What is Statistical: Definition and 654 Discussions
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.
Homework Statement
One class application of correlation and regression involves the association between the temperature and the number of times cricket chirps in a minute. Listed below are numbers of chirps in 1 minute and the corresponding temperature in degrees Fahrenheit.
Chirps in 1...
According to my thermal physics book , from classical thermodynamics for a reversible change :
dU = dQ - dW
From stat mech ,
dU = \sum_i \epsilon _i dn_i + \sum_i n_i d \epsilon _i
Then by drawing some arguments from quantum mechanics the writer relates the change in volume and...
For some given statistics (e.g. Fermi-Dirac or Bose-Einstein), once we know the average number of particles at state r, it is easy to calculate the dispersion by calculating
\overline{(\Delta n_r)^2} = -\frac{1}{\beta}\frac{\partial \bar{n}_r}{\partial \epsilon_r}
and the total number of...
I took classical (engineering) thermodynamics a few years ago, and this semester I am taking a statistical thermodynamics class from the physics department. We are using the book "Fundamentals of statistical and thermal physics" by Reif, which seems to be a pretty good book.
Unfortunately I...
http://www.ph.ed.ac.uk/~pmonthou/Statistical-Mechanics/documents/SM15.pdf
At the top of page 59 in the above link, I can't see where the first three bullet points that follow
E(T=0)=\int_0^{\epsilon_f} g(\epsilon) d \epsilon
come from?
can anyone help?
In my notes,
http://www.ph.ed.ac.uk/~pmonthou/Statistical-Mechanics/documents/SM3.pdf
on page 1 we are told we're dealing with systems of fixed total E but in the expilicit example on page 3, do we not find 4 different values for the total energy. how is this possible?
Homework Statement
Find an expression of the statistical weight for a system of N particles of corresponding energy levels with 'gi' degeneracy.
Homework Equations
W = N! / No! N1!2! ...Ni!
The Attempt at a Solution
The expression is then :
W = N! goNo g1N1 g2N2 ... giNi / No...
Homework Statement
A box is separated by a partition which divides its volume in the ratio 3:1. The larger portion of the box contains 1000 molecules of Neon gas, the smaller one contains 100 molecules of He gas. A small hole is made in the partition, and one waits until equilibrium is...
I have been doing a lot of work recently on quantitatively modeling error in models versus experiment.
One of the problems I currently have is with a model and a sparse phase space of experimental measurements and how to determine the accuracy of the model for the entire phase space. One the...
Hello,
I am stuck on the first part of this question. There are several parts that follow that depend on this bit, and I know I can do them if I can just work this out. Any help would be gratefully received.
Homework Statement
Consider an isolated system of N identical spin-0 bosons...
Homework Statement
The pass mark of an examination is set at 40. The following table shows some summary statistics of the marks (x) of the 200 students who took the examination.
Mark Range: x >= 40, x < 40
Number of students: 160, 40
Mean: 64.0, 32.0
Standard deviation: 6.0, 4.0
The...
(1) Derive the expression for \muPG for a monatomic molecule (like argon) which has no internal structure and only has translational energy.
where \muPG is the perect gas contribution to the chemical potential.
(2)Treating the molecule N2 as a linear, rigid molecule is an excellent...
What is the physical or statistical meaning of the following integral
\int^{a}_{o} g(\vartheta) d(\vartheta) = \int^{\infty}_{a} g(\vartheta) d(\vartheta)
where g(\vartheta) is a Gaussian in \vartheta describing the transition frequency fluctuation in a gaseous system (assume two-level and...
I have to take that next year and am nervous about it. I took it as an undergraduate, but most of it went right over my head. Now I have to take the graduate-level course. Can anyone recommend a good book for me to read in preparation for that class?
In adjacent containers of volume V1 and V2, there contains a gas of N molecules. The gas is free to move between the containers through a small hole in their common wall.
What is the probability to find k molecules in the V1?
Probability of one molecule to be in V1 is given by...
I am trying to show that the change in number of accessible microstates, and therefore the change in the multiplicity function g of a simple system is
g=\left( \frac{\tau_F^2}{\tau_1\tau_2}\right)^{\frac{mC_V}{k_B}}
where the system is two identical copper blocks at fundamental...
Greetings
Next semester I'll have solid state physics, statistical physics and and introductionary course in elementary particles. So I was wondering maby those of you who had those subjects could recommend me some books.
I really liked books by Griffiths so far. Especially the one on...
Homework Statement
Hi all.
The partition function for fermions is (according to Wikipedia: http://en.wikipedia.org/wiki/Partition_function_(statistical_mechanics)#Relation_to_thermodynamic_variables_2) given by:
Z = \prod\limits_i {\left( {1 + \exp \left[ { - \beta \left( {\varepsilon _i -...
Hi!
I am a software programmer and I would like to make a performance analysis of my software. One of the objectives that we need to achieve is to create a module for accurately computing the payment. Now, my expected result is that all computations are accurate. What statistical tool should...
Imagine a jet of fluid (perhaps air) impinging on a flat plate. It could be said that the jet has a slightly higher mean velocity in the direction normal to the flat surface (we'll arbitrarily call this X).
From a classical thermodynamic point of view it could be said that the gas has a higher...
Homework Statement
Hi all.
I've read some about the MaxEnt method (maximum entropy), and one question of mine is still unanswered: Why is it that we wish to maximize the entropy?
When looking at a gas with average energy U, the higher the entropy, the "less" we know about the gas because...
is there any video lectures source which is about statistical physics or quantum physics like MIT Lectures. I found some lectures about quantum physics but these are only related with it and it couldn't be considered as essential source.
Homework Statement
I have measurements of some response of a gene, and two factors: the gene, g=1...G and whether the patient/subject has a certain disease, t=1,2.
the full model is
y_{gtk}=\mu+\alpha_g+\beta_t +(\alpha\beta)_{gt}+\epsilon_{gtk}
I know that to see if genes have...
Homework Statement
Flip N fair coins. The distribution for different numbers of heads and tails should be peaked at N/2. When N is very large, the peak will be very high. Let x = N(head)-N/2, required to find an expression for this distribution near the peak, i.e. x<<N.
Homework Equations...
Hi,
This question is probably a dumb one but I admit that I am quite perturbed with this issue.
Indeed, I don't understand why canonical ensembles like the microcanonical ensemble or
the canonical one are called "equilibrium ensemble".
I do agree that they correspond to steady measures of...
Dear all,
I'm reading "Equilibrium Statistical Mechanics" by Atlee Jackson (Dover) which is very good. What could be a next step? In the web everybody speaks highly of "Introcution to modern statistical physics" by Chandler. What about the books by Hill (Dover) or Principles of Statistical...
Please help:
Homework Statement
A simple model of a Geyser is an underground huge lake connected to the surface
by a small tube.
let the depth (and the tube length) be 90m.
proove that the ratio between the mass of the water in the lake and the gas which sparks from the geyser is...
I came across this interesting Havard article at http://www.cfa.harvard.edu/~huchra/hubble.plot.dat listing over 600 Hubble survey results dating back to about 1929 and up to 2008. I put all the data into into a spreadsheet and using over 500 Hubble estimates from 1960 onwards I got an average...
Hi,
I was having a discussion with a friend about the probability of life in the universe outside of earth.
From what i know, there are 70 000 000 000 000 000 000 000 stars and god knows how many planets. Given the low probability of life spawning on a single planet but the huge amounts of...
I'm planning on taking a statistical thermodynamics grad course this Fall. Being an ME I am very familiar with classical thermo but what other topics should I be keen on? Any specific field of mathematics like abstract alg(only math I have not taken) I should be worried about?
Also, I am...
I've recently gotten interested in statistical physics. Notions derived from this area are frequently applied in learning theory and other areas of biology which are of interest to me.
I'm wondering if anyone knows of a good book or web resource for learning about statistical physics that...
statistical mechanics -- why is temperature not a mechanical variable
Hi, I have heard that temperature is not a mechanical variable. That is, that even if you knew the positions and momenta of all the particles in some system, you still couldn't calculate the temperature, because temperature...
hello, I'm trying to learn statistical mechanics with feynman's book
he seems to have a lot of times where he thinks something is very clear and then it will take me a few pages of working things out to get what he has jumped to.
i'm having one of those moments on page 15.
i was wondering if...
Does anyone know of any research programs out there that are considering physical structure formation in terms of emergent, self-organizing statistical manifolds, and does so without starting from some reasonable first principles without any structure or preconception of manifold, or ad hoc...
Hello
This attachment is a practice paper I am doing. I know how to do everything except for questions 11, 12 and 13 so I would appreciate it if someone could please show me the process for working them out. thanks in advance.
Almost no one has answered.To take some external help, i write all questions here, maybe someone have an answer.
1)An aluminium cube cooled down to 90 Kelvin.At this point there is too many microstates available.how much heat we need, to increase these microstates in factor of 10^{10} ...
Homework Statement
Given the width of an energy band of electrons is typically ~10eV, calculate the number of states per unit volume within a small energy range KT about the centre of the band at room temperature.
K is boltzmanss constant = 8.617*10^-5 eV/K
Homework Equations
volume...
Homework Statement
The neutnral carbon atom has a 9-fold degeerate ground level and a 5-fold degenerate excited level at an energy 0.82 eV above the ground level. Spectroscopic measurements of a certain star show that 10% of the neutral carbon atoms are in the excited level, and that the...
Homework Statement
If we assume entropy is a function of the multiplicity, \Omega, (S=k*f(\Omega)) show that that function f(\Omega) is ln(\Omega).Homework Equations
The Attempt at a Solution
\Omega can be written as N!/ni!. By using stirling's approximation, this becomes \Omega=...
Homework Statement
average energy per particle u = (Eo + E1 e^(-B deltaE)) / (1 + e^(-B deltaE))
B = 1/T
Homework Equations
Possibly relevant: e^x = 1 + x^2 / 2! + x^3 / 3! ...
The Attempt at a Solution
It tells me the average energy is about u = Eo + (deltaE)e^(-B delatE) as...
I'm having some difficulties with a problem. Based on the constraints, I have found that the average energy per particle is u = (Eo + E1 e^(-B deltaE)) / (1 + e^(-B deltaE)). I know this is correct. However, I am having problems solving as T approaches 0 and infinity. B = 1/T
It tells me the...
Hi all,
I am hoping someone can recommend a useful statistical test. I have a set of data on an x-y plot which varies about the y=0 line in a seemingly random way. Each data point has a y error bar, which appears to be, in general, smaller than the standard deviation of the data.
I would...
Homework Statement
Find Condensation Temperature of ideal Bose gas with internal degrees of freedom. Assume only ground and first excited state of internal spectrum need to be taken into account.
Homework Equations
The Attempt at a Solution
I know j(T) for internal degrees of...
[SOLVED] statistical mechanics 6.1
Homework Statement
http://ocw.mit.edu/NR/rdonlyres/Physics/8-044Spring-2004/00E63135-AD4E-4F76-9917-349D5439ABF4/0/ps6.pdf
The answer to Problem 1 part c is 2N. I disagree. I think it should be N because if you specify the z component of the system then you...
Homework Statement
http://ocw.mit.edu/NR/rdonlyres/Physics/8-044Spring-2004/85482B93-6A5E-4E2F-ABD2-E34AC245396C/0/ps5.pdf
I am stuck on Problem 5 part a. They say that the relevant state variables are H,M,T, and U. Obviously the first law of thermodynamics still holds: dU = dW+dQ (does...
[SOLVED] statistical physics
Homework Statement
http://ocw.mit.edu/NR/rdonlyres/Physics/8-044Spring-2004/85482B93-6A5E-4E2F-ABD2-E34AC245396C/0/ps5.pdf
I am working on number 3 part a.
I am trying to calculate C_P.
From the first law of thermodynamics: dQ = dU -dW = dU +PdV (does anyone...
[SOLVED] statistical physics
Homework Statement
http://ocw.mit.edu/NR/rdonlyres/Physics/8-044Spring-2004/AC9B128C-9358-4177-BFE6-A142E0FD897B/0/ps4.pdf
I am working on Problem 3.
So I want to calculate the integral of dW along each of those paths. But how can I relate dW to dV? dW is equal...
A complete set of lecture notes for an upper-division thermodynamics and statistical mechanics course. Topics covered include elementary probability theory, classical thermodynamics, the thermodynamics of the atmosphere, heat engines, specific heat capacities of gases and solids, the Maxwell...
[SOLVED] statistical physisc
Homework Statement
http://ocw.mit.edu/NR/rdonlyres/Physics/8-044Spring-2004/AC9B128C-9358-4177-BFE6-A142E0FD897B/0/ps4.pdf
I am working on Problem 1a. I am really confused about this question. Do I set the two equations equal to each other and solve for something...