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.
Hello all,
I need to find the statistical significance (95% confidence interval) for 3 independent samples. From what I understand, in order to find the statistical significance for more than 2 samples, ANOVA (ANalysis Of VAriance) is employed.
ANOVA fits in nicely with what I want to do...
This shouldn't be too hard but I'm struggling.
Consider N identical particles in a box of volume V. The relation between their linear momentum and kinetic energy is given by E = c\left|\overrightarrow{p}\right|, where c is the speed of light. So, the Hamiltonian of the system is...
I've seen an approximation in a statistical mechanics book, given without any proof:
\frac{<N!>}{<(N-n)!>} = <N^n> (1 + \mathcal{O}( \frac{1}{<N!>} ))
I've been trying to work out a proof, but I'm simply stuck :yuck: any ideas how it can be proved?
I think I get the concept of fermi-dirac and bose-einstein statistics can follow the derivation of their distribution functions as per Stastical physics by Guénault but I'm having severe trouble trying to apply them:redface:.For what I imagine is a simple question, two identical particles are...
Hi.
Can anyone explain the meaning of this formula from statistical physics to me:
S = -k\sum_r{p_r\ln p_r}
Ok, I know that S is the entropy, the p's are probabilities of some sort - but somehow this is not satisfactory :-)
Supposing I have a face-down card containing one of the 26 english alphabet, but I don't know which one. Hence, the entropy is kln 26.
But, if I were to open the card, now I know exactly what the alphabet is. Hence, the entropy now is zero.
How can this be in view of the second law?
I am trying to make the connection from statistical mechanics to thermodynamics for the isothermal isobaric ensemble. Partition function = (sum of)exp(-BEj-gamma*Vj).
I have followed T.L. Hill [Statistical Mechanics, p. 67] but can not understand how he justifies dE=(sum of)EdP, rather than...
Hi.
I'm having trouble with this statistical physics thing again. I am given this exercise:
Problem 9 – A spin model
In a solid at temperature T the atoms have spin 1 so that the m quantum number takes on the values m = 0, ±1. Due to an interaction with the electrostatic field in the...
I don't like to assume things without checking..
Is the distribution really a 'normal distribution' aka Gaussian? Is this an experimental result or a 'good enough' assumption?
I understand that a microstate is one possible distribution of energies that make up the system's total energy. But I can't understand why, in a monatomic gas for example (where there is only translational kinetic energy of atoms), there is a finite number of states. Surely that would mean that...
Heres the problem:
When asked to explain the meaning of "statistically significant at the 0.05 level," a student says, "This means there is only probability 0.05 that the null hypothesis is true." Is this an essentially correct explanation of statistical significance? Explain.
Help...
Hi.
I need some useful and conceptual book in advanced statistical physics.
I can understand the mathematics in some famous book such as Pathria, but I have trouble understanding some concepts in that area.
Any suggestion?!
Thanks in advance.
Somy
If 408 people try to guess someone's birthday, how do you calculate the chance of 13 of them being right? http://www.greasypalm.co.uk/gpforum/forum14/363.html
I was searching for references on statistics and found this -
StatSoft Electronic Texbook - http://www.statsoft.com/textbook/stathome.html
Elementary Concepts in Statistics - http://www.statsoft.com/textbook/esc.html
Basic Statistics - http://www.statsoft.com/textbook/stbasic.html
Hello Gentlemen,
Here's a statistical question for you:
A man and a woman mate to produce a child that either has blue eyes (B) or green eyes (b), where blue eyes are dominant to green eyes.
As you remember from biology, there are 3 different possibilities for the genotype - BB, Bb, or...
Let be an statistical system of N particles with their Hamiltonian..
H=\sum_{i=0}^{N}\frac{p_{i}^{2}}{2m}+V(q1,q2,...,qN)
then you could obtain their equations of motion in the form:
dp_{i}/dt=-dH/dp_{i} and dq_{i}/dt=p_{i}/m
but of course if N is big you could take years and...
This isn't a homework question. I'm studying this from a book (Thermal Physics by Kittel & Kroemer) currently. Up til now I've had no problem following it. There's one derivation that's got me a little stumped however. I had thought my calculus was proficient enough, but I'm just not seeing...
it's just not sinking in.. i know a cell in phase space has 6 dimensions, 3 for momentum and the other 3 for position.
but i'd like to understand it(phase space). can someone give me an example maybe or tell me why this constuct is needed?? or a link to a very good description?
Ok I am having a really annoying time with my father here. Now statistically, you are much more likely to get in an accident in a car then in a small airplane per mile traveled. Now my father thinks that its better to say that its more dangerous to have a small airplane since you are more likely...
I understand that there are a couple of statistical models out there that describe physical systems. One I know is Fermi-Dirac statistics. What are the other models, what are their key features and when are they applied? When working with a system, how can you be sure you should be using this...
Hi, I am just starting a 3rd year course in Statistical Mechanics, and am a bit confused about the operator trace, Tr. I know there is a trace for quantum operators, as well as one in classical physics, but i am not sure how to calculate either, or their physical meaning. Any help would be...
I have studied statistical mechanics using F. Reif's book, and learned a lot, but there are still a couple very fundamental questions which still elude me. If anyone would be willing to share some insight, I would really appreciate it!
So Reif early on discusses the fundamental postulate of...
I'm searching for a statistical mechanics textbook. It should be roughly at or above the advanced undergraduate level, have a decent set of problems, be reasonably thorough, and be ideal for self-study. I would be grateful for any recommendations.
You have two tools:
1. A straight edge
2. The ability to judge any distance to within 20%, or any angle to within 20 degrees.
What can you construct with these tools, with an unlimited? In particular, is there a method for constructing an approximate circle that tends towards perfect...
McNally's Statistical Constant: No set of statistics can be more than 6.25% accurate
This may be represented by the mathematical equation -
A(SS) = A(D) x E(M) x B(J -->N) x I(I <--N)
Where
A(SS) is the ACCURACY of any (SET of STATISTICS)
A(D) is the ACCURACY of the (DATA)...
I wish to know about web sites or other resourses from where i can get solutions for all the end of the chapter problems for the book on statistical Mechanics by R K Pathria
I am mixed up about thermal equilibrium in statistical physics. And I hope you excuse me if I use unconventional words, I am from Sweden, my book is in german and I try to express myself in english.
In my book (Noltings "Grundkurs theoretische Physik, Band 6") thermal equilibrium is defined...
1. If X1, X2, X3, X4 are a random samle from a normal distribution with mean 17, then what is the distribution of 2(X - 10) / S where X should be X bar.
Our notes are just awful for this topic.
Any tips how to proceed with this one, and what is S?
2. Let X1, X2, ... be a sequence of...
Does "equilibrium" imply max. entropy in statistical mechanics?
I've gotten myself confused thinking about the meaning of "equilibrium" in statistical mechanics. I thought I remembered that an isolated system at equilibrium is equally likely to be in any possible microstate, which means there...
In my Statistical Physics and Entropy module, we did something about atoms in a box with an imaginary partition down the middle, so atoms could either be on the left or the right. If there were 4 atoms in the box, the system would have 5 macrostates. If there were 8, there would be 9.
Is is...
Hi all.
So something's bothering me;
Given the def. S=-k(sum on r: pr*ln(pr)) for the entropy, find the entropy of N>>1 free particles moving in a 2d box with energy E.
Now, don't I have a continuum of states here? How do I do that?
Thanks :smile:
Please Help- A problem in statistical mechanics
In a gas at STP, let p(r)dr be the probability that an atom has a nearest neighbor between distances r and r+dr. Find p(r).
I am struggling with this question. For STP I can find the particle density. But where do I go from there? Do I need...
I have a chemical reaction at equilibrium " cis <--> trans " at 300K
The energydifference between the two states is 4,7 kJ/mol and cis has the highest energy. I want to find out how many molecules that is in cis- and how many that is in trans-state?
Thankful for tips!
Regards
Daniel
Now this may sound rather stupid, but I'm just curious:
Of the people who interact with and use this site, how many are
INTPs or INTJs? Do those temperaments form the majority (or the largest subgroup) of this site's users? These are two of the possible 16 results of the ""Keirsey...
This may be too much work, so I understand if no one tackles the problem...
I'm doing an analysis of the power usage of a building. Sheet 2 of the attached spreadsheet shows typical data (we have 4 years of it). The top graph is the kWh used every 30 minutes of every day in January, 2003...
I got into a discussion about the "arrow of time" recently, and one point I brought up is that for a system governed by time-symmetric laws, if you place no special restrictions on the initial conditions but instead pick an initial state randomly from the system's entire phase space, you will be...
According to QM, a diatomic gas molecule possesses rotational energy levels given by En = (1/2I)(h^2)n(n + 1). h is meant to be h-bar, Planck's constant over 2π here and I = moment of inertia. Energy level n has a degeneracy of 2n + 1.
Find the partition function of the rotational motion of a...
I have two questions, I've completed. I am partially sure that the answers I've obtained are correct, and all I really want is confirmation on whether they are correct or not. If not, what am I doing wrong?
Question 1:
If the joint probability distribution of X1 and X2 is given by...
Consider 2 systems of spin 1/2 paramagnets, which may point either up or down wrt a magnetic field. The first system contains 8 paramagnets and the second contains 6 paramagnets.
Suppose the energy of the combined system is constrained such that the total number of spins pointing up in the 2...
Hi everyone, I have two questions from my latest homework set that are driving me nuts, so here goes:
1) "Recalling that the Fermi-Dirac distribution function applies to all fermions, including protons and neutrons, each of which have spin 1/2, consider a nucleus of 22Ne consisting of 10...
Imagine a number of finite universes superimposing to realize our apparent physical situation. For us, their many accessible states are expressed by the behavior of corresponding probability waves.
Coexisting universes, including even those representing different physics (yet retaining a...
Lately I've been reading a lot of statistical physics and I really enjoy it. I was curious what sorts of physics more study in this area would lead me to. Condensded matter? Solid state?
I would just like to know what exactly is the basis of the Chaos Theory; has it anything to do with the idea of statistical anentropy? Could someone give me an introduction to these topics? (I am a 15 year old going into a Honors Physics course in high school, thinking maybe I would like to...
http://nedwww.ipac.caltech.edu/level5/Sept03/Feigelson/paper.pdf
PhyStat 2003: Statistical Problems in Particle Physics, Astrophysics, and Cosmology
Statistical Challenges in Modern Astronomy
E. D. Feigelson
Department of Astronomy & Astrophysics, Penn State University, University...
I have got a bit of problem here. I have got 4 sets of results for a biology experiment, and I want to find the correlation between them. So I read about Spearman's rank coefficient but it only works for two sets of results. Can somebody help please?