Error Analysis and Propagation
In experimental physics, no measurement is perfectly accurate. Error analysis is the systematic study of uncertainties to quantify the reliability of experimental data. Understanding how these errors propagate through calculations is essential for reporting valid scientific results.
1. Representation of Errors
If a physical quantity is measured as \(x\) with an uncertainty \(\Delta x\), we define the following types of error:
- Absolute Error: The magnitude of the uncertainty, denoted as \(\Delta x\).
- Relative Error: The ratio of absolute error to the measured value, \(\frac{\Delta x}{x}\).
- Percentage Error: \(\frac{\Delta x}{x} \times 100\%\).
2. Propagation of Errors
When calculating a result derived from multiple measured quantities, errors accumulate according to algebraic rules.
A. Sum and Difference
For \( Z = A + B \) or \( Z = A - B \), the absolute errors add up:
\[ \Delta Z = \Delta A + \Delta B \]
B. Product and Quotient
For \( Z = A \cdot B \) or \( Z = \frac{A}{B} \), the relative (fractional) errors add up:
\[ \frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B} \]
C. Power Functions
For a quantity raised to a power, \( Z = A^n \), the relative error is multiplied by the power \(n\):
\[ \frac{\Delta Z}{Z} = n \cdot \frac{\Delta A}{A} \]
3. General Formula
Combining these rules, if a quantity \(Z\) depends on \(A\), \(B\), and \(C\) as \( Z = \frac{A^p B^q}{C^r} \), the maximum percentage error is:
\[ \% \text{Error in } Z = p(\%A) + q(\%B) + r(\%C) \]
Derivation of Error Formulas
The rules for error propagation are not arbitrary; they are derived directly from differential calculus. By treating errors as small changes (differentials) in a function, we can mathematically prove how uncertainties combine in sums, products, and powers.
1. General Principle
If a quantity \(Z\) depends on measured values \(A\) and \(B\), the error \(\Delta Z\) is derived from the total differential \(dZ\): \[ dZ = \frac{\partial Z}{\partial A} dA + \frac{\partial Z}{\partial B} dB \] For maximum error, we sum the absolute values of the individual contributions: \[ \Delta Z \approx \left| \frac{\partial Z}{\partial A} \Delta A \right| + \left| \frac{\partial Z}{\partial B} \Delta B \right| \]
2. Derivation for Sums and Differences
Let \(Z = A + B\). Differentiating gives \(dZ = dA + dB\).
Let \(Z = A - B\). Differentiating gives \(dZ = dA - dB\).
Since errors can be positive or negative, the worst-case scenario occurs when they add up. Thus, for both cases:
\[ \Delta Z = \Delta A + \Delta B \]
3. Derivation for Products and Quotients
We use logarithmic differentiation for products. Let \(Z = A \cdot B\).
Taking logs: \(\ln Z = \ln A + \ln B\).
Differentiating:
\[ \frac{dZ}{Z} = \frac{dA}{A} + \frac{dB}{B} \]
Similarly for quotient \(Z = A/B\), we get \(\ln Z = \ln A - \ln B\), leading to \(\frac{dZ}{Z} = \frac{dA}{A} - \frac{dB}{B}\). Converting to maximum error form:
\[ \frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B} \]
4. Derivation for Power Rule
Let \(Z = A^n\). Taking logs gives \(\ln Z = n \ln A\).
Differentiating both sides: \[ \frac{dZ}{Z} = n \frac{dA}{A} \]
This proves that the relative error is multiplied by the power \(n\):
\[ \frac{\Delta Z}{Z} = n \frac{\Delta A}{A} \]
Derivation of Error \(\Delta f\) in Orthonormal Coordinates
When a physical quantity \(f\) depends on two independent variables \(R_1\) and \(R_2\) measured along orthonormal directions, the error propagation follows the rules of vector calculus. Here is the derivation of the total uncertainty \(\Delta f\).
1. The Total Differential
Let \(f(R_1, R_2)\) be the distribution function. We approximate the change in \(f\) due to small changes in \(R\) using the first-order Taylor expansion (Total Differential):
\[ df = \frac{\partial f}{\partial R_1} dR_1 + \frac{\partial f}{\partial R_2} dR_2 \]
Here, \(\frac{\partial f}{\partial R_i}\) represents the sensitivity of the function to changes in the \(i\)-th variable.
2. Orthonormality and Independence
Since \(R_1\) and \(R_2\) are along orthonormal directions, their variations are statistically independent. This means the covariance term \(\langle dR_1 dR_2 \rangle\) is zero.
To find the standard uncertainty (Root Mean Square error), we square the differential and take the expectation value:
\[ (\Delta f)^2 = \left( \frac{\partial f}{\partial R_1} \Delta R_1 \right)^2 + \left( \frac{\partial f}{\partial R_2} \Delta R_2 \right)^2 \]
3. Final Result
The final value of the function, including the central value \(f_{\text{central}}\) and the calculated uncertainty \(\Delta f\), is expressed as:
\[ f = f(R_1, R_2) \pm \sqrt{ \left( \frac{\partial f}{\partial R_1} \Delta R_1 \right)^2 + \left( \frac{\partial f}{\partial R_2} \Delta R_2 \right)^2 } \]
Statistical Analysis: Mean, Variance & Standard Error
In experimental physics, repeating a measurement \(n\) times allows us to minimize random errors. This post derives the formulas for the mean, the standard deviation of the sample, and the standard error of the mean to determine the final reported value.
1. Sample Mean (\(\bar{R}\))
For a set of observations \(R_1, R_2, \dots, R_n\), the arithmetic mean is the best estimate of the true value:
\[ \bar{R} = \frac{1}{n} \sum_{i=1}^{n} R_i \]
2. Variance and Standard Deviation of Sample
The spread of the individual data points is measured by the variance (\(\sigma^2\)) and standard deviation (\(\sigma\)). We use \(n-1\) for an unbiased estimate:
\[ \sigma = \sqrt{ \frac{1}{n-1} \sum_{i=1}^{n} (R_i - \bar{R})^2 } \]
3. Uncertainty in the Mean (Standard Error)
The mean itself is a random variable. Its variance is \(n\) times smaller than the variance of the individual data points.
Variance of Mean: \[ \text{Var}(\bar{R}) = \frac{\sigma^2}{n} \]
Standard Deviation of Mean: \[ \sigma_{\text{mean}} = \frac{\sigma}{\sqrt{n}} \]
4. Final Reported Value
The final result is expressed as the Mean \(\pm\) Standard Error:
\[ R = \bar{R} \pm \frac{\sigma}{\sqrt{n}} \]
Precision vs. Accuracy
In scientific measurements, "precision" and "accuracy" are not synonyms. Accuracy describes how close a measurement is to the true value, while precision describes how consistent repeated measurements are with each other. Understanding the distinction is vital for error analysis.
1. Defining the Terms
Accuracy
Accuracy is the degree of closeness of a measured value to the actual (true) value. It indicates the absence of Systematic Errors.
\[ \text{Accuracy} \propto \frac{1}{| \text{True Value} - \text{Measured Value} |} \]
Precision
Precision is the degree to which repeated measurements under unchanged conditions show the same result. It relates to the resolution of the instrument and Random Errors.
\[ \text{Precision} \propto \frac{1}{\text{Standard Deviation } (\sigma)} \]
2. The Target Analogy
- High Precision, Low Accuracy: Measurements are consistent (tightly clustered) but incorrect (far from the true value). This usually indicates a calibration issue.
- Low Precision, High Accuracy: Measurements are scattered, but their average is close to the true value. This indicates high random noise but good calibration.
Proof of Central Limit Theorem
The Central Limit Theorem (CLT) asserts that the sum of a large number of independent, identically distributed variables approaches a normal distribution. The most rigorous way to prove this is using Moment Generating Functions (MGFs) and Taylor Series expansions.
1. Setup and Assumptions
Let \( X_1, X_2, \dots, X_n \) be a sequence of independent and identically distributed (i.i.d) random variables with:
- Mean: \( E[X_i] = \mu \)
- Variance: \( Var(X_i) = \sigma^2 \)
- Moment Generating Function: \( M_X(t) \) exists.
We define the sum \( S_n = \sum_{i=1}^n X_i \) and consider the standardized variable \( Z_n \):
\[ Z_n = \frac{S_n - n\mu}{\sigma\sqrt{n}} \]
Goal: We want to show that as \( n \to \infty \), the MGF of \( Z_n \) approaches the MGF of the Standard Normal Distribution \( N(0,1) \), which is \( e^{t^2/2} \).
2. Defining a Standardized Variable
To simplify the math, let's define a new variable \( Y_i \) which is the standardized version of a single observation:
\[ Y_i = \frac{X_i - \mu}{\sigma} \]
The properties of \( Y_i \) are:
- \( E[Y_i] = 0 \)
- \( Var(Y_i) = E[Y_i^2] - (E[Y_i])^2 = 1 \)
We can now rewrite \( Z_n \) in terms of \( Y_i \):
\[ Z_n = \frac{1}{\sqrt{n}} \sum_{i=1}^n Y_i \]
3. The Moment Generating Function (MGF)
The MGF of the sum \( Z_n \) is defined as \( M_{Z_n}(t) = E[e^{t Z_n}] \). Substituting our expression for \( Z_n \):
\[ M_{Z_n}(t) = E \left[ \exp \left( t \frac{1}{\sqrt{n}} \sum_{i=1}^n Y_i \right) \right] \]
Since the \( Y_i \) are independent, the expected value of the product is the product of the expected values:
\[ M_{Z_n}(t) = \left( E \left[ e^{\frac{t}{\sqrt{n}} Y_1} \right] \right)^n = \left( M_Y \left( \frac{t}{\sqrt{n}} \right) \right)^n \]
4. Taylor Series Expansion
We expand the MGF \( M_Y(x) \) about zero using the Taylor series. Recall that \( M_Y(0) = 1 \), \( M_Y'(0) = E[Y] = 0 \), and \( M_Y''(0) = E[Y^2] = 1 \).
\[ M_Y(x) = 1 + x E[Y] + \frac{x^2}{2!} E[Y^2] + O(x^3) \]
Substituting \( x = \frac{t}{\sqrt{n}} \):
\[ M_Y \left( \frac{t}{\sqrt{n}} \right) = 1 + \frac{t}{\sqrt{n}}(0) + \frac{t^2}{2n}(1) + O \left( \frac{1}{n^{3/2}} \right) \]
\[ M_Y \left( \frac{t}{\sqrt{n}} \right) \approx 1 + \frac{t^2}{2n} \]
5. The Limit
Now we take the limit of the MGF for \( Z_n \) as \( n \to \infty \):
\[ \lim_{n \to \infty} M_{Z_n}(t) = \lim_{n \to \infty} \left( 1 + \frac{t^2}{2n} \right)^n \]
Using the standard calculus limit identity \( \lim_{n \to \infty} (1 + \frac{x}{n})^n = e^x \), where \( x = t^2/2 \):
\[ M_{Z_n}(t) \to e^{t^2/2} \]
Since \( e^{t^2/2} \) is the Moment Generating Function of the standard normal distribution, by the uniqueness theorem of MGFs, \( Z_n \) must be distributed as \( N(0,1) \).
Q.E.D.