# What is Mean: Definition and 1000 Discussions

There are several kinds of mean in mathematics, especially in statistics:
For a data set, the arithmetic mean, also known as average or arithmetic average, is a central value of a finite set of numbers: specifically, the sum of the values divided by the number of values. The arithmetic mean of a set of numbers x1, x2, ..., xn is typically denoted by

x
¯

{\displaystyle {\bar {x}}}
. If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is the sample mean (denoted

x
¯

{\displaystyle {\bar {x}}}
) to distinguish it from the mean, or expected value, of the underlying distribution, the population mean (denoted

μ

{\displaystyle \mu }
or

μ

x

{\displaystyle \mu _{x}}
).In probability and statistics, the population mean, or expected value, is a measure of the central tendency either of a probability distribution or of a random variable characterized by that distribution. In a discrete probability distribution of a random variable X, the mean is equal to the sum over every possible value weighted by the probability of that value; that is, it is computed by taking the product of each possible value x of X and its probability p(x), and then adding all these products together, giving

μ
=

x
p
(
x
)
.
.
.
.

{\displaystyle \mu =\sum xp(x)....}
. An analogous formula applies to the case of a continuous probability distribution. Not every probability distribution has a defined mean (see the Cauchy distribution for an example). Moreover, the mean can be infinite for some distributions.
For a finite population, the population mean of a property is equal to the arithmetic mean of the given property, while considering every member of the population. For example, the population mean height is equal to the sum of the heights of every individual—divided by the total number of individuals. The sample mean may differ from the population mean, especially for small samples. The law of large numbers states that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean.Outside probability and statistics, a wide range of other notions of mean are often used in geometry and mathematical analysis; examples are given below.

View More On Wikipedia.org
1. ### B Mean Free Path And Ball Bearings In A Box

I recently came across the concept of 'mean free path', and some similar concepts in thermodynamics (the depth of my understanding is very shallow though, which is why I'm here). This is very much a shower though, so: Suppose I have a box filled with some some metal ball bearings, which I...
2. ### Test whether the diets are different from one another at ##α=5\%##

Looking at stats today, In my working i have; Let ##H_0 = μ_1=μ_2## v/s ##H_1 = μ_1-μ_2≠ 0## then, ##\bar x = \dfrac{134+83+...+123}{12}=120## ##\bar y = \dfrac{70+118...+94}{7}=101## ##t=\dfrac{\bar x- \bar y}{S_p ⋅\sqrt {\dfrac{1}{n_1}+\dfrac{1}{n_2}}}## ##t=\dfrac{120-101}{21.21...
3. ### A Finding Minimal Mean Distance Curves on the Unit Sphere

**Problem:** Find parametric equations for a simple closed curve of length 4π on the unit sphere which minimizes the mean spherical distance from the curve to the sphere; the solution must include proof of minimization. Can you solve this problem with arbitrary L > 2π instead of 4π? There...
4. ### B Arithmetic mean of an infinite number of points?

So I was thinking about arithmetic, geometric and harmonic means when I had a thought. Let's say we have a curve y = x^2. We want to find the AM of the points on the curve between x=1 and x=2 i.e. y = 1 and y = 4. To make thing easier, we'll start with just the endpoints and keep adding...
5. ### B How is the harmonic mean affected by additional data points?

We have a collection of 8 discrete data points. They are: 10, 20, 30, 20, 30, 40, 30, 40 In increasing order: 10, 20*2, 30*3, 40*2 The harmonic mean of this data series is 22.86 I read on Wikipedia that the harmonic mean is skewed towards the smaller values i.e. smaller values will affect the...
6. ### B What are the energy eigenvalues of a harmonic oscillator?

Is it a total energy of a vibrating molecule? So is it a sum of potential and kinetic energy? Or it is only a total energy of a vibrational motion of the molecule? Or is it only a potencial energy, when it is related to a dissociation curve? I am confused.
7. ### B What does this distance of an atom mean?

I can look at it as if a vibrational motion of the atoms was a simle harmonic motion. So I can consider one of the two atoms to be at rest and the second one to vibrate. Its deviation can be written as ##x(t)=r(t)-r_0##. When I know that the hydrogen molecule stops exiting when the range of...
8. ### Does anyone have a table of Earth mean temperatures?

I would like a table of Earth mean temperatures going back as far as available. Where can I find that?
9. ### What Is the Significance of PF Signatures in Historical Physics Debates?

Physics did not begin in London in 1686. It began 2 centuries later in 1874 in Kristiania. @fresh_42 Sorry for the blatant off-topic, please explain this text in your signature. Thanks!
10. ### I What does it mean that symmetries do not hold globally?

Perhaps this is a stupid question but, if Lorentz symmetry and time translational symmetry are not global in an expanding universe, wouldn't that mean that is possible that other Hubble spheres outside our observable universe could have other symmetries or an absence of the Lorentz symmetry? I...
11. ### B 'Universe Breaking' results from JWST -- What does this mean?

Hello. I was wondering if the experts here would like to comment on this news? Thus far I've been unable to find a link to any science paper. Thank you, Cerenkov.
12. ### B In uniform acceleration, mean velocity ##\bar v = \frac{v_0+v}{2}##?

Question : For uniformly accelerated motion ##a(t)=a_0\;\; \forall \text{times}\;t##, we can say that the average velocity for the entire motion ##\bar v = \frac{v_0+v}{2}##, where ##v(t)## is the final velocity at some time ##t## and ##v_0## is the initial velocity. How do we show that? Issue...
13. ### I Lack of Symmetry in X & T: Explained for Beginners

Hello everyone, I'm reading "Special Relativity For the Enthusiastic Beginner" (by David Morin) and so far I really like the book, but I have a little bit of a hiccup in understanding terminology regarding stated symmetries or the lack there of in reference to the Galilean transformation when...
14. ### What does a numerical rating mean? Mechanical equipment or device

A certain mechanical appliance or device is given a rating that says, a certain operation or a certain movement can be performed some specified thousand times. What does the rating really mean? Was that movement or operation performed until the piece failed? Does it mean some advanced test...
15. ### Chemistry Derive an expression for mean free path from survival eqn exp(-x/λ)

If the distance between the centres of two molecules is σ, then imagining a a cylinder with radius σ the number of molecules can be given by πσ²cn where c = average velocity. So mean free path can be given by λ = c/πσ²cn = 1/nπσ². But do I derive it from exp(-x/λ)?
16. ### JavaScript What do the x and y values mean here? Can you show in a figure?

function checkBoundaryCollision() { if (x - ballRadius <= 0 || x + ballRadius >= canvasWidth) { dx = -dx; } if (y - ballRadius <= 0 || y + ballRadius >= canvasHeight) { dy = -dy; } } Full code here:
17. ### What does weak chemisorption mean?

I am reading a research paper that says weak chemisorption of silver with olefins basically means more effective olefin/paraffin separation. If silver weakly binds to olefins, wouldn't that make a lousy separator?
18. ### Are both sample spaces the same or do they mean different things?

I came up with two different forms of the sample space S, but I am not sure if they mean the same thing or the first one could mean something different. H stands for heads showing up and T stands for tails showing up. $$S = \{ \{i,j\}: i \in \{H,T\}, j \in \{H,T\} \}$$  S = \{ (i,j) : i...
19. ### I What is the significance of recoupling in spin network theories?

Hi Pfs My question is not technical. I found papers about recoupling theories about spin networks. i see what they are about but why this word "recoupling"? Was there before a decoupling and recoupling whar?
20. ### I Time-Like Events: Spatial Order Not Relevant?

Hi, I'm having trouble understanding that for time-like events where the order of time is absolute, these events can be in different spatial orders depending on reference frame. Can someone provide an example please? Thank you!
21. ### DC motors and Induction motor: What does "no load conditions" mean?

TL;DR Summary: So I'm just confused when the question asks me to solve for the no load speed of DC motors and induction motors. Does no load condition mean that the output torque (Tout) is zero? This is what I was assuming so far for both DC and induction motor. Is no load condition the same...
22. ### I Prove that the geometric mean is always the same

Given are a fixed point ##P## and a fixed circle ##c## with the radius ##r##. Point ##P## can be anywhere inside or outside the circle. I now draw two arbitrary lines ##l_1## and ##l_2## through the point ##P## in such a way, that both lines intersect with the circle ##c## in two distinct...
23. ### In a ALU in a simple processor, what does the operation BNEQZ mean?

TL;DR Summary: So I'm doing this coding assignment that involves a simple processor with an ALU. I have a list of operation of the ALU but I don't understand one operation called BNEQZ. I have the list of operation below and an example of it being used to form some loop. Would be very grateful...
24. ### Input & Output Explained: What Does It Mean & Why Is It Relevant?

Can someone explain the question so I can attempt a solution? What does it mean by input and output and why is that relevant?
25. ### What does the identity mean here?

I don't understand why the identity is mentioned in the group's definition and how I am supposed to incorporate it into the table. I honestly have missed some lectures on Linear Algebra, and I can't find any examples or definitions for this in the prof's notes. I'd appreciate some help for sure...
26. ### I What do the Roman Numerals mean in Spectroscopic Data?

A basic question. Looking at the NIST spectroscopic data, what exactly is, for example, Ar I vs Ar II vs Ar III? If Ar I is unionised Argon, then is Ar II an Ar- ion or an Ar+ ion? (and whichever way around it works, how do we denote the opposite ionisation? If they are all ionized, is there...
27. ### I Measuring Time & Spatial Distances: Timelike vs Spacelike

My book writes " The geometric distinction between timelike and spacelike distances is mirrored in the devices used to measure them. A clock is a device that measures timelike distances; a ruler is a device for measuring spacelike ones. Two nearby points or a timelike world line are timelike...
28. ### A What does Rovelli mean with "oriented and ordered graph"?

Hi Pfs Rovelli writes this in his book (Qunatum Gravity) about spin networks: Given an oriented and ordered graph there is a finite disgrete group of maps that change its order or orientation and that can be obtained as a diffeomorphism. A link is equipped the source and target functions. this...
29. ### Find a 95% confidence interval for population mean

I am refreshing on this... I think there is a mistake on the circled part in red...right? not correct symbol for sample mean...This is the part that i need clarity on. The other steps to solution are pretty easy to follow...as long as one knows the t-formula and also the knowledge to...
30. ### What does G mean in general relativity?

In Newtonian mechanics, G is simply a proportionality constant or the force with which two bodies of unit mass attract each other. However, GR doesn't treat gravity as a force. So how is G defined in GR? Is it a property of spacetime or just some useless mathematical artefact? What does G...
31. ### What does the differential equation answer mean?

Hi, last semester I "solved" a full differential equation and the answer was (see the picture). What does it mean? Can I make a graphic with it or what? I really don't get it. *Arrows are just a continuation of the main formula*
32. ### B What does "1 diopter of astigmatism" mean?

The measurement for astigmatism is diopter. But what does 1 diopter of astigmatism mean? Thanks a lot if you help me figure this out!
33. ### Nuclear Fusion Breakthrough....What Does This Mean?

Nuclear Fusion Breakthrough Confirmed: California Team Achieved Ignition BY JESS THOMSON ON 8/12/22 AT 9:20 AM EDT https://www.newsweek.com/nuclear-fusion-energy-milestone-ignition-confirmed-california-1733238 For those of us not literate in the field, how significant is this and what are the...
34. ### Find the correct value of the mean in the given problem

Find the problem and solution below; Now the part i do not seem to understand from the given solution is the negative value i.e to be specific ##-0.06## In my understanding we shall have ##5## possibilities with each giving the correct mean value of ##3.48## which implies ##+0.06## from the...
35. ### I Connection between mean and median

I have 100 random real (even rational with only one decimal, like average temperatures of months at a particular weather station) numbers. With them I compute the arithmetical mean and the median. It is a (very) small probability they are the same number within let's say 0,1 or 0,2. Question...
36. ### A Probability flux integrated over all space is mean momentum?

In Sakurai Modern Quantum Mechanics, I came across a statement which says probabiliy flux integrated over all space is just the mean momentum (eq 2.192 below). I was wondering if anybody can help me explain how this is obtained. I can see that ##i\hbar\nabla## is taken as the ##\mathbf{p}##...
37. ### I Spatially Homogeneous Scalar Field on Spacetime: Showing $\nabla^2 f$

If ##f## is a "spatially homogeneous" scalar field on spacetime ##ds^2 = dt^2 - a^2(t) \delta_{ij} dx^i dx^j## then show that ##\nabla^2 f = \ddot{f} + 3H \dot{f}##. Should be easy if I knew what the condition on ##f## is, i.e. ##\nabla^2 f = \partial_{\mu} \partial^{\mu} f = \ddot{f}- a^{-2}(t)...
38. ### MHB TEST OF HYPOTHESIS INVOLVING THE POPULATION MEAN 𝝁 WHEN THE VARIANCE IS UNKNOWN

1. You are working in a company facing attrition problems of the customer service representatives for the past five years. The company president proposed that if the attrition rate is at least 10 per month, then the salary scale, compensation package, and professional development programs for...
39. ### MHB Test of hypothesis involving the population mean 𝝁 when the variance is known

1. The Head of the Mathematics Department announced that the mean score of Grade 11 students in the second periodical test in Statistics was 89, and the standard deviation was 12. One student believed that the mean score was less than this, randomly selected 34 students, computed their mean...
40. ### I How to 'reverse' derive correct value when given MEAN + SD

I have a scatterplot that I'm trying to extract. I found 54 values out of 55. There is one "missing" value, probably because it is overlapped and I can't actually see. I have the MEAN and Stand Dev of the 55 values. Is there a way to reverse find the one that is missing? I mean, is there a...
41. ### What does it mean that MySQL is a server application?

I would like to practice SQL and install MySQL, one of the most popular open source DMBS. MySQL workbench can also to be installed and used as a visual GUI to interact with MySQL which in turns interacts with the database. My question: is MySQL a server application? If so, what does it mean? I...
42. ### I Why Can We Use This Mathematical Formula for Mean Value of Measurement?

Hi, I'm new to the quantum world, and would like to know why mathematically can we say that for mean value of measurment: <T>=<phi|T|phi> ?

44. ### Find the mode, median and mean of the number of children

See attached question and markscheme. Solutions Now they give the Mode as ##1## i am not getting this... My understanding 'maybe its the English used' is that we have ##0## children living in 3 houses, 7 houses have ##1## kid each bringing total number of kids to 7!, 5 houses have ##2##...
45. ### Understanding the Mean value theorem

Can we have two tangents (two turning points) within the given two end points just asking? I know the theorem holds when there is a tangent to a point ##c## and a secant line joining the two end points. Or Theorem only holds for one tangent point. Cheers
46. ### I What does "upright" mean in geometric optics?

if someone want to explain to me what is an upright image ? , and what are the other adjectives to define an image in geometric optics and their meaning , Thanks .
47. ### What does "Low strength but good strength-to-weight ratio" mean?

Good strength-to-weight-ratio
48. ### B Does more energy mean more work?

As we always say eat food and rest because tomorrow you have to go to work. When we throw a ball up, it gains a max PE at some height. More the height more energy it has which makes it do more work. Is this right to say?
49. ### I Finding Momentum Mean & Variance from Wavefunction

I've a Gaussian momentum space wavefunction as ##\phi(p)=\left(\frac{1}{2 \pi \beta^{2}}\right)^{1 / 4} e^{-\left(p-p_{0}\right)^{2} / 4 \beta^{2}}## So that ##|\phi(p)|^{2}=\frac{e^{-\left(p-p_{0}\right)^{2} / 2 \beta^{2}}}{\beta \sqrt{2 \pi}}## Also then ##\psi(x, t)=\frac{1}{\sqrt{2 \pi...
50. ### I How Is Uncertainty Calculated for the Mean of a Gaussian Function in MATLAB?

Hello! If I have N points (x,y) which I know they are described by a Gaussian i.e. y(x) is a Gaussian of unknown mean and standard deviation, and each y has an associated error of ##\sqrt{y}##, is there a general formula for the uncertainty on the mean of this function? Thank you!