Counting Statistics and Error Prediction in Radiation Measurements

Foundations of Counting Statistics and Error Prediction

• Radiation events are inherently stochastic in nature, meaning they occur randomly over time. Because of this, it is necessary to analyze radiation measurement data using statistical methods.

• The study of counting statistics and error prediction is based on the book Flynn Naw, tablet three.

• Error prediction is derived from standard relations used for error propagation.

• In any research or equipment operation measurement, reporting a single mean value is insufficient. Data should be reported as the mean plus or minus a scatter band (e.g., $x \bar{} \pm \sigma$).

Characterization of Experimental Data

• When measuring decays over a specific time interval (e.g., in seconds), data is collected as a series of counts.

• An example dataset of measured decays includes the following values: $8, 10, 3, 13, 9, 7, 12, 9, 6, 10, 10, 11, 7$.

• Three primary properties are used to characterize such a dataset:

  1. Experimental Mean ($x_e$): The average value of the measurements.

  2. Distribution Function ($f(x)$): Defined as the number of occurrences of a particular value $x$ divided by the total number of measurements ($n$).

  3. Sample Variance ($s^2$): A measure of the internal scatter within the recorded data.

  4. Standard Deviation ($\sigma$): The square root of the sample variance, defining the scatter band for the reported measurement.

Mathematical Framework for Data Properties

• Experimental Mean ($x_e$):

  • Calculated as the sum of all measurement values divided by the total number of measurements:     $x_e = \frac{\sum_{i=1}^n x_i}{n}$

• Distribution Function ($f(x)$):

  • Also known as the frequency distribution function or data distribution function.

  • It represents normalized occurrences. For a dataset of $n$ measurements, the sum of all distribution function values must equal one:     $\sum_{x=0}^{\infty} f(x) = 1$

  • The experimental mean can be calculated as the first moment of the distribution function:     $x_e = \sum_{x=0}^{\infty} x f(x)$

• Residuals ($d_i$):

  • The difference between an individual measurement and the experimental mean:     $d_i = x_i - x_e$

  • The sum of all residuals in a dataset is always zero:     $\sum_{i=1}^n d_i = 0$

• True Mean ($\bar{x}$):

  • The true mean represents the actual value if one were to measure every possible instance (e.g., inspecting all 10,000 fuel rods produced by a factory).

  • Because inspecting every item is expensive and time-consuming (total quality control), companies use sampling (e.g., inspecting 20 rods), which gives the experimental mean ($x_e$).

• True Deviation ($\epsilon_i$):

  • The difference between a measurement and the true mean:     $\epsilon_i = x_i - \bar{x}$

• Sample Variance ($s^2$):

  • Based on the true mean, variance is defined as:     $s^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2$

  • Since the true mean is often unknown, the sample variance is calculated using the experimental mean:     $s^2 = \frac{1}{n - 1} \sum_{i=1}^n (x_i - x_e)^2$

  • The use of $n-1$ instead of $n$ accounts for the fact that the experimental mean is an estimate. As $n$ becomes very large, the difference between the two definitions becomes negligible.

  • Sample variance measures internal scatter and does not depend significantly on the number of measurements $n$; the scatter band of a system remains relatively constant regardless of whether 20 or 40 measurements are taken.

Statistical Models for Trials and Success

• Statistical models are used to determine if measured sample variance matches predicted variance. Significant discrepancies indicate systematic errors, procedural issues, or system artifacts.

• Models are defined by three parameters:

  1. $n$: The number of trials.

  2. $p$: The probability of success in a single trial.

  3. $x$: The number of successful events.

Binomial Distribution

• Generally complex to solve for radiation measurements because $n$ is typically very large.

• It assumes probability $p$ is constant and does not change with time or the number of trials.

• Distribution Function:     $P(x) = \frac{n!}{x! (n - x)!} p^x (1 - p)^{n - x}$

• Mean ($\bar{x}$):     $\bar{x} = n \times p$

• Predicted Variance ($\sigma^2$):     $\sigma^2 = n \times p (1 - p) = \bar{x} (1 - p)$

• Example: Rolling a Die:

  • Rolling a die 10 times ($n = 10$).

  • Success defined as rolling a 3, 4, 5, or 6 ($p = \frac{4}{6} = 0.667$; non-success $1-p = \frac{2}{6} = 0.333$).

  • Mean success: $x_{bar} = 10 \times 0.667 = 6.67$.

  • Variance: $\sigma^2 = 10 \times 0.667 \times 0.333 = 2.22$.

  • Standard Deviation: $\sigma = \sqrt{2.22} = 1.49$.

Poisson Distribution

• Simplified model widely used for radiation measurements where success probability $p$ is very small.

• Often used when the mean value ($\bar{x}$) is less than 20.

• Distribution Function:     $P(x) = \frac{(np)^x e^{-np}}{x!} = \frac{\bar{x}^x e^{-\bar{x}}}{x!}$

• Mean ($\bar{x}$):     $\bar{x} = n \times p$

• Predicted Variance ($\sigma^2$):     $\sigma^2 = \bar{x}$

• Standard Deviation:     $\sigma = \sqrt{\bar{x}}$

• Example: Birthday Survey:

  • Survey 1,000 people ($n = 1,000$) to see if their birthday is today ($p = \frac{1}{365}$).

  • $\bar{x} = 1,000 \times \frac{1}{365} = 2.74$.

  • Calculated probabilities $P(x)$:

    • $P(0) = 0.064$

    • $P(1) = 0.177$

    • $P(2) = 0.242$

    • $P(3) = 0.221$

    • $P(4) = 0.152$

    • $P(5) = 0.083$

    • $P(6) = 0.038$

  • The Poisson distribution for small means is asymmetric.

Gaussian (Normal) Distribution

• A continuous function used when $p \ll 1$ and the mean $\bar{x} > 20$.

• Distribution Function:     $P(x) = \frac{1}{\sqrt{2 \pi \bar{x}}} e^{-\frac{(x - \bar{x})^2}{2 \bar{x}}}$

• Normalized Deviation ($t$):

  • Defined as the deviation divided by the standard deviation:     $t = \frac{x - \bar{x}}{\sigma}$

  • The Gaussian distribution in terms of $t$ is:     $G(t) = \sqrt{\frac{2}{\pi}} e^{-\frac{t^2}{2}}$

• Confidence Levels and Standard Deviations:

  • $0.674 \sigma$: 50% probability.

  • $1.00 \sigma$: 68.3% probability.

  • $1.64 \sigma$: 90% probability (often used in experiment design).

  • $1.96 \sigma$: 95% probability.

  • $3.00 \sigma$: 99.7% probability.

Practical Examples and Comparative Analysis

Example: Probability of 8 Heads in 12 Coin Tosses

• $n = 12$, $x = 8$, $p = 0.5$.

• Using Binomial Distribution:     $P(8) = \frac{12!}{8! 4!} (0.5)^8 (0.5)^4 = 0.121$

Example: Inspecting Corn for Insects

• A farmer pulls 200 ears of corn ($n = 200$). Probability of insects $p = 0.01$.

• Binomial Analysis:

  • Probability of exactly 3 ears having insects:     $P(3) = \frac{200!}{197! 3!} (0.01)^3 (0.99)^{197} = 0.181355$

  • Mean: $\bar{x} = 2$. Standard deviation: $\sigma = \sqrt{200 \times 0.01 \times 0.99} = 1.407$.

• Poisson Analysis:

  • $P(3) = \frac{2^3 e^{-2}}{3!} = 0.18045$

  • Mean: $\bar{x} = 2$. Standard deviation: $\sigma = \sqrt{2} = 1.414$.

Example: Radioactive Decay Over Time

• Given decay constant $\lambda = 1.3188 \times 10^{-7} \, \text{min}^{-1}$.

• Success defined as 120 decays ($x = 120$) in 1 minute ($\Delta t = 1 \, \text{min}$) from a sample of $10^9$ atoms ($n = 10^9$).

• Average success $\bar{x} = n \times p = 10^9 \times (1.3188 \times 10^{-7}) = 131.88$.

• Poisson probability: $P(120) = 0.02096$ (approx. 2%).

• Gaussian probability: $P(120) = 0.0203$.

Chi-Squared Value and Single Measurements

• Chi-Squared ($\chi^2$):

  • A method to compare predicted variance and experimental variance:     $\chi^2 = \frac{(n - 1)s^2}{\bar{x}}$

  • Alternatively expressed as:     $\chi^2 = \sum_{i=1}^n \frac{(x_i - \bar{x})^2}{\bar{x}}$

• Single Experimental Values:

  • If only one experimental value $x$ is available, the true mean and experimental mean are unknown.

  • In this common situation, the standard deviation is approximated as $\sqrt{x}$.

  • The result is reported as: $x \pm \sqrt{x}$.