Measurement Systems and Statistical Analysis

Overview of Measurement Systems

• Measurement System Components

  • Sensor: Needs calibration and is connected to a transducer.

  • Transducer: Converts sensed data into an electrical signal.

  • Signal Conditioning Stage

  • Includes filters and amplifiers.

  • Output Stage

  • Typically involves a data acquisition system for sampling data.

  • Feedback Control System

  • Incorporates a controller that receives signals from the output stage and feeds it back to the process being measured.

Measurement Uncertainty and Errors

• Importance of Errors and Uncertainties in Measurements

  • Aim to reduce errors through:

    • Improved experimental procedures.

    • Repeated measurements.

  • Error Definition: Difference between the measured value and the true value.

  • The true value is generally unknown; approximations can be derived from prior measurements or theoretical predictions.

Accuracy and Precision

• Definitions of Accuracy

  • Accuracy: The ability to indicate the true value exactly.

  • Absolute Error (ε): Defined as $ext{True Value} - ext{Indicated Value}$.

  • Relative Accuracy (a): Defined as $a = 1 - rac{ε}{ ext{True Value}}$.

  • Accuracy can only be determined when the true value is known, typically during calibration with a precise voltage source.

• Definitions of Precision

  • Precision: Ability to indicate a particular value across repeated independent measurements.

  • Precision Error: Measure of random variations during repeatability trials.

  • Possible for a system to be precise (consistent measurements) yet inaccurate (consistently wrong value).

Measuring and Estimating True Value

• Estimating True Values with Statistical Theory

  • Conduct multiple measurements (n times) of a variable (x).

  • Sample Set: ${x<em>1, x</em>2, \ldots, x_n}$; $n = ext{sample size}$.

  • Assume negligible bias error for estimation.

  • Sample Mean (\bar{x}): $\bar{x} = \frac{1}{n} \sum<em>{i=1}^{n} x</em>i$ as the most probable estimate of true value.

  • Uncertainty (u(x)): Gives a measure of the uncertainty (confidence interval) associated with the measurement.

  • Determining the Band of Uncertainty: $x_{true} \in (\bar{x} - u(x), \bar{x} + u(x))$.

Basic Statistical Analysis for Measurements

• Sample Variance (S²): $S² = \frac{1}{n} \sum<em>{i=1}^{N} (x</em>i - \bar{x})²$.

  • Can also be expressed as $S² = \text{Average of } x² - (\bar{x})²$.

• Root Mean Square (RMS): $RMS = \sqrt{\bar{x}²} = \sqrt{\frac{1}{n} \sum<em>{i=1}^{n} x</em>i²}$.

Variance and Confidence Interval

• Intuition about Variance and Confidence Interval

  • As variance increases, confidence interval increases.

  • As sample size (n) increases, uncertainty decreases.

  • A higher probability (p) results in a wider confidence interval for true value estimation.

  • Standard Deviation (σ): Defined as the square root of variance. Standard coefficient of variation (SCV) is defined as $SCV = \frac{s}{\bar{x}}$.

Data Distribution and Histogram

• Creating a Histogram

  • Find maximum ($x<em>{max}$) and minimum ($x</em>{min}$) values in the data set.

  • Define bin width ($Δx$) as $Δx = \frac{x<em>{max} - x</em>{min}}{k}$.

  • Normalized Relative Frequency ($N/N_{total}$): Measure of probability density function (PDF).

Transition from Histogram to Probability Density Function (PDF)

• Definition of PDF and its limits as n approaches infinity and $Δx$ approaches zero.

• Probability that ($x$) is within an interval given by the integral of the PDF over that interval.

Cumulative Distribution Function (CDF)

• Defined as $F(x<em>1) = \int</em>{-\infty}^{x<em>1} p(x)dx$, which indicates probability that $x$ is less than $x1$.

Parameters of the Probability Density Function (PDF)

• Mean/Expected Value: $E[x] = \int_{-\infty}^{\infty} x p(x)dx$.

• Median: Value that splits the PDF in half, defined through cumulative distribution.

• Mode: Value at which the PDF peaks (highest frequency).

Variance and Standard Deviation for Continuous Distributions

• Defined for continuous distribution as $σ² = \int_{-\infty}^{\infty} (x - μ)² p(x)dx$.

• Standard Deviation: $σ = \sqrt{σ²}$.

Random Variables and Probability Density Function

• Random Variable (X): Represents outcomes of measurements that can fluctuate.

  • Sample Space (S): Set of all possible values for the random variable.

  • Realizations of X observed in measurement are denoted by lowercase x.

• Probability Density Function (PDF) (p(x)): Models the uncertainty of the random variable through repeated measurements.

Discrete Distributions and Future Topics

• Introduction to discrete distributions, focusing on random variables within repeated measures.

  • Examples include Bernoulli trials modeled by binomial distribution for occurrence/non-occurrence measurements.

• Next class will focus on Bernoulli experiments and extend to binomial distribution as a modeling framework.