Statistical Definition and 659 Threads

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.

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  1. Agent Smith

    B Statistical Hypothesis Testing

    ##H_0##: The probability of an obese person using chopsticks = the probability of a normal-weight person using chopsticks ##H_a##: The probability of an obese person using chopsticks ##\ne## the probability of a normal-weight person using chopsticks "Partial" Chi-Square Test: I focused only on...
  2. Agent Smith

    B Statistical Errors, Type I and Type II

    Reached Hypothesis testing in my statistics notes (high school level). It reads ... 1. Type I Error: Rejecting the null (hypothesis), ##H_0##, when ##H_0## is true. The risk of a Type I error can be reduced by lowering the significance level ##\alpha##. The downside is this increases the...
  3. Agent Smith

    B Test of statistical significance

    The following appears as part of an intro to statistical significance. I don't quite get it and hence the appeal for clarification. Someone claims that a newly invented medical test T is 99% accurate. To check their claim the test is conducted among 100 subjects and in 95 the result is...
  4. Mayhem

    B How are critical values in statistical tests obtained?

    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...
  5. gleem

    I Observational Studies: What to Believe

    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...
  6. MatinSAR

    Statistical mechanics: particles in magnetic fields

    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##...
  7. N

    Other Which Springer books to buy? (QM, GR and statistical mechanics)

    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...
  8. tworitdash

    I Making Sense of Notation Confusion in Statistical Digital Signal Processing

    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...
  9. V

    Why Do MCNP5 Statistical Tests Fail Despite Increasing Particle Numbers?

    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.
  10. A

    How much statistical mechanics is enough for a physicist?

    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...
  11. A

    A Why Statistical Theories of Mental Test Scores?

    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...
  12. LCSphysicist

    Statistical mechanics and problem with integrals

    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...
  13. Lynch101

    B Statistical Independence in Quantum Mechanics

    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...
  14. N

    A Derivation of Statistical Mechanics

    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##.
  15. K

    I The Equilibrium Macrostate: Reif's Statistical Physics

    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...
  16. K

    I Probability in statistical mechanics

    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...
  17. Philip Koeck

    I Reading course in statistical physics

    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...
  18. ipsky

    Intro Physics Opinions on books for (self) studies in statistical physics

    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...
  19. B

    Statistical physics, using the ideas of Fermi Energies, etc. for a star

    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...
  20. Philip Koeck

    A Volume constraint in micro-canonical derivation of statistical physics

    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...
  21. M

    MHB Calculating statistical values from given data

    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...
  22. Arman777

    A Understanding an Approximation in Statistical Physics

    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...
  23. F

    I Scaling and Standardization in Statistical Analysis

    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...
  24. Arman777

    A Calculating the statistical properties of the given PDF

    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
  25. V

    I Statistical analysis of COVID reinfection

    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...
  26. T

    Physics Non-equilibrium statistical physics and complex systems

    Is Non-equilibrium statistical physics and complex systems a good area of study to go into? Is it a well respected field? Thank you
  27. P

    Understanding basic statistical mechanics formulas

    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...
  28. Frabjous

    Classical The Statistical Foundations of Entropy by Ramshaw

    Does anyone have any thoughts on this book? https://www.amazon.com/dp/9813234121/?tag=pfamazon01-20
  29. Demystifier

    A The minimal statistical interpretation is neither minimal nor statistical

    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...
  30. L

    A Exponential statistical process to be characterised

    I'd be grateful for any formulation that describes this statistical process
  31. A

    I Gibbs paradox: an urban legend in statistical physics

    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...
  32. K

    Probability density in statistical Mechanics

    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...
  33. A

    Does a statistical mechanics of classical fields exist?

    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...
  34. qnach

    A Partition function in number theory and statistical mechanics

    Is the partition function used in number theory and statistical mechanics the same thing?
  35. AndreasC

    Difficulty with Lagrange multipliers in Kardar's Statistical Physics book

    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...
  36. thaiqi

    Deriving Statistical Behavior of Particles via Classical Mechanics

    Hello, using computation simulation, can the statistical behavior of many particles be derived through deterministic classical mechanics?
  37. P

    I Tell me how scientific journals scrutinize statistical work

    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...
  38. dontknow

    A Statistical physics : Irreversibility

    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...
  39. S

    Courses Statistical Physics vs QFT vs General relativity

    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...
  40. C

    Do any electric lights have statistical lifetimes X~Exp(λ)?

    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...
  41. L

    Dispersion relation statistical mechanics

    I know that this is the general case of the Einstein and Debye Model.
  42. Haorong Wu

    I Exploring Entropy in Intro to Statistical Physics by Huang

    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 (...
  43. Pispi Choudhury

    Courses Math & Physics Courses for Quantum & Statistical Field Theory

    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.
  44. tanaygupta2000

    Statistical Mechanics: Two systems reaching an equilibrium temperature

    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) =...
  45. tanaygupta2000

    Statistical Mechanics Occupation number

    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...
  46. tanaygupta2000

    Statistical Mechanics: Four non-interacting particles are confined in a box

    Regarding the first part, I proceeded as: nx ny nz 4 0 0 => E1 = 16C 0 4 0 => E2 = 16C 0 0 4 => E3 = 16C 3 1 0 => E4 = 10C 3 0 1 => E5 = 10C 0 3 1 => E6 = 10C 1 3 0 => E7 = 10C 0 1 3 => E8 = 10C 1...
  47. T

    A Three Body Problem - Statistical (almost) Solution

    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
  48. PeterDonis

    I Does the statistical weight of data depend on the generating process?

    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...
  49. A

    I Interpretations of QM vs. statistical physics as an "interpretation"?

    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...
  50. A

    Does classical statistical physics predict newer things vs. thermodynamics?

    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...
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