Experimental Methods and Data Analysis

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Questions and Answers

What is a crucial reason for generating experimental data?

  • It simplifies the process of data collection.
  • It is easy to create visual representations.
  • It lacks any practical applications.
  • It helps in understanding measurements and predicting outcomes. (correct)

Which of the following is NOT a basic rule for calculation accuracy?

  • Match the number of decimal places in addition or subtraction to the least accurate number.
  • Identify the value with the least significant figures when dividing.
  • Use the least number of significant figures when multiplying.
  • State the final answer in whole numbers. (correct)

What should all tables present when displaying data?

  • Units of measurement and clear structure. (correct)
  • Graphs corresponding to each column.
  • Color-coded values for ease of interpretation.
  • Aesthetically pleasing designs.

What is a common flaw found in data graphs?

<p>Inconsistent scales. (B)</p> Signup and view all the answers

What can be fitted through any set of data points?

<p>A curve. (A)</p> Signup and view all the answers

What is the main focus of data analysis techniques discussed?

<p>Understanding and filtering data effectively. (B)</p> Signup and view all the answers

Which term refers to the methods used to adjust a raw dataset for accuracy?

<p>Calibration. (C)</p> Signup and view all the answers

What graphical representation is considered most prone to errors?

<p>Three-dimensional bar charts. (A)</p> Signup and view all the answers

What is one disadvantage of using higher-order polynomials for curve fitting?

<p>They can overfit the data. (B)</p> Signup and view all the answers

What is a common method to fit generic functions in MATLAB?

<p>fminsearch (D)</p> Signup and view all the answers

What do residuals represent in curve fitting?

<p>The difference between the model fit and observed values. (B)</p> Signup and view all the answers

Which of the following statements about R-squared is true?

<p>A higher R-squared generally indicates a better fit. (D)</p> Signup and view all the answers

In the context of calibration, what is the dependent variable?

<p>Desired quantity. (B)</p> Signup and view all the answers

What might plotting residuals after a curve fit provide?

<p>Insight into the degree of fit. (A)</p> Signup and view all the answers

What does the function polyfit in MATLAB calculate?

<p>Polynomial fits using least-squares algorithms. (B)</p> Signup and view all the answers

Which statement about the fifth-order polynomial fit is correct?

<p>It perfectly fits the data points. (D)</p> Signup and view all the answers

What does the term 'detrending' refer to?

<p>Removing trends to analyze the remaining components. (D)</p> Signup and view all the answers

How is the total sum of squares (SST) defined?

<p>The summed square deviations of actual values from their mean. (D)</p> Signup and view all the answers

What parameter does the first raw moment represent?

<p>Mean (D)</p> Signup and view all the answers

Which moment corresponds to the variance in statistics?

<p>Second central moment (D)</p> Signup and view all the answers

What does skewness measure in a distribution?

<p>The alignment of data relative to the mean (D)</p> Signup and view all the answers

How is kurtosis commonly defined in terms of a normal distribution?

<p>It has a value of zero for a normal distribution (D)</p> Signup and view all the answers

What is the aim of the method of least squares in curve fitting?

<p>To minimize the sum of squared distances (B)</p> Signup and view all the answers

What is the general expression for calculating the third standardised moment (skewness)?

<p>$\frac{1}{N \sigma^3} \sum (X_i - \bar{X})^3$ (B)</p> Signup and view all the answers

Which of the following is not a type of moment mentioned?

<p>Cumulative moments (C)</p> Signup and view all the answers

In the context of data distribution, what does a negative skew indicate?

<p>A longer tail on the left side of the distribution (B)</p> Signup and view all the answers

What does the fourth standardised moment (kurtosis) measure?

<p>The sharpness of the peak of the distribution (B)</p> Signup and view all the answers

What mathematical approach is used to find the minimum in the least squares method?

<p>Partial derivatives (A)</p> Signup and view all the answers

When standardizing moments, which parameter is used to normalize the moment?

<p>Standard deviation (D)</p> Signup and view all the answers

Which statement regarding higher-order polynomials in curve fitting could be seen as a disadvantage?

<p>They can lead to overfitting (A)</p> Signup and view all the answers

What is the formula for the second raw moment?

<p>Mean squared value of data (C)</p> Signup and view all the answers

What does a positive kurtosis indicate about a distribution?

<p>It has a sharper peak compared to a normal distribution (C)</p> Signup and view all the answers

What is the outcome when calculating $y = 2.5 + 2.58$ using 5 divided by 3?

<p>6.7 (D)</p> Signup and view all the answers

How does increasing the number of samples affect the error of the mean in averaging?

<p>It reduces the error. (C)</p> Signup and view all the answers

What is the primary assumption in ensemble averaging?

<p>The only difference between measurements is noise. (C)</p> Signup and view all the answers

What does Root Mean Square (RMS) represent?

<p>A measure of the power contained in an oscillating signal. (B)</p> Signup and view all the answers

What is a key characteristic of a moving average?

<p>It introduces a delay in the measurement. (C)</p> Signup and view all the answers

In calculating the median, what must be done with the samples prior to determining the median value?

<p>They must be sorted in ascending order. (C)</p> Signup and view all the answers

What is decimation in the context of data analysis?

<p>Calculating a mean for each block of samples. (A)</p> Signup and view all the answers

What effect does noise have on the results of averaging?

<p>It can distort the average value. (C)</p> Signup and view all the answers

Which of the following defines the moving median filter?

<p>It is similar to a moving average but applies a different calculation. (C)</p> Signup and view all the answers

What is the implication of averaging a large number of terms?

<p>It can result in a better approximation of the mean. (A)</p> Signup and view all the answers

In the context of significant figures, what determines the precision of the result in a calculation?

<p>The least precise measurement used in the calculation. (C)</p> Signup and view all the answers

What mathematical operation is essential for 'moving averaging' calculations?

<p>Averaging and convolution. (B)</p> Signup and view all the answers

For oscillating signals, which equation represents a sinusoidal function in relation to the power grid?

<p>$X_t = A sin(ωt)$ (C)</p> Signup and view all the answers

What is the primary objective of applying ensemble averaging?

<p>To minimize the effect of random noise. (C)</p> Signup and view all the answers

Flashcards

Data Presentation

Graphs and tables are common methods for presenting experimental data. Units and clear structure should be used in tables. Graphs should also be clear and easy to read.

Data Analysis

Methods for understanding collected data; includes calculating averages, applying filters, fitting curves, and calibrating data sets.

Curve Fitting

Drawing a line or a curve through plotted data points to represent a relationship that best describes the data.

Significant Figures

The number of digits that contribute to the accuracy of a measurement or calculation; multiplication and division use the fewest significant figures in the calculation and addition and subtraction use the fewest decimal places.

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Calibration

Process of adjusting a measurement instrument or raw data to a known, standard value.

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Experimental Data

Measurements and observations collected from experiments.

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Functional Relationship

A predictable relationship between measurements and their outcomes that can be used for predicting and designing.

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Bad Graphs

Graphs that are poorly designed or presented and do not effectively convey the data.

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Significance Arithmetic Rule 1

When multiplying measured values, the result should have the same number of significant figures as the least precise measurement.

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Significance Arithmetic Rule 2

When adding measured values, the result should have the same precision (number of decimal places) as the least precise measurement.

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Arithmetic Mean

The sum of all values divided by the number of values.

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Time-Averaged Value

The average value of a signal over a period of time.

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Ensemble Averaging

Averaging the results of multiple independent measurements to reduce noise.

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Moving Averaging

Calculate the average of a subset of recent data points.

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Moving Median

Calculates the middle value of a subset of recent data points.

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Decimation

Reducing the amount of data by averaging over blocks of measurements.

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Root Mean Square (RMS)

A measure of the power of a fluctuating signal, calculated by squaring the values, taking the average, and then taking the square root.

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Median

The middle value when data is sorted.

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Mean Absolute Deviation

Average of the absolute deviations from the mean.

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Noise Reduction

Averaging over more samples reduces the errors from noise

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Oscillating signal

Signal with periodic fluctuations around a certain value

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RMS Value

Effective value of an alternating current, or a rapidly fluctuating signal. This value is representative of the power

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Higher-order Polynomials

Polynomials with a degree greater than 2, e.g., cubic (degree 3) or quartic (degree 4).

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Polyfit Function

A function in MATLAB that uses least-squares to fit data with polynomials.

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Fitting Exponential Trends

Using a mathematical function of the form y = a*e^(bx) + c to represent data.

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Fminsearch Function

A MATLAB function for finding the minimum of a function, useful for fitting any generic function to data.

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Residuals in Curve Fitting

The differences between the actual data points and the values predicted by the fitted curve.

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R-squared Value

A statistical measure (between 0 and 1) that indicates how well the curve fits the data, with 1 being a perfect fit.

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Total Sum of Squares (SST)

The sum of the squared deviations of each data point from the mean value.

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Residual Sum of Squares (SSR)

The sum of the squared differences between the actual data points and the values predicted by the curve.

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Detrending Data

Removing the underlying trend from data, leaving only the fluctuations.

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What are Moments in statistics?

Moments are statistical parameters that describe the shape of data distributions. They help us understand the central tendency, spread, and asymmetry of our data.

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What is a Raw Moment?

A raw moment measures how the data points are spread around zero. The first raw moment is the mean.

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What is a Central Moment?

A central moment measures how data points are spread around the mean. The second central moment is the variance.

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What is a Standardised Moment?

A standardised moment measures the spread of data in terms of standard deviations. This allows us to compare data from different populations.

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What is Skewness?

Skewness measures the asymmetry of a distribution. It tells us whether the tail of the distribution is longer on the positive or negative side of the mean.

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Negative Skew

A distribution with a longer tail on the negative side of the mean.

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Positive Skew

A distribution with a longer tail on the positive side of the mean.

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What is Kurtosis?

Kurtosis measures the 'peakedness' or 'tailedness' of a distribution. It tells us how concentrated the data is around the mean.

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Negative Kurtosis

A distribution with a flatter peak and broader tails than a normal distribution.

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Positive Kurtosis

A distribution with a sharper peak and narrower tails than a normal distribution.

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What is Curve Fitting?

Curve fitting is the process of finding a mathematical function that best represents the relationship between data points.

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What is Linear Least Squares?

A method of curve fitting where we find the line that minimizes the sum of the squared distances between the data points and the line.

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How does Least Squares Work?

Least squares calculates the sum of the squared distances between the data points and the fitted line, then uses calculus to minimize this sum.

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What is a Polynomial?

A polynomial is a mathematical expression with terms containing variables raised to non-negative integer powers.

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What are Higher-Order Polynomials?

Polynomials with a degree greater than 2, e.g. cubic (degree 3), quartic (degree 4).

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Study Notes

Experimental Methods

  • The lecture covers experimental methods, including data analysis and presentation
  • Contact information for lecturers is provided: Dr. Mark Quinn ([email protected]) and Dr. Andrew Kennaugh ([email protected])

Data Analysis and Presentation

  • The lecture covers standard procedures for data presentation
  • Basic data analysis techniques and filters are discussed
  • Curve fitting algorithms are part of the lecture
  • Calibration will also be discussed

Why Do We Care?

  • Experimental data is useless without methods to understand it
  • Functional relationships between measurements and outputs are useful for prediction and design
  • Calculating averages and producing effective graphs are essential

Presentation

  • Graphs and tables are the most common methods to present data
  • Tables should include units and a clear layout
  • Graphs are often produced poorly

Calculation Accuracy

  • Two basic rules are presented for calculations:
    • When multiplying/dividing, use the value with the fewest significant figures
    • When addition/subtracting, use the least precise decimal place

Averaging

  • Ordinarily, the arithmetic mean is used to calculate the average
  • When applied to time-history data, the result is the time-averaged value
  • The approximation of the mean becomes more accurate with more data points
  • A static sample with noise is given as an example
  • Reducing noise by repeated measurements (ensemble averaging) is also discussed.

Averaging Methods (Specific examples given)

  • Averaged response of 1000 experiments is an example
  • Signal noise decreases by about 1/√N, when N is the number of samples. This gives diminishing returns, so very important but not ideal in the long run.

Ensemble Averaging

  • Repeated measurements of the same experiment can be averaged, assuming major differences in experimental data are just noise.
  • Importance of aligning all experiments to a common start point (aligning to t0)
  • The ensemble average of 100 measurements are provided

Moving Averaging

  • A sub-sample of the measurement is taken
  • The mean of these (L) samples is calculated
  • Resulting data is smoothed, but it does introduce an unavoidable time lag

Moving Median

  • A median filter offers better results than moving average filter, particularly in situations with sharp edges.

Decimation

  • Oversampling in time can be reduced by decimation
  • Average values for blocks of samples is calculated
  • Simple downsampling is possible

Oscillating Signals

  • Real-world examples like the UK power grid, which oscillates according to a sine function X(t) = Asin(wt), are given
  • Mean of this signal is addressed

Root Mean Square (RMS)

  • The time average of fluctuations, RMS, provides a measure for power of oscillations
  • RMS is presented as a metric for understanding power fluctuations contained within an oscillating signal or waveform
  • The RMS voltage of an example signal is used to demonstrate the practical application of RMS

Curve Fitting

  • Various methods and algorithms are available
  • Linear least-squares is presented as one of them
  • Using higher-order polynomials has drawbacks; there are less disadvantages for lower-order fits and exponential functions
  • There is a Matlab function called polyfit to fit polynomials using least-squares
  • This applies to any length polynomial data type
  • Alternative methods for curve fitting in Matlab

Residuals

  • Residuals represent the difference between an estimated value of ŷ₁ and an actual value y₁
  • This is a measure for error and uncertainty in the ability of a curve to predict (or fit) data points
  • R-squared is a common measure for residuals for fitting curves to data

Detrending

  • A signal can be broken down as a steady component plus an oscillating component
  • This is an essential technique for understanding and separating the underlying trends from noisy data in a set of graphs
  • Calculations are given for finding the mean and fluctuating temperatures from a set of examples using data.

Calibration

  • Generating a function
  • Example given is a pressure transducer

Questions

  • What is R-squared and what does it tell you?
  • What is t₀ in the context of ensemble averaging?

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