Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments.When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements. In contrast, an observational study does not involve experimental manipulation.
Two main statistical methods are used in data analysis: descriptive statistics, which summarize data from a sample using indexes such as the mean or standard deviation, and inferential statistics, which draw conclusions from data that are subject to random variation (e.g., observational errors, sampling variation). Descriptive statistics are most often concerned with two sets of properties of a distribution (sample or population): central tendency (or location) seeks to characterize the distribution's central or typical value, while dispersion (or variability) characterizes the extent to which members of the distribution depart from its center and each other. Inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena.
A standard statistical procedure involves the collection of data leading to test of the relationship between two statistical data sets, or a data set and synthetic data drawn from an idealized model. A hypothesis is proposed for the statistical relationship between the two data sets, and this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets. Rejecting or disproving the null hypothesis is done using statistical tests that quantify the sense in which the null can be proven false, given the data that are used in the test. Working from a null hypothesis, two basic forms of error are recognized: Type I errors (null hypothesis is falsely rejected giving a "false positive") and Type II errors (null hypothesis fails to be rejected and an actual relationship between populations is missed giving a "false negative"). Multiple problems have come to be associated with this framework, ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis. Measurement processes that generate statistical data are also subject to error. Many of these errors are classified as random (noise) or systematic (bias), but other types of errors (e.g., blunder, such as when an analyst reports incorrect units) can also occur. The presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems.
Hi, I'm a rising senior and interested in developing nano-sensors/imaging, and I'm also thinking of going into industry after grad school.
I'm debating on whether to take Data mining or Statistical Physics/Thermodynamics and wasn't sure which will be more suited for my needs.
Both courses are...
Homework Statement
If Hamiltonian ##\hat{H}## is hermitian show that operator ##\hat{\rho}=\frac{e^{-\beta\hat{H}}}{Tr(e^{-\beta\hat{H}})}## is statistical.Homework Equations
In order to be statistical operator ##\hat{\rho}## must be hermitian and must have trace equal ##1##...
I red griffiths many times but even now there is something I can't understand. It's about statistical interpretation. In his book chapter 3.4 he says
"If you measure an observable Q(x,p) on a particle in the state ψ(x,t), you are certain to get one of the eigenvalues of the hermitian operator...
I am considering majoring in statistics. The school (a mediocre state uni, by the way) calls the major "statistics", but the curriculum is that of an "applied statistics" program, in that it requires you to choose an emphasis that you could apply statistics to. I'm going to choose science...
When do these two fields of physics overlap other than when we discuss entropy of supersymmetric black holes? Does statistical mechanics or statistical physics support any aspect in string theory or vice verse? I have read some place that the string theory reduces to the standard model by doing...
Homework Statement
Consider a classical one-dimensional nonlinear oscillator whose energy is given by \epsilon=\frac{p^{2}}{2m}+ax^{4}
where x,p, and m have their usual meanings; the paramater, a, is a constant
a) If the oscillator is in equilibrium with a heat bath at temperature T...
There had been a case in the UK where a woman's two babies died one after the other. Then some apparent statistician concluded 'If the chance of that occurring is 1 in a million, then she must have killed her babies'. Later, a very long court of law had been doing research on it and she appeared...
Homework Statement
A cavity contains black body radiation at temperature at T=500K. Consider a optical mode in the cavity with frequency w=2.5x10^(13) Hz. Calculate;
(a)the probability of finding 0 photons in the mode.
(b)the probability of finding 1 photon in the mode
(c)the mean number of...
1. Homework Statement
An average value of N measurements is defined as:
ravg = 1N ∑ ri where i = 1 and it sums up to N measurements
By using this expression in the master formula for a general function z = (x1, x2,...) error propagation, we find the δravg error propagation:
(δravg)2 =...
Homework Statement
If ##Z## is homogeneous function with property
##Z(\alpha T,\alpha^{-\frac{3}{\nu}}V,N)##
and you calculate Z(T,V,N). Could you calculate directly ##Z(\alpha T,\alpha^{-\frac{3}{\nu}}V,N)##.
Homework Equations
##Z(T,V,N)=\frac{1}{h^{3N}N!}(2\pi m k...
Hi all, quick query please:
My nearest university is a good one (with a good department) but offers only an MSc Mathematics, not an MSc Statistics. But the course options are such that it can be virtually all statistics and probability. E.g. modules include Mathematical Biology, Medical...
I've been reading, in my own time, a first course in thermodynamics and they present a quantum treatment of statistical mechanics (discrete energy levels), but on the article for the partition function on wikipedia, I find out that there is a classical treatment of statistical mechanics as well...
The probabilistic nature of QM seems apparent(in theoretical formulation as well as experimentally), then why do most of the physicists give the credit of in-determinism to the nature itself and look for many worlds, abruptly collapsing observer conscious experiment setups, and the likes...
That's the only equation I know on the topic so far. I'm an undergrad physics major and one of my courses this term is statistical mechanics which I am finding to be overwhelmingly difficult to understand. Apparently the course is considered "graduate level," although they still require us to...
I was studying statistical mechanics and found some concept that are difficult for me to grasp. Any kind of help will be highly appreciated.
My questions are :
1. In equilibrium which quantity should be maximized? Is it the number of Microstate or the way that a particular Microstate is...
Homework Statement
A system of N particles has three possible energy levels namely; 0, E and 4E. How many particles does one expect in the second state at temperature T?
Homework Equations
It's a sample problem for our finals. Our Text book is Statistical Mechanics by Roger Bowley and...
So, there are two things in Quantum Mechanics that I understand are axioms: the first is the schrodinger equation, which cannot be derived. Okay fine, we have to start somewhere. The second axiom is that the integral from a to b of the wavefunction-mod-squared gives the probability of finding...
Hello.
I developed an algorithm to detect events in time domain and I want to know the efficiency of the algorithm.
The problem is related with the time duration of the data.
Each file has data with a time duration of hundreds of minutes and I have dozens of files.
Instead of...
Hello,
Trying to make sense of the odds that I (or you) exist, from what I read people say its basically 0, which doesn't make sense to me since I am here after a short ~14 billion years of our universe existing.
Given IF the universe and time are infinite, my odds of existing again, no...
Homework Statement
A polymer consits of ##10^{20}## monomers, each 2 nm long. One end of the polymer is hanged at the ceiling on the other end, we have a load of ##4\cdot 10^{-10}g##.
Calculate the average potential energy!Homework Equations
The Attempt at a Solution
Well... I don't know...
Homework Statement
A system of N distinguishable particles is arranged such that each particle can exist in one of the two states: one has energy \epsilon_{1}, the other has energy \epsilon_{2}. The populations of these states are n_{1} and n_{2} respectively, (N = n_{1}+n_{2}). The system is...
Hi, I'm completely lost in this discipline. Does anyone know of a good book with solved exercises and some examples, particularly on the 3 types of ensembles (microcanonical, canonical and large canonical), quantum mechanics statistical physics, and fermions/bosons gases? Thank you.
Homework Statement
The statistical and spectral functions for bosonic operators \phi_a are:
G_{ab}(t,\vec{x})=\frac{1}{2}\langle \{\phi_a(t,\vec{x}),\phi_b(0,\vec{0})\}\rangle ,
\rho_{ab}(t,\vec{x})=\langle [\phi_a(t,\vec{x}),\phi_b(0,\vec{0})]\rangle .
The expectation values are in...
We don't have rigorous approach to statistical mechanics but have an intuitive one.We follow Concepts of Modern Physics by Sir Arthur Beiser.
It is given that
n(ε)=g(ε)f(ε) Here ε is the energy state.
where according to my understanding
n(ε)=number of particles which are present in an...
I am planing to study thermal physics on my own as I got an incompetent faculty who doesn't know anything about this subject. I want to have a detailed understanding about the topic. So can somebody suggest me which book I should follow from the following list.
1. Schroeder, Thermal...
I'm taking stat mech right now (Kittel book) and...I might hate it. I love e&m, quantum, classical, relativity, but I like almost none of what we're doing in this class. It's not the type of physics I'm interested in and it's abstract in a way I don't like (I normally like abstraction). I know...
Stochastic Differential Equation
Hi there,
I am trying to solve (analytically) a stochastic differential equation of the form:
\frac{d^2}{dt^2}x +\left(k(t)+\delta k \ t\right)x)=0
Here \delta k is a random (gaussian) white noise. Note, that in the differential equation it is multiplied...
Homework Statement
There are 5 Bosonic particles N = 5 populating 2 degenerate energy levels E1 and E2 such that:
E1 < E2, N2 ≤ N1
and the respective statistical weights are
g1 = 3 and g2 = 2.
.
What are the possible macrostates of this system?The Attempt at a SolutionI'm not sure if the...
I am supposed to do an experimental report in my Social Psychology class. I am analyzing the factors affecting the lack of females in engineering and other math-intensive fields. I measured men and women's life interests, life values and attitudes to gender roles in our survey because I think...
Prove that the following two statements of the second law are equivalent (each one can be derived from the other):
(a) “It is impossible for heat to flow spontaneously from a colder to a warmer body without causing other changes”
(b) “It is impossible to convert all the heat taken from a...
Usually in the problems I have done, I found the partition function by simply summing exponential of the eigenenergies, but lately I have started to wonder why this approach is correct. Do we not want to sum over the energies of all possible states as we did in the classical case. In that case...
I'm taking Stat Mech at the graduate level and I find the course incredibly frustrating in that it just feels like a whole lot of mathematical formalism without any examples. I've looked in Paritha, Huang, and Salinas and all these books seem the same in that they just looks like a bunch of...
Homework Statement
Consider two spin-1/2 subsystems with identical magnetic moments (μ) in equal fields (B). The first subsystem has a total of NA spins with initially "a" having magnetic moments pointing against the field and (NA - a) pointing along the field, so that its initial energy is...
A glass bulb contains air at room temperature and at a pressure of 1 atmosphere. It is
placed in a chamber filled with helium gas at 1 atmosphere and at a room temperature. A
few months later, the experimenter happens to read in a journal article that the particular
glass of which the bulb is...
An Introduction To Thermodynamics And Statistical Mechanics by Stowe is the course textbook for my Statistical Physics course I am having trouble with the book as it does NOT provide good examples and explanations nor does it have solutions just answers.
I was wondering if there is Schaums or...
In a aislate system, the probability on a microcanonical state \Gamma is
p(\Gamma ) = 1/K , if E<H<E + ΔE, and 0 on otherwise
with K = \int_{\Gamma : E<H<E+ΔE} d \Gamma
a) Show that ΔE →0, then
p(\Gamma) = \frac{\delta (E-H)}{\int_{\Gamma : H=E} \delta(E-H)}
b) Show that if...
Homework Statement
Consider a system consists of two subsystem A and A' in which A contains 3 spins and A' contains 2 spins. Suppose that, when the systems A and A' are initially separated from each other, measurements show the total magnetic moment of A to be - 3{\mu _0} and the total...
Can anyone help me with this, please?
It's about how you go about trying to decide if two distributions are consistent, statistically speaking; specifically, what statistical test, or tests, is (are) most appropriate to use.
Here's the data:
N(A) N(B) G/R/P
0043 0046 #101
0264 0235...
Homework Statement
Consider a random walker in one dimension, which can take right or left steps with equal probability. Assume that the walker starts at the location k>0 and there is an absorbing wall at point labelled as 0, that means if the walker reaches 0 the process stops and he stays...
I'm looking for a book on statistical physics. One's I've used already are Huangs Intro one and Sachs/Sens/Sexton. They're both great and the problems are great but neither provide answers nor are there any solutions to be found online so I have no idea if I understand the material.
Could...
I have a set of data that was recorded from an engine that we are testing. We've noticed lately that a particular pressure value will sometimes spike with no apparent explanation, as seen in the attached graph. The pressure in question is passively regulated by a pump, but it is also dependent...
I have a question about statistical operator. In statistical physics you deal with temperature. So for example ##\hat{\rho}=\frac{1}{Z}e^{-\beta \hat{H}}## where ##\beta=\frac{1}{k_BT}##. In definition there is temperature. And also equivalent definition is
##\hat{\rho}=\sum_i w_i|\psi_i\rangle...
Homework Statement
Consider the case of a gas in the atmosphere. Assume that the temperature is a constant. Based
on the Maxwell Boltzmann distribution, at sea level the atmosphere contains 78.1% N2, 21%
O2, 0.9% Argon, and 0.036 CO2. What are the ratios at the the top of Everest? (Molecular...
Here are to challenge questions given to the class to answer to be turned for a free slice of pizza. We have to days, and he said we can get help, and I'm taking him at his word. Here they are.
1). Suppose a room is divided into the front fifth, the middle fifths, and the back two fifths...
Homework Statement
In a large system of distinguishable harmonic oscillators, how high does the temperature have to be for the probable number of particles occupying the ground state to be less than 1?
Homework Equations
N(E_{n})=NAe^{\frac{-En}{K_{B}T}}
E_{0}=0 (the textbook...
For the people who are in either of these fields, which math did you encounter most frequently in the field? Abstract Algebra, Analysis, Probability, Statistics, Calculus, or other? I have taken introductory courses for both topics at university and both seem to involve a wide range of math...
The derivation of the momentum/position Heisenberg Uncertainty Principle (HUP) is based on the statistical interpretation which says that if we have a lot of quantum systems in identical states, and measure the momentum in half of them and get a distribution with standard deviation σp, and...