error
Quantization Error
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Central Limit Theorem
Central Limit Theorem
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Frequency Distribution
Frequency Distribution
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Probability Density Function (PDF)
Probability Density Function (PDF)
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Mode
Mode
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Median
Median
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Standard Deviation
Standard Deviation
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Study Notes
Data Presentation, Uncertainty, Errors, and Statistics
- Data presentation and statistical analysis are critical in engineering analysis, especially when dealing with errors present in real-world scenarios.
- Focus is given to understanding errors, uncertainty, and statistical methods
Prerequisites
- Requires a basic knowledge of algebra
- Knowledge of functions from EA Topic 1.1
- Knowledge of differentiation from EA Topic 1.2
- Knowledge of integration from EA Topic 1.3
Learning Outcomes
- Present data with context to errors.
- Understand data errors and uncertainty.
- Describe error affecting measurements.
- Understand random variables.
- Understand variability of random variables,
- Estimate expected errors using data or uncertainty models.
- Understand probability density functions.
- Estimate uncertainty for data points or averages.
- Estimate uncertainty in summed or multiplied random variables.
- Choose error representations for data.
- Use fitting to estimate parameters from plotted points.
Introduction to Uncertainty in Data
- Considers uncertainty in resistance measurements as an example of error.
- Calculating resistance involves voltage and a current measurement.
- Uncertainty in resistance comes from current and voltage measurement uncertainty.
Sources of Errors and Uncertainty
- Sources of error include quantum noise, thermal noise, electromagnetic interference, and quantisation error for a quantity such as current.
- Quantum noise arises from current existing as flow of electrons.
- The number of electrons is integer value, not fraction, introduces error.
- Thermal noise is caused by material temperature and free electrons are in motion even without current.
- Electromagnetic interference occurs when stray fields induce currents in the circuit, corrupting measurements.
- Quantisation error exists when digital ammeters convert analogue values.
More Errors and Uncertainties
- Further errors are noise in the amplifier or voltage source variations might affect current.
- The central limit theorem explains a model can be used with uncertainties.
- Errors can be random, but systematic errors are also possible.
Describing Errors (I) – Frequency Distributions
- Measured value consists of true value and error
- Random error means error values are different for measurement and cannot be predicted.
- Random errors can undergo mathematical description or estimation.
Experiment Examples
- Experiment A has measurements of current closer to 1 mAmp than Experiment B's measurements.
- Experiment A exhibits less variation and more certainty
- Random variation measured quantitatively
Estimating Variation
- Spread/Range: the likely spread
- Inter-Quartile Range: The Inter-Quartile Range
- Variance and Standard Deviation: Variance and Standard Deviation
- Frequency distribution or data values used in calculations.
- Random data generated by MATLAB can be divided into bins to count frequency.
Types of Random Variables
- Continuous random variables can take any value in a range.
- Discrete random variables only take integer values .
Continuous Random Variables
- Discrete random variable examples are photon counts or circuit board faults.
- Counting the values which fit variable allows frequency distribution estimation
- Distribution graphed in a bar chart.
- Wider frequency implies greater uncertainty by occurring over wider range.
- Estimates of variable uncertainty through measuring frequency.
Describing Errors (II) – Probability Density Functions (PDFs)
- Histograms estimate uncertainty, but theoretical forms can be known.
- PDFs come from normalising the frequency distribution.
Continuous Random Variables and PDFs
- Divide each frequency count from histogram data by 𝑁∆ to normalise continuous variables
- 𝑁 is total data values
- ∆ is width of each bin
- Each rectangle represents probability to find variable in bin range
- PDF becomes one which reflects probabilities and its value.
Discrete Random Variables and PDFs
- Normalisation also gives probabilities.
- Divide frequency counts in discrete variables by N, total data values.
- The probability of particular value becomes the line height
- Sum of values are equal to one.
Properties of PDFs
- Properties of random variable deduced if PDF known.
- The mean of symmetrical PDF can be predicted by locating the peak position.
- Symmetrical PDF mean is same as other measures/averages like mode and meadian.
- Mode: The most-likely value to occur
- Median: Half the distribution higher, half lower the value.
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Description
Learn how to present and analyze data in engineering while accounting for errors. Understand different types of errors, uncertainty, and statistical methods. Explore random variables, probability density functions, and error estimation techniques for data and averages.
