An Incomplete Medical Physics Review

Statistics and biostatistics

mean = $\bar{x} = \sum{x_i} /n$

median = middle number in ordered data points

standard deviation = $\sigma = \sqrt{ \frac{\sum{(x_i-\mu)^2}}{N} }$ where $\mu$ is the mean and $x_i$ is each of $N$ data points 

coefficient of variation (cov) = $\sigma / \mu$ (relative standard deviation)

variance = $\sigma^2$ 

standard error: measure of accuracy of a measurement. Standard deviation of a sampling distribution.

Bayes' theorem: Probability of an event given prior knowledge of related conditions
$P(A\mid B)=\frac{P(B\mid A)\,P(A)}{P(B)}$
Probability of event A given that B is true depends on probability of A, of B, and of B given that A is true.

Binomial, Poisson, Gaussian distribution functions

Binomial: probability of getting exactly $x$ successes in $N$ trials given a single trial probability $p$
$P(x) = \frac{N!}{x!(N-x)!}p^x(1-p)^{N-x}$
$\bar{x} = pN$ and $\sigma = \sqrt{pN(1-p)}$ and for small p: $\approx \sqrt{pN} \approx \sqrt{\bar{x}}$

Useful when there is high counting efficiency.

Poisson statistics: probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant rate, e.g., radioactive decay
Probability of $n$ events given an average of $\lambda$ events: $P = \frac{\lambda^n\,e^{-\lambda}}{n!}$ 
Expected value = $\lambda$ and variance = $\lambda$

Approximate standard deviation for a single measurement: $\sigma \approx \sqrt{x}$
Fractional error (cov): $\sigma/x \approx 1/\sqrt{x}$
E.g. fractional error of 40,000 counts ~ 1/sqrt(40,000) = 1/200 = 5%

Useful for low detection efficiency and when $p$ or $n$ is not known directly.

Gaussian: normally distributed
$\text{PDF} = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}}$

Standard deviation: (σ) For a normal distribution, 68.3% of measurements are within 1 σ, 95.5% within 2 σ, 99.7 within 3 σ.
Full width half max:  $\text{FWHM} = 2.355 \sigma$

Poisson, binomial and Gaussian distributions become identical when $p$ is small and mean is large.

Confidence interval: Interval for the probability of containing the true mean. E.g., there is a 68.27% probability that the true mean is within 1 σ of a measurement and 95.45% probability it is within 2 σ.
95% confidence for measurement of 40,000 cts is 40,000 ± 1.96 σ = 40,000 ± 392 cts. Thus the 95% confidence interval is from 39,608 to 40,392 cts.

Probability that mean is within %	Interval around measurement
50	±0.674 σ
68.3	±1 σ
90	±1.64 σ
95	±1.96 σ
95.5	±2 σ
99	±2.58 σ
99.7	±3 σ

(Spatial) Resolution

Spatial resolution is often measured with a point source and in images exhibits a gaussian distribution of intensity: the point spread function (PSF). The value of the resolution is typically given as the FWHM of the PSF, where FWHM = 2.355σ.
Another way to display the response of a system to spatial frequency is to create a modulation transfer function (MTF) - essentially how well the system can recover the true intensity for a range of frequencies. An ideal system would have MTF = 1 for all frequencies, but it more typically falls off gradually as the frequency increases. The MTF is calculated as the Fourier transform of the PSF.  See plots below.

Modulation: $M = (I_{max} - I_{min}) / (I_{max} + I_{min})$ where $I$s are the min and max radiation intensities in the test pattern. If $I_{min} = I_{max}$ there is no contrast. If $I_{min}=0$ there is maximum contrast. 

$\text{MTF}(k) = M_{out}(k) / M_{in}(k)$ where $M_{out}$ is the modulation of the output (image)  and $M_{in}$ is the modulation of the original pattern for each frequency $k$.

Propagation of error

If the standard deviation of x is σ, the standard deviation of x/c is σ/c so long as c does not have random error.
Two numbers with error are added/subtracted: $x_1 + x_2 \rightarrow \sigma = \sqrt{\sigma_1^2 + \sigma_2^2}$
Other cases: ${\displaystyle s_{f}={\sqrt {\left({\frac {\partial f}{\partial x}}\right)^{2}s_{x}^{2}+\left({\frac {\partial f}{\partial y}}\right)^{2}s_{y}^{2}+\left({\frac {\partial f}{\partial z}}\right)^{2}s_{z}^{2}+\cdots }}}$

Random errors: unknown, unpredictable (stochastic, noise). Usually have a gaussian distribution. 

Systematic errors: incorrect measurements that are affect measurements in the same way each time. E.g., problems with the measuring instrument, Can be an offset error (e.g., instrument not zero when quantity is zero) or multiplicative/scale factor (e.g. instrument does not measure linearly when it should). 

Precision: How much the individual values deviate around the true value.

Accuracy: How close the measured value is to the true value.

Random errors limit precision but less so accuracy. Systematic errors might not affect precision but usually affect accuracy.

Linear regression

Model relationship of dependent variables to explanatory variables.  Parameters of the model are estimated from data, typically using the least squares method. 

Least squares: minimize the square difference between the data and the model estimation when varying input parameters.