Statistics are often employed in the field of physics, for example in the sub fields of:

• Quantum Mechanics
• Thermal Physics
• Condensed Matter
• BioPhysics
• GeoPhysics

An example of statistics in QM is expectation values. The expectation value (or average) of an observable “A” in a given quantum state is calculated as: \[ \langle A \rangle = \int \psi^*(x) A \psi(x) \, dx \] -The result represents the average result you would obtain if you took many measurements of A on identical quantum systems.

-There is another statistical element inherent to the equation in the variable ψ(x) which represents the probability amplitude.

A fundemental concept is the Maxwell-Boltzman distribution, which describes the distribution of speeds of particles in an ideal gas at a given temperature.

This can be shown graphically, it will show the distribution. The X-axis being speed in m/s, the Y-axis is the probability density or the likelihood of finding the particle. The blue line represents the function for the MB distribution.

In condensed matter physics, phonons are quantized collective excitations of atoms in a crystalline structure. These vibrations play a crucial role in determining the thermal, mechanical, and electrical properties of materials. Statistical mechanics are used to understand this.

Consider a one-dimensional lattice of atoms arranged in a regular pattern. Each atom can vibrate with a certain frequency, corresponding to the energy of the phonon. The energy levels of the phonons follow a statistical distribution determined by the Boltzmann factor:

\[ P(E) = \frac{1}{Z} e^{-\frac{E}{k_B T}} \]

A common example of the use of statistics in Biophysics is in the analysis of DNA sequencing data.

Let’s consider a scenario where we have obtained DNA sequencing data from a sample of individuals, and we want to analyze the frequency of different genetic variants (alleles) at a specific locus in the genome. We can do this using imagined data and creating a histogram.

A common example of the use of statistics in geophysics is in the analysis of earthquake magnitudes and frequencies.By analyzing earthquake data using statistics, geophysicists can characterize seismic activity, assess the likelihood of future earthquakes, and make informed decisions regarding earthquake preparedness and mitigation measures. This statistical analysis provides valuable insights into the seismic behavior of a region and contributes to our understanding of earthquake processes and their potential impacts on society and infrastructure.

We can demonstrate this with the use of imagined data and a histogram plot: