What is Statistical: Definition and 656 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.
In science, statistics are constantly used to give 'rigorous' interpretations of data sets. In this process, tests are often employed to verify a property that is being investigated. For example, normal distribution or randomness. Usually an algorithm is employed on the data set and a test...
The behavioral sciences use statistical methods to link behavior to specific outcomes. Medical science also tries to find relationships between various behaviors and their health effects. Such studies use statistical methods unfamiliar to most physical scientists.
Regularly, studies with the...
Let’s consider that the total energy of this system is represented as ##E=-2mB##.
Question 1: how many microstates correspond to this energy level?
We have ##2^4=16## microstates.
++++ Total magnetic moment: ##4m## Energy: ##-4mB##
- - - - Total magnetic moment: ##-4m## Energy: ##4mB##...
Hello, Springer books are on sale this week so I wanted to buy some textbooks to support my studies and (eventual) future career.
I'm an undergrad (in europe) and my courses next year will be QM, GR and statistical mechanics, so I was looking for books about these topics, but any suggestion on...
I started my research in statistical digital signal processing two years ago, so I need to familiarize myself with all the notations people use in probability and statistics. I come from a deterministic science background. I name my variables based on what they mean. A velocity is a v , a...
Can anyone tell me how I can solve the problem of non-verification of statistical tests done by MCNP5 (relative error, VOV, figure of Merite, slope). I tried to increase the number of particles generated in order to hope to verify the tests but it did not work.
How much statistical mechanics do I need to know to study QFT, astrophysics, black hole thermodynamics, and other advanced topics? And where should I study it in your opinion? So far I have only read Tong's notes however I don't think it is enough. Some quantum statistical mechanics is also...
In 1968, Lord and Novick published a book called Statistical Theories of Mental Test Scores. I wonder why they used the adjective 'statistical'. Does this suggest that the theories mentioned are not psychological theories and, if so, what could be the meaning of such theories? Should these...
So we have a system of N non interacting particles, on a d-dimensional space, the system is in contact with a bath of temperature T. The hamiltonian is $$H = \sum_{l = 1}^{N} (A_{l}|p_{l}|^{s}+B_{l}|q_{l}|^{s})$$.
What is the avarage energy?
Now, i have some problems with statistical...
Very basic question here, about statistical independence in quantum mechanical experiments. The quote from PD below is what prompted the question.
When we talk about "some kind of pre-existing correlation" are talking about a simple correlation in the sense of the correlation of sunglasses and...
Moderator's note: Spin-off from previous thread due to topic change.
Because it doesn't work. Bohmian time evolution doesn't involve the coarse graining steps that are used in his calculation. A delta distribution remains a delta distribution at all times and does not decay into ##|\Psi|^2##.
Reif, statistical physics
"The equilibrium macrostate of a system can be completely specified by very few macroscopic parameters. For example, consider again the isolated gas of ##N## identical molecules in a box. Suppose that the volume of the box is ##V##, while the constant total energy of...
Suppose we've an isolated box having ##N## classical distinguishable particles in it, the box being hypothetically divided into two parts, left and right with both parts identical.
Its said that the probability of having the configuration of ##n## particles in the left side is given as...
This is the beginning of an online reading course of the book "Statistical physics" by Reif, volume 5 in the Berkeley physics course, using PF.
We'll start with chapter 3 and loop back to the initial 2 chapters if necessary.
All questions should be specifically about what is written in the this...
Nearly two decades after I graduated with an engineering degree, I'm currently studying for a master's with a particular emphasis on conceptual/theoretical statistical physics. Based on my interests and stylistic preferences, I'm using the following books to build my understanding of physical...
a) V=(4/3)pi(r^3)
N=M/m_n (M=mass of neutron star, m_n=mass of neutron)
Subbed into E_f = (hbar^2 / 2m) (3(pi^2)N / V)^(2/3).
T_F = E_F / k_B
b) dU = (dU/dS)_s dS + (dU/dV)_s dV
p = -(dU/dV)_s dV
V=(4/3)pi(r^3) -> r = cubedroot(3V/4pi)
subbed into U_g = -(3/5)(G M^2 / r)
take (dU/dV)
plug into...
Another question about the use of the micro-canonical ensemble in deriving distributions.
On the Wikipedia-page the authors mention that the total volume of the system has to be constant.
See...
Hey! :giggle:
Analyst has collected the following data on the performance of the $X$ stock for $10$ different years.
a) Calculate the arithmetic mean, the median, the mode, the standard deviation, the coefficient of variability and of asymmetry. You interpreted your results.
b) Does the...
In a book that I am reading it says
$$(V - aw)(V - (N-a)w) \approx (V - Nw/2)^2$$
Where ##V## is the volume of the box, ##N## is the number of the particles and ##w## is the radius of the particle, where each particle is thought as hard spheres.
for ##a = [1, N-1]##
But I don't understand how...
Hello everyone,
When working with variables in a data set to find the appropriate statistical model (linear, nonlinear regression, etc.), the variables can have different range, standard deviation, mean, etc.
Should all the input variables be always standardized and scaled before the analysis...
For instance if we are given only a PDF in the form of ##p(x)##, how can one calculate the characteristic function, the mean, and the variance of these PDF's ?
Any site or explanation will be enough for me
Hi, I'm a physicist so I have a basic knowledge of probability and hypothesis testing etc. I would like to more sophistically calculate from available data in my country whether ones Covid infected people have a statistically significant different probability of reinfection than people who are...
Firstly, I would like to check my understanding of the first formula:
Using velocity distribution = f(v), speed distribution = fs(v):
fs(v) = f(vx)f(vy)f(vz)dxdydz, since dxdydz = 4pi*v^2*dv, fs(v) = 4piv^2f(v)
The second formula is the confusing one:
What does it mean? What is the...
Those days I'm in the mood of criticizing the Ballentine's statistical interpretation, also known as the minimal statistical interpretation. Here I will argue that it is, in fact, neither minimal nor statistical.
The main culprit is that Ballentine repeatedly insists that there is no wave...
Hi,
I recently discovered that there is no real paradox in the question of the mixing of classical distinguishble particles. I was shocked. Most books and all my professors suggest that an extensible entropy could not be defined for distinguishble particles.
I believe that many of you will be...
First of all, I've calculated the partition function:Z=1h3∫e−βH(q,p)d3pd3q=1h3∫e−β(p22m−12mrω2)d3prdrdθdz=2πL(2mπh2β)3/2e12βmω2R2−1ω2mβThe probability of being of one particle in radius $r_0$ is:
p(r=r0)=1Z∫e−βHd3pd3q=∫1Z2πL(2mπh2β)3/2eβmrω22rdr
So I've thought that because, by definition, the...
The usual presentation of classical statistical mechanics are based on the Liouville equation and phase space distribution. This, in turn, is based on the Hamiltonian mechanics of a system of point particles.
Real undulatory systems, specially non-linear ones, have to be complex to study...
Alright, so I did some progress and then I got stuck. After some time I went to check the solution. Up to some point, it's all well and good:
I understand everything that is happening up to the point where he takes the partial derivative of S wrt ρ(Γ). I don't understand how he gets the...
I have been reading statistics for a while (I am a physics major but also a stat-enthusiast), and one of the topics that drew my attention was the misrepresentation, or to be precise, misinterpretation of the data. This came up while reading about Simpson's paradox and the likes. When I see...
I was reading mehran kardar (books and lectures) it says the concept of irreversibility comes from an assumption (in which we increase the length scale by interaction disctance between two particles).
So My question is the concept of irreversibility is still valid in the case of 1 particle...
Good day,
I'm starting my master in physics, and it's time for me to choose my courses.
I will take one or two of the following three courses, which are: Statistical Physics, QFT and General relativity.
Now, I'm finding it very hard to decide as on the one hand, I'm interested in QFT and...
I've come across a number of problems in elementary probability theory and statistics that can be exemplified as follows:
Naturally, real lamps decay over time, so their lifetimes can't be memoryless. With that being said, is the exponential distribution a good approximation for the...
Hi, I am currently reading Introduction to statistical physics by Huang. In the section of entropy, it reads
But what if I choose ##R-P## as a closed cycle? Then in a similar process, I should have ##\int_{R} \frac {dQ} {T} \leq \int_{P} \frac {dQ} {T}## and ##S \left ( B \right ) - S \left (...
Summary:: What are the relevant mathematics/ mathematical physics courses for studying quantum field theory and statistical field theory?
I'm a physics undergraduate currently in my junior(third) year, thanks.
First I found partition functions of both the systems and hence total energies of them using above formulas.
Z(A) = (1 - e-ε/kT)-1 and Z(B) = (1 + e-ε/kT)
Then I equated these values to the given values of total energies.
I got:
For System A, T(A) = ε/kln(2) > 0
For System B, T(B) =...
Upto now I've only dealt with the problems regarding non - degenerate energy states.
Since bosons do not follow Pauli's Exclusion Principle, three bosons can be filled in two energy states (say E1 and E2) as:
E1
E2
1 boson
2 bosons
2 bosons
1 boson
3 bosons
0 bosons
0 bosons
3...
Looks like a good step forward. (At least to someone far outside the field. :biggrin:)
Abstract and paywall article in Nature.
https://www.nature.com/articles/s41586-019-1833-8
Full preprint at:
https://arxiv.org/abs/1909.05272
The specific example I'm going to give is from a discussion I am having elsewhere, but the question itself, as given in the thread title and summary, is a general one.
We have two couples, each of which has seven children that, in order, are six boys and one girl (i.e., the girl is the youngest...
Personally I tend to believe all (or almost all) of the interpretations of QM are unsatisfactory simply because they tell us something that we already know but do not tell us something we don't know. That is, they do not predict new phenomena or principles or properties of matter, etc. that can...
I'm wondering if the passage from a classical thermodynamic theory, i.e. which does not resort to an atomistic theory and methods of probability and statistics, to classical (i.e. non-quantum) statistical mechanics, led to new discoveries and especially if it was able to explain properties of...
Hi All,
I have recently read about a fallacy that seems to be based on looking at a non-representative subsample of the population. I would like to know if this goes by a name and if it has been formalized. It just seems the problem is that of considering a variable within a subpopulation and...
First, i'd like to apologize for the vague title. Unfortunately my understanding of the question is equally vague. I think the dXd matrix is meant to be a covariance matrix, so the above equation would be some complex constant multiplied by the covariance matrix. The Tr would referring to the...
Why is a thermally isolated process that occurs sufficiently slow is necessarily adiabatic and not just reversible process ? Here I mean that the definition of adiabatic process is no change in the entropy of the subsystem, and a reversible process is define by no change of the total entropy of...