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# Equilibrium shape of string

1. Apr 11, 2014

### CAF123

1. The problem statement, all variables and given/known data
An inextensible string of length $\ell$, whose mass per unit length is $\rho$ is hung between two points at the same height a distance $d$ apart ($d < \ell)$. Using the Euler-Lagrange equations and the first integrals, find the shape of the string when it is in equilibrium.

Relate the constants of integration to $\ell$ and $d$.

2. Relevant equations
Need to find a function$F$ such that $I[y] = \int_{x_1}^{x_2} F(y,y',x)\,dx$ and $I[y]$ is minimized. For $F \neq F(x),$ $$y' \frac{\partial F}{\partial y'} - F = \text{const}$$

3. The attempt at a solution
Since l>d, the string will be elongated or distorted between the two ends. Suppose the shape is described by $y(x)$. In equilibrium, the net force on each segment of the string will vanish. Equivalently, the potential energy of each segment of the string is at a minimum and this is easier to deal with.

Take the zero of potential to be at the highest point of the string between the two supports. Then a small segment d$\ell$ of the string will be of length $\sqrt{1+y'^2}$. So $dm = \rho d\ell = \rho \sqrt{1+y'^2}$. Since zero of potential was assigned at the highest point, all points on the string are below this, so the potential energy of a segment is then $U = - \rho \sqrt{1+y'^2}gy$, with the y axis vertical. Is this correct?

If I then use Euler-Lagrange, since $U \neq U(x)$, I use the formula in the relevant equations to get that the equilibrium length satisfies $y/\sqrt{1+y'^2} = \text{const}$, which when integrated gives $y$ ~ $\cosh x$, which seems reasonable given the form of this graph.

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Last edited: Apr 11, 2014
2. Apr 11, 2014

### BvU

Looks good to me. Cosh is also called Catenary

3. Apr 11, 2014

### CAF123

My integral is $$\int_{y_o}^{y} \frac{\text{d}y'}{\sqrt{\left(\frac{y'}{C}\right)^2 - 1}} = \int_{x_o}^x \text{d}x',$$ where $C$ is the constant on the RHS of the Euler-Lagrange first integral.

I set $y_o = x_o = 0$ arbritarily in my set up. Carrying the integration through, I obtain $C(\theta(y) - \theta(y_o)) = x - x_o$ using the substitution $y = C \cosh \theta$. My problem is there does not exist a $\theta$ satisfying $y_o = 0 = C \cosh \theta$, so I cannot change the integration limits.

4. Apr 12, 2014

### CAF123

In terms of relating the constants of integration to l and d (the second part of the question), I can see one boundary condition that is y(x = 0, d) = 0. (it was given both ends were at the same height at both ends, and in my set up I chose these points to coincide with the x axis). I only had one integration constant, so this should be enough but I did not get any relation involving l.

5. Apr 12, 2014

### CAF123

Is there something wrong with setting $x_o = y_o = 0$? Physically I think no, but if I am to use the substitution I did then maybe I need to change this.

6. Apr 14, 2014

### BvU

A few questions to get you unstuck again:
Whatever happened to $\ell$ itself ? Can't just disappear !
Why the integral from $y_0$ to $y$ (meaning from 0 to 0 ?) and not from $y'_0$ to $y'_{x_2}$ ?
And (from post 4): "y(x = 0, d) = 0" is really two boundary conditions, rght ?
Then: you introduce $\theta$ with $y = C \cosh \theta$; what is $\theta$ ?
Previously you had $y$ ~ $\cosh x$; would $y$ ~ $\cosh (x-d/2)$ be OK too ?

From post 1: $dm = \rho d\ell = \rho \sqrt{1+y'^2}$, but you mean $\rho \sqrt{1+y'^2}\, dx$, right ? You can't have a differential on the left and not on the right hand side !
Similarly: $U = - \rho \sqrt{1+y'^2}gy$, is a value on the left and function on the right. You mean $U = \int_0^d F dx \$ with $F = - \rho \sqrt{1+y'^2}gy\$, right ?

7. Apr 14, 2014

### CAF123

Indeed, so a small segment of string of length dl contributes $dU = -\rho \sqrt{1+y'^2} gy \,dx$ to the total potential energy of the system. To obtain the total potential energy of the system, integrate over all x. I.e $U = \int_0^d -\rho \sqrt{1+y'^2} gy \,dx$. Now, since the integrand is not a function of x, I can still apply the relevant Euler-Lagrange first integral.

Yes, I think I need to incorporate it into the boundary conditions.
$y_o=0$ and $y$ stood for some arbritary y in the range from 0 to the y value corresponding to the lowest point of the string.
Yes.
I did not really think of a physical realization of $\theta$, I was using the substitution in order to solve the integral.

8. Apr 14, 2014

### BvU

Ah, I'm sorry, you are doing the right thing. I got confused by your using $y'$ for $dy/dx$ on one occasion and for the name of an integration variable on another.

Indeed the "equilibrium length" (i.e. the functional that gives an extremum for $U$) satisfies the differential equation $${\rho g \, y \over \sqrt{1+y'^2} } = c,$$ which solves to $${dx\over dy} = { \bar c \over\sqrt{y^2-\bar c^2}}$$with $\bar c ={c\over \rho g}$.

Then you set $y = \bar c \,\cosh \theta$, hence $dy = \bar c \, \sinh \theta d\theta$ to get an integral over $\bar \theta$ :
$$x = \bar c \int_{\theta_0}^\theta {\sinh \bar\theta\over \sinh \bar\theta}\, d\bar\theta = \bar c \, \theta + b$$

But now I've painted myself in the same corner as you. There must be something wrong with our y: The minimum surface of revolution has the same differential equation and the same cosh answer, but there is of course a condition y > 0 (see e.g. page 6 of http://math.hunter.cuny.edu/mbenders/cofv.pdf) that we are lacking. We can shift the x such that e.g. b = 0, but I don't know about y.

And I also need $\ell$ back in, probably via the $\int ds$...

9. Apr 15, 2014

### BvU

Found a solution that indeed makes use of the constraint, but I don't know if it is going to make you happy.

Credit to Frank Porter at CalTech for this link that combines Euler-Lagrange with Lagrange multipliers method for optimization with the $\int d\bar \ell = \ell$ constraint.

$\bar c$ and b as integration constants now get company from a third unknown (the Lagrange multiplier $\lambda$), so that y0=0, y2=0 and $\ell$ can be accomodated comfortably.

10. Apr 15, 2014

### CAF123

I think I can use the boundary conditions to determine $\bar{c}$ and $b = -\bar{c}\theta_0$, but the problem is I cannot do this in practice because there is no theta satisfying coshθ=0.

Also, I am yet to find a boundary condition involving l. (as per the question requirements).

Edit: we typed past each other - thanks, but the solution uses Lagrange multipliers and I didn't cover this in my course. I am speaking to my professor on Thursday and I will ask him then unless someone else chips in. Then I will let you know.

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text:
MAT334--2020S > Quiz 1

Q1: TUT 0201

(1/1)

Ziyi Wang:
Questions:
Write the equation of the perpendicular bisector of the line segment joining $-1+2i$ and $1-2i$.
Let $z = x + iy$.
Since $z$ is an arbitrary point on the perpendicular bisector of the line segment joining $-1+2i$ and $1-2i$, we know that the distance between $z$ and $-1+2i$ is same as the distance between $z$ and $1-2i$.
Then, we can calculate that:
$$|z-(-1+2i)| = |z - (1-2i)|$$
$$|(x + iy)-(-1+2i)| = |(x + iy) - (1-2i)|$$
$$|(x + 1)+(y-2)i| = |(x - 1) + (y + 2)i|$$
$$\sqrt{(x+1)^2 + (y-2)^2} = \sqrt{(x - 1)^2 + (y + 2)^2}$$
$$(x+1)^2 + (y-2)^2 = (x - 1)^2 + (y + 2)^2$$
$$x^2 + 2x + 1 + y^2 - 4y + 4 = x^2 - 2x + 1 + y^2 + 4y + 4$$
$$4x = 8y$$
$$y = \frac{1}{2} x$$
Then, we write the equation in complex number notation:
$$Re((\frac{1}{2} + i)z) = 0$$

format: plain


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# Defining dynamic LyX shortcuts

I use LyX especially for writing math, and I created my own bind file for my preferred shortcuts.

Among other commands, I defined shortcuts that insert a command-sequence, for inserting long data easily. for example, I have the following shortcut assigned:

\bind "C-S-L x 1 n" "command-sequence self-insert x; math-subscript; self-insert 1; char-forward; self-insert ,; math-insert \ldots; self-insert ,; self-insert x; math-subscript; self-insert n; char-forward;"

This sequence generates a list of variables, named x_1 to x_n, separated by dots and commas. As can be seen, I use a shortcut that defines exactly the variable type I want to insert, and the needed range for the sequeunce. Soon enough, I realized I want all sorts of these sequences that only differ in their variable name ('x', here) and their range (1 to n, here).

My question is - is there any way to define a 'dynamic' shortcut, in that sense it will generate the wanted sequence according to three given arguments, instead of defining each of these as a separate shortcut?

Thanks for the help.

-

Why don't you use math macros? In LyX menu: Help/Math/22.2 Math macros. If you still need a special shortcut afterwards, you just bind the shortcut to inserting the command of your macro and that should simplify editing even more.

-
thanks for the tip. I still prefer to use command sequences as I described above, since inserting them gives exactly the requested expression. Using math macro, though, as far as I get it, only lets me create some kind of patterns. After inserting a macro I have to go back and fill in the data manually - and that's exactly what I don't want to do (mostly for typing speed reasons). Besides, macro are only available in the document where they were defined - and that means redefining them for each document, but binding a command sequence is available globally. –  Mosheke Jul 13 '12 at 16:12
@Moasheke: I am not sure I understand what you mean. A math-macro defines a new command with the name of the math macro. A macro can have zero arguments, i.e., it defines a static math expression. It can however also have arguments so that your macro can be arbitrarily parametrised. Now let's say you need all sorts of expressions of the template <var>_1,\ldots,<var>_<range>. You define a macro with two arguments for <var> and <range> and all occurences of those will get automagically filled in. Macros can be reused as well. –  walkmanyi Jul 13 '12 at 22:43
I understand that macros can be set to be fixed expressions ("zero arguments") or some kind of templates (taking arguments). Still, when using a macro, this template is inserted and I have to manually fill in the arguments. Is there any way to give the arguments together with the macro name in advance - so as soon as the expression is generated it will hold the given args? otherwise it means going (almost) from box to box filling them up - which doesn't take too long, but I just want to make sure. Also, how do I use macro in another document without redefining it every time? tnx. –  Mosheke Jul 14 '12 at 23:08
Macros: I think the best way to find out is to play with it. Again: macros are in fact just new commands. I.e., macro is not only a template. Consider it an arbitrary expression optionally with placeholders for parameters of your interest (variable name, index range). As for the second question regarding macro re-use, you are right, you would need to redefine it again and again. But in my case copy&paste are working just OK for the set of macros I use often. –  walkmanyi Jul 14 '12 at 23:25
Is there some way to get LyX to not replace \foo from a two argument macro \foo{#1}{#2} with the LyX graphical representation before both arguments have been typed in as well? –  Rasmus May 10 at 21:29


text: Just a hint, for now: if I understand you correctly, you're trying to use v1, v2, v3 as the indeterminates of a (boolean) polynomial whose coefficients are themselves boolean polynomials. Try to implement that...


text:
tisdag 17 mars 2020

DigiMat Web Education

The Corona crisis asks for school mathematics education on the web.

DigiMat Basic is web based
which will soon be available on edX as MOOC as an expansion of DigiMat Pro now running as
DigiMat is constructive mathematics opening to interactive learning in both individual and group form,  without need of traditional class room teacher instruction,

Stay tuned! The World is not what it used to be.

tisdag 3 mars 2020

Drag Crisis and Slip at Reynolds Number 1 million

This is a continuation of the previous post identifying three types of contact between a fluid and a fixed smooth solid wall:
1. laminar slip/small skin friction
2. laminar no-slip
3. turbulent no-slip
where DFS Direct Finite Element Simulation uses 1 while standard CFD uses 2 and 3.

No-slip forms a thin boundary layer connecting fluid with zero velocity on the wall with free flow velocity away from the wall. Slip allows fluid particles to glide along a smooth solid wall without boundary layer at small skin friction.

Standard CFD uses no-slip with thin boundary layers beyond direct computational resolution thus requiring wall models for turbulent flow, which have shown to be elusive. Standard CFD therefore is not truly predictive and thus not very useful.

DFS uses slip/small friction as an effective boundary condition, which does not form a boundary layer. This makes DFS computable, with true predictive capability demonstrated.

The appearance of slip/small friction connects to the so called drag crisis observed to occur in slightly viscous bluff body flow with drag drastically dropping at a Reynolds number $Re\equiv\frac{UL}{\nu}$ of around 1 million (or 500.000), where $U$ is typical flow speed, $L$ typical length scale and $\nu$ kinematic viscosity. With $U=1$ and $L=1$, the drag crisis thus connects to $\nu\approx 10^{-6}$ or $Re =10^6$.

For Reynolds numbers below drag crisis the effective boundary condition can be viewed to be no-slip, which forces early separation into a large turbulent wake and large drag.  For Reynolds numbers above drag crisis separation is delayed to form a narrow wake with small drag,  which the analysis of DFS shows to connect to the appearance of an effective slip/small friction boundary condition.

Let us seek to follow this transition, thus starting before drag crisis with a laminar no-slip layer of width $d=\sqrt{\nu}$ and shear $\frac{1}{\sqrt{\nu}}$ with free stream velocity $U=1$, and corresponding Reynolds number based on $L=d$ of size $\frac{1}{\sqrt{\nu}}$.

A laminar no-slip layer is an example of shear flow, which shows to develop into a turbulent no-slip layer for Reynolds numbers of size $10^3$ as described in detail in the book Computational Turbulent Incompressible Flow. This connects to a drag crisis at $\nu =10^{-6}$ with $\sqrt{\nu}=10^{-3}$.

In a first step a laminar no-slip low shear layer thus develops into a turbulent no-slip high shear layer which in a second step can develop into an effective slip/small friction condition as an effect of plastic yield in high shear turbulent flow, with a corresponding maximal shear force of size $\sqrt{\nu}=10^{-3}$ appearing as small skin friction of size 0.001.

The transition from laminar no-slip to turbulent no-slip to slip can be followed in the flow over a convex surface which as laminar no-slip flow separates, because the pressure gradient normal to the boundary is small in a laminar shear layer,  and so develops into a turbulent no-slip layer which can reattach by effectively forming a slip layer with pressure gradient preventing separation.

Summary: Drag crisis connected to slip occurring at a macroscopic Reynolds number of about $10^6$ with a shear of $1000$ and corresponding skin friction $0.001$, can thus be connected to
• transition from laminar no-slip at $Re =10^6$ to turbulent no-slip with shear exceeding $10^3$,
• transition from turbulent high shear with layer to effective slip skin friction $0.001$ as an effect of visco-plastic flow.

måndag 2 mars 2020

Laminar Slip Layer vs Turbulent No-Slip Layer: Change of Paradigm

 A turbulent no-slip  boundary layer is uncomputable and lacks mathematical model. A troublesome concept. Modern fluid dynamics has been obsessed with the problem of tackling this problem, without success. The result is CFD which is not predictive  and thus not very useful.

DFS Direct Finite Element Simulation as a new paradigm in Computational Fluid Dynamics CFD exhibits a new basic phenomenon of
• laminar slip boundary layer
to be compared with the basic elements identified by Prandtl as the Father of modern fluid mechanics of:
• laminar no-slip boundary layer,
• turbulent no-slip layer.
The appearance of a laminar slip boundary is connected to the so called drag crisis occurring in bluff body slightly viscous flow such as air and water at a Reynolds number $Re\approx 500.000$ with the drag of a bluff body drastically dropping beyond $500.000$.

The reduction is the result of delayed separation with reduced wake as an effect of a shift from a laminar no-slip boundary layer, which trips the flow to early separation,  to effectively a laminar slip boundary layer, which allows a different form of separation as 3d rotational slip separation without tripping.

The appearance of a turbulent no-slip layer is typically artificially induced in experiments through a transversal ribbon/strip attached to the body thus effectively changing the shape of the body, which trips the flow into separation and turbulent wake. The idea is that this way force the experiment to fit with a preconceived notion by Prandtl of a turbulent no-slip boundary layer, but this is against the most basic principle of science to fit theory to observation and not the other way around.

The result of using an effective laminar slip boundary condition without any artificial tripping, is that fluid flow beyond the drag crisis is computable by DFS because impossible computational resolution of thin turbulent boundary layers required in Prandtl CFD,  is no longer needed. A non-computable turbulent no-slip boundary is thus replaced by a computable laminar slip layer.

DFS shows to accurately predict fluid flow beyond the drag crisis by computing best possible turbulent solutions of Euler's equations as first principle physics without parameters with slip as wall model and a turbulence model as emergent from computation. This makes CFD computable from being uncomputable to all Prandtl followers, and thus represents a veritable change of paradigm.

A key to the breakthrough is the concept of laminar slip boundary layer of a fluid which is viscous-plastic with fluid particles sliding along a smooth wall with skin friction coefficient of size 0.001 at drag crisis and decreasing beyond.

DFS shows that slightly viscous flow is not Newtonian with a constant (small) viscosity since the emergent turbulence model in DFS does not reflect a constant viscosity, nor does the viscosity-plastic slip boundary condition.

This gives perspective on the Clay Navier-Stokes problem which concerns a Newtonian fluid seemingly without relevance for slightly viscous flow as the main challenge of fluid mechanics.

fredag 28 februari 2020

Banach Documentary: Digital Math: Body and Soul

Together with Per Enflo and Johan Jansson I participate in a documentary about the great Polish mathematican Stefan Banach, to be shown in Polish TV in March. The film was shown to an invited audience at Fokus in Östervåla Fokus 22/2 followed by a discussion about Banach and our connections to his work. The historical event is recorded at the Per Enflo web site and featured on Icarus Digital Math.

The film has appeared in festivals of documentary film and will be shown at KTH in the Spring and maybe also on Swedish State Television. Stay tuned.

The title of the film is
• Banach: Between Spirit and Matter
with a connection the inscription on the grave stone of Steinhaus, who discovered Banach's talent and became his teacher:
• Mathematics connects Soul to Matter
which is basically the same as the leading theme of the series of books:
as made clear in the film.

DFS: Change of Paradigm in CFD

DFS Direct Finite Element Simulation is change of paradigm of Computational Fluid Dynamics CFD by correctly predicting the forces acting on a body moving through a slightly viscous fluid such as air or water with the shape of the body as only input, through computation of best possible solutions to Euler's equations expressing first principle physics without parameters.

DFS takes CFD out of the conundrum of finding turbulence and wall models, which despite efforts over more than 100 years has not led to true predictive capability. Standard CFD is typically fitted to match observation but does not deliver correct prediction without prior (wind tunnel) observation and so is not very useful for design.

DFS combines the Euler equations in the fluid domain with a slip boundary condition on the smooth wall of the body modeling vanishing viscous skin friction. DFS shows to correctly predict drag as form/pressure drag within experimental precision and thus shows that the contribution from skin friction is negligible. This is in direct contradiction to standard CFD which attributes $50\%$ or more of drag to skin friction for slender bodies.

As an example we consider the case of drag and lift coefficients $C_D$ and $C_L$ for the basic test case of a long Naca0012 wing, as function of angle of attack $\alpha$. DFS delivers the following results for $0\le \alpha\le 15$ well below stall:
• $C_L(\alpha ) \approx = 0.1\times\alpha$,
• $C_D(\alpha ) \approx = 0.004 + 0.001\times\alpha$.
This fits wind tunnel experiments (without artificial tripping) by Ladson within experimental precision.

The Ladson value $C_D=0.005$ for $\alpha =0$ instead of $0.004$ with DFS, stands out as a limit case for which extrapolation from $\alpha\ge 2$ as in DFS may well be more relevant than direct measurement with tripping as an issue ($C_D=0.008$ with tripping).

We see a linear variation of both $C_L$ and $C_D$ with the angle of attack $\alpha$ as an expected effect of changing geometry.  For lift it connects to effective downwash scaling with $\alpha$ and for drag with an effective frontal area also scaling with $\alpha$

The efficiency of the wing is measured by the lift $L$ to drag $D$ quotient $\frac{L}{D}=\frac{C_L}{C_D}$ ranging from 33 for $\alpha =2$ over 60 for $\alpha =6$ to 75 for $\alpha =15$, thus with steadily increasing $\frac{L}{D}$ before stall.

The common view is that for a short wing $C_D$ has a contribution scaling with $C_L^2$ thus quadratically in $\alpha$  due to a wing tip effect, which suggests that for a long wing $C_D$ is constant as being dominated by skin friction, however without support in observation.

Summary:
• DFS shows that for slightly viscous flow beyond the drag crisis for Reynolds number around $500.000$, total drag is mainly form/pressure drag with a very small (at most $10\%$) contribution from skin friction.
• Standard CFD attributes instead $50\%$ or more to skin friction for an airplane or ship.
The consequence for design is a change of paradigm from an old standard bogged down by unsuccessful attempts to decrease skin friction, to a new standard focussing on form, where possibilities for improvements are many.

The dogma of $50\%$ skin friction is upheld by tripped experiments where e.g. a ribbon is fastened on the body transversal to the flow to generate turbulence increasing drag which is then attributed to skin friction, while it effectively instead corresponds to a change of form. This way observation is fitted to theory prescribing massive skin friction, while in correct science theory is fitted to observation.

onsdag 19 februari 2020

Prandtl's Tripped Science vs Boeing Max

 Prandtl making tripped experiments

 Danger of tripping

Ludwig Prandtl is viewed as the Father of Modern Fluid Mechanics because he offered a resolution of the pressing problems of fluid mechanics in the beginning of the 20th century including d'Alembert's paradox through his discovery of the laminar and turbulent boundary layer in wall bounded fluid flow.

The legacy of Prandtl is described in Prandtl-Essentials of Fluid Mechanics edited by Herbert Oertel, Springer 2004, with the following introduction
• The development of modern fluid mechanics is closely connected to the name of its founder, Ludwig Prandtl.
• In 1904 it was his famous article on fluid motion with very small friction that introduced boundary-layer theory.
• His article on airfoil theory, published the following decade, formed the basis for the calculation of friction drag, heat transfer, and flow separation.
• Prandtl was particularly successful in bringing together theory and experiment, with the experiments serving to verify his theoretical ideas.
• It was this that gave Prandtl’s experiments their importance and precision. His famous experiment with the tripwire, through which he discovered the turbulent boundary layer and the effect of turbulence on flow separation, is one example.
• The tripwire was not merely inspiration, but rather was the result of consideration of discrepancies in Eiffel’s drag measurements on spheres.
• Two experiments with different tripwire positions were enough to establish the generation of turbulence and its effect on the flow separation. For his experiments Prandtl developed wind tunnels and measuring apparatus, such as the Göttingen wind tunnel and the Prandtl stagnation tube.
• His scientific results often seem to be intuitive, with the mathematical derivation present only to provide service to the physical understanding, although it then does indeed deliver the decisive result and the simplified physical model.
• According to a comment by Werner Heisenberg, Prandtl was able to “see” the solutions of differential equations without calculating them.
To give the highlighted parts perspective recall that when I was awarded the Prandtl Medal in 2014 by ECCOMAS, I stated that I would receive the medal under the condition that it would be expressed that the New Theory/Computation of Flight developed together with Johan Hoffman and Johan Jansson showed that Prandtl had misled modern fluid mechanics into a fruitless search for the origin lift and drag of an airplane wing in a boundary layer so thin that it could never be resolved in computation. This was not allowed to be expressed and the result was that I did not accept to receive the medal. The story can be read here.

The New Theory of Flight supported by refined computations since 2014 shows that contrary to Prandtl wall bounded slightly viscous flow can be modeled by a slip boundary condition without any boundary layer, which makes the flow computable as time variable turbulent flow. There is thus now massive evidence that Prandtl was wrong, seriously wrong.

Signs that there is something fishy with Prandtl's boundary layers as the origin of drag and lift can be seen in the above highlights:
1. Prandtl use a tripwire to change the flow to fit what he could "see" without mathematics and computation.
2. His results were intuitive.
The effect of artificially tripping the flow in experiments has led to the misconception that skin friction drag is a major part of total drag with form/pressure drag a minor part, viewed to be relevant  also for an airplane wing without tripping device. The New Theory gives hard evidence that this is seriously misleading by computing drag and lift with slip in close accordance to observations.

The lesson is that if you rely on intuition rather than correct mathematics and are ready to trip experiments to fit, then you can end up with something with little connection to reality. Evidently Prandtl did so. The consequences are severe with the Boeing Max debacle a result of misconceived engineering computation following Prandtl.

PS The following question/answer appears on FAQ at Secret ion Flight:

Q30: Why is the flow tripped by a wire, strip or ribbon in wind tunnel measurements of drag of wing, when a real wing does not have any tripping device and the tripping thus appears to be artficial?

A30: The rationale presented is that the tripping will force the development of a turbulent boundary layer with substantial skin friction,  which according to Prandtl should be present. The tripping is thus done to artificially fit reality to theory, which is opposite to the basic principle of science to fit theory to reality. In the New Theory, which fits with untripped experiments, the flow of air meets the wing with a slip boundary condition modeling vanishing skin friction.

tisdag 18 februari 2020

Fundamentals of Aerodynamics by John D Anderson as Old Theory of Flight

The book Fundamentals of Aerodynamics by John D Anderson describes the standard theory of flight as Old Theory of Flight with basic ingredients expressed in Chapter 4:
• The purpose of this chapter is to present theoretical methods for the calculation of airfoil properties.
•  In most of this chapter we will deal with inviscid flow, which does not lead to predictions of airfoil drag; indeed, d’Alembert’s paradox says that the drag on an airfoil is zeroclearly not a realistic answer.
• However, if we lived in a perfectly inviscid world, an airfoil could not produce lift.
• Indeed, the presence of friction is the very reason why we have lift. These sound like strange, even contradictory statements to our discussion in the preceding paragraph.
• What is going on here? The answer is that in real life, the way that nature insures that the flow will leave smoothly at the trailing edge, that is, the mechanism that nature uses to choose the flow shown in Figure 4.18c, is that the viscous boundary layer remains attached to the surface all the way to the trailing edge.
• Nature enforces the Kutta condition by means of friction. If there were no boundary layer (i.e., no friction), there would be no physical mechanism in the real world to achieve the Kutta condition.
• So we are led to the most ironic situation that lift, which is created by the surface pressure distribution—an inviscid phenomenon, would not exist in a frictionless (inviscid) world. In this regard, we can say that without friction we could not have lift.
We read that the Old Theory is strange, contradictory, not realistic and ironic. The New Theory of Flight presented on Secret of Flight shows that this characterisation is correct. It is now time to allow the Old Theory to retire since it is physically incorrect and no longer is needed as a facade when there is a physically correct theory.

måndag 17 februari 2020

DFS vs standard CFD: Form vs Skin Friction Drag

Direct Finite Element Simulation DFS, as a new revolutionary methodology/software for Computational Fluid Dynamics CFD, computes best possible turbulent solutions to Euler's equations as first principle physics without parameters.

DFS gives results in close agreement with observations as true prediction without adjustment of parameters to match each computation with observation in non-predictive mode. In particular, DFS uses a slip boundary condition on a smooth solid wall as a model of vanishingly small skin friction.

DFS thus computes the drag of a bluff body as form/pressure drag with vanishingly small contribution from skin friction, in close agreement with observation.

For a Naca0012 airfoil at zero angle of attack DFS delivers a drag coefficient $C_D =0.006$ as form/pressure drag, which fits well with untripped measurements (red) by Abbott and von Doenhoff:

The figure also shows measurement (blue) by Ladson with artificial tripping not present for a real wing with larger $C_D\approx 0.008$, thus not applicable to a real wing which does not carry any tripping device.

We now compare DFS with standard CFD where we find the following account in the standard reference Fundamentals of Aerodynamics 5th ed by John D Anderson p. 381 with reference in particular to Lombardi, G., Salvetti, M. V. and Pinelli, D.: Numerical Evaluation of Airfoil Friction Drag, J. Aircraft, vol. 37, no. 2, March–April, 2000, pp. 354–356:
• total drag 0.00623
• skin friction drag 0.00534 ($85\%$ of total)
• DFS: form/pressure drag $95-100\%$ of total drag.
• Standard CFD: form/pressure drag $15\%$ of total drag.
We see a vast difference of form/pressure drag with a factor 6! There is no way both DFS and standard can be correct.

We have massive evidence that DFS without parameters gives correct drag. The conclusion can only be that standard CFD does not capture anything like the truth.

Standard CFD includes turbulence and wall models with many parameters, with the wall model delivering large skin friction ($85\%$) of total drag. The parameters are then adjusted to give total drag in accordance with observations, which means that form/pressure drag comes out as a small portion ($15\%$) of total drag.

The conclusion can only be that standard CFD is not useful, acknowledged by many users, since by the necessity of parameter fitting is not predictive and does not capture true physics.

The reason standard CFD does not capture physics is rooted in the wall model used, which prescribes a separation pattern which is not physical. In DFS the separation is not prescribed by a model and instead follows the physics.

It is clear that aerodynamic design will be very different if based on predictive DFS with form/pressure drag dominating skin friction drag, instead of as now non-predictive standard CFD postulating dominating skin friction in contradiction with physics.

It is the same story in ship hydromechanics with a current consensus of $70\%$ skin friction drag and  correspondingly small form/pressure drag, misleading design into (resultless) efforts to reduce skin friction.

PS More precisely John D Anderson reports the following results from Lombardi at al for NACA0012 at zero angle of attack at $Re =3\times 10^6$:

We see standard CFD delivering skin friction drag even larger than observed total drag.

Good Example: NRK Denies Denial

Klimarealsitene reports that Norwegian State Radio NRK has declared a new Strategy on Climate Alarmism:
• Dekningen skal i hovedsak handle om hvordan, og ikke om, det skal handles for å tilpasse seg eller dempe den globale oppvarmingen.
• De stadfester at menneskeskapte klimaendringer er reelt, og at NRK skal legge dette til grunn for journalistikken.
• NRK skal være oppmerksomme på den «falske balansen». Hvis de slipper til klimafornektere, skal de stille de rette motspørsmålene.
In English:
• NRK is to report only about how to stop global warming, not if it is needed or meaningful.
• NRK declares that humans control the climate.
• Climate denialism will only be allowed to be voiced if directly countered with complete denial by NRK
We now await Swedish State Radio SR to follow with a similar declaration to meet the mission of public radio, although effectively such a policy is already implemented.

söndag 16 februari 2020

Strong State: Scaring or Comforting?

 Swedish Social Democracy 2020: Stronger Safer
A Strong State pretending to have a mission, can use one of two basic strategies:
1. Fear-Mongering: Imminent Threat! Alarm! The State protects you.
2. Comforting: No Threat! No Alarm! The State takes care of you.
There are many examples of 1. through history collapsing to state terror. Swedish Social Democratic Society peaking in the 1950s is an example of 2.

Today we see a Swedish Social Democratic Society, which has switched mode from 2 to 1, in the name of climate alarmism set on a road supported by a new climate law to be the first fossil free welfare state by 2050 in another great leap forward, as an example of a new brave world to be followed by the entire world.

Once that new brave world is reached, Swedish Social Democratic Society will return to mode 2. Only 30 years of great leap state terror with the end as usual justifying the means.

But is climate alarmism starting to crack already today? Yes, there are signs like this one.


text:
# How can I remove the last "Continued from previous page" from my longtable?

Right now I have a long table spanning across both pages. At the bottom of the portion of the first page it says "Continued on next page", which is good, but this also appears at the bottom of the portion on the second page. Here's the code I'm using:

\begin{longtable}{cc}
\caption{Reward surfaces of all MuJuCo environments} \\
\hline
\multicolumn{2}{c}%
{\tablename\ \thetable\ -- \textit{Continued from previous page}} \\
\hline
\hline \multicolumn{2}{r}{\textit{Continued on next page}} \\
\endfoot
\hline

\includegraphics[width=2.75in,keepaspectratio]{./plots/mujuco/antepisode_rewards_3dsurface.png} & \includegraphics[width=2.75in,keepaspectratio]{./plots/mujuco/halfcheetahepisode_rewards_3dsurface.png} \\
\includegraphics[width=2.75in,keepaspectratio]{./plots/mujuco/hopperepisode_rewards_3dsurface.png} & \includegraphics[width=2.75in,keepaspectratio]{./plots/mujuco/humanoid@3x3episode_rewards_3dsurface.png} \\
[...]

\end{longtable}

• you need an \endlastfoot without that Sep 10, 2021 at 16:36

Please see the following MWE resulting in two pages. Please see that I marked important code changings with <=====.

\documentclass{article}

\usepackage{longtable}
\usepackage{graphicx}

\begin{document}
\begin{longtable}{cc}
\caption{Reward surfaces of all MuJuCo environments} \\
\hline
\multicolumn{2}{c}%
{\tablename\ \thetable\ -- \textit{Continued from previous page}} \\
\hline
\hline
\multicolumn{2}{r}{\textit{Continued on next page}} \\
\endfoot
\hline % <============================================================
\endlastfoot % <========================================================
\includegraphics[width=2.75in,keepaspectratio]{example-image} & \includegraphics[width=2.75in,keepaspectratio]{example-image-a} \\
\includegraphics[width=2.75in,keepaspectratio]{example-image} & \includegraphics[width=2.75in,keepaspectratio]{example-image-b} \\
\includegraphics[width=2.75in,keepaspectratio]{example-image} & \includegraphics[width=2.75in,keepaspectratio]{example-image-a} \\
\includegraphics[width=2.75in,keepaspectratio]{example-image} & \includegraphics[width=2.75in,keepaspectratio]{example-image-b} \\
\end{longtable}
\end{document}

I added command \endlastfoot to create a special footer for the last page of the table, containing only \hline.

The resulting second pdf page is


text:
If the resistance path from the wall outlet end of the neutral wire to ground is 23 ? what is the voltage of that point (relative to ground) if a 26 W compact fluorescent bulb is connected to the outlet?

Thank you for any help.


text:
Journal Article @Article Artikel in Fachzeitschrift
 Show entries of: this year (2020) | last year (2019) | two years ago (2018) | Notes URL
 Action: login to update Options: Library locked Goto entry point

 Author, Editor(s)
 Author(s): Berberich, Eric Kerber, Michael Sagraloff, Michael dblp dblp dblp
 BibTeX cite key*: bks-cgta-2009

 Title
 Title*: An efficient algorithm for the stratification and triangulation of an algebraic surface

 Journal
 Journal Title*: Computational Geometry: Theory and Applications (CGTA) Journal's URL: www.elsevier.com/locate/comgeo Download URL for the article: http://dx.doi.org/ Language: English

 Publisher
 Publisher's Name: Elsevier Publisher's URL: http://www.elsevier.com Publisher's Address: Amsterdam, Netherlands ISSN: 0925-7721

 Vol, No, pp, Date
 Volume*: 43 Number: 3 Publishing Date: April 2010 Pages*: 257-278 Number of VG Pages: 20 Page Start: 257 Page End: 278 Sequence Number: DOI: 10.1016/j.comgeo.2009.01.009

 Note: (LaTeX) Abstract: We present a method to compute the exact topology of a real algebraic surface $S$, implicitly given by a polynomial $f\in\mathbb{Q}[x,y,z]$ of arbitrary total degree~$N$. Additionally, our analysis provides geometric information as it supports the computation of arbitrary precise samples of $S$ including critical points. We compute a stratification $\Omega_S$ of $S$ into $O(N^5)$ nonsingular cells, including the complete adjacency information between these cells. This is done by a projection approach. We construct a special planar arrangement $\mathcal{A}_S$ with fewer cells than a cad in the projection plane. Furthermore, our approach applies numerical and combinatorial methods to minimize costly symbolic computations. The algorithm handles all sorts of degeneracies without transforming the surface into a generic position. Based on $\Omega_S$ we also compute a simplicial complex which is isotopic to~$S$. A complete C++-implementation of the stratification algorithm is presented. It shows good performance for many well-known examples from algebraic geometry. URL for the Abstract: Categories, Keywords: Algebraic surfaces, exact geometric computation, topology computation, cylindrical algebraic decomposition HyperLinks / References / URLs: Copyright Message: Personal Comments: Download Access Level: Public

 Correlation
 MPG Unit: Max-Planck-Institut für Informatik MPG Subunit: Algorithms and Complexity Group Audience: popular Appearance: MPII WWW Server, MPII FTP Server, MPG publications list, university publications list, working group publication list, Fachbeirat, VG Wort

BibTeX Entry:

@ARTICLE{bks-cgta-2009,
AUTHOR = {Berberich, Eric and Kerber, Michael and Sagraloff, Michael},
TITLE = {An efficient algorithm for the stratification and triangulation of an algebraic surface},
JOURNAL = {Computational Geometry: Theory and Applications (CGTA)},
PUBLISHER = {Elsevier},
YEAR = {2010},
NUMBER = {3},
VOLUME = {43},
PAGES = {257--278},


text:
# Linear Correlation Coefficient Formula

To find out the relation between two variables in a population, linear correlation formula is used. To see how the variables are connected we will use the linear correlation. Also known as “Pearson’s Correlation”, a linear correlation is denoted by r” and the value will be between -1 and 1.

The elements denote a strong relationship if the product is 1. Similarly, if the coefficient comes close to -1, it has a negative relation. If the Linear coefficient is zero means there is no relation between the data given.

Where “n” is the number of observations, “xi” and “yi “are the variables.

Also Check: Correlation Coefficient Formulas

### Solved Examples

Question 1: Calculate the linear correlation coefficient for the following data. X = 4, 8 ,12, 16 and Y = 5, 10, 15, 20.

Solution:

Given variables are,

X = 4, 8 ,12, 16 and Y = 5, 10, 15, 20

For finding the linear coefficient of these data, we need to first construct a table for the required values.

 x y x2 y2 XY 4 5 16 25 20 8 10 64 100 80 12 15 144 225 180 16 20 256 400 320 Σ x = 40 Σ y =50 480 750 600

According to the formula of linear correlation we have,

$r(xy)=\frac{(4\times 600)-(40\times 50)}{\sqrt{4(480)-40^{2}}\sqrt{4(750)-50^{2}}}$

$r(xy)=\frac{2400-2000}{\sqrt{1920-1600}\sqrt{3000-2500}}$

$r(xy)=\frac{400}{\sqrt{320}\sqrt{500}}$

$r(xy)=\frac{400}{17.89\times 22.36}$

$r(xy)=\frac{400}{400} = 1$

Therefore, r(xy) = 1


text:
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In the following Question, Rs. 33,000 is paid off in 12 instalments, such that each installment is Rs. 100 more than the preceding one. Find the amounts of the first and the last instalments.

Last updated date: 15th Mar 2023
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Hint: The given question is in A.P., therefore we can consider the value of the number of terms to be 12, common difference to be 100 and then we can use these values to find the sum of n terms using the formula.

${S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$.

Let, since we do not know the value of the first instalment, let the first instalment be a.
n = 12
d = 100
${S_{12}} = \dfrac{{12}}{2}\left( {2a + 11d} \right)$
$33000 = 6\left( {2a + 1100} \right)$
$2a + 1100 = 5500$
$2a = 4400$
$a = 2200$

So, the first instalment is Rs. 2200
Last instalment is 2200+11(100) = 2200+1100=3300

Note: In these questions where there is a common difference we can directly apply arithmetic progression formulas to find the answer.


text:
## Introduction

Measurement of the prostate gland in vivo is commonly required in the management of prostate disorders, both benign and malignant. Measurement is especially useful in the diagnosis and management of adenocarcinoma and benign prostatic hyperplasia (BPH). In the former, knowledge of total prostatic volume is necessary in the calculation of prostate specific antigen (PSA) density (PSAD)—a key indicator of the likelihood that elevated PSA is due to malignancy1,2. Accurate measurement may affect cancer treatment strategies1,2. However, the ability to make consistent and accurate measurement of the prostate is also useful in the diagnosis and management of BPH3,4.

Early attempts to estimate total prostatic volume by digital rectal examination (DRE) alone were replaced by more accurate measurement with transrectal ultrasound (TRUS) that showed excellent correlation with gross pathologic specimens using planimetric and calculated ellipsoid volume formula (EVF) techniques5. Similarly, accuracy compared to pathologic specimens has also been established using 3-plane magnetic resonance imaging (MRI)6,7,8,9. While the forgoing techniques have shown reasonable success, details describing exact landmarks of the boundaries measured on imaging are rarely provided to assure general reproducibility. Difficulty in identifying landmarks such as the bladder neck and prostatic apex on ultrasound has been mentioned10,11. Most studies performed with TRUS or MRI underestimate overall pathologic measurement of size by weight6,7,12,13,14 although this may be due to inclusion of the seminal vesicles (SV), vas deferens (VD) stumps and periprostatic tissue in weighing the pathologic specimens8. Measured against an expert panel, in vivo measurement with TRUS overestimates post-operative small specimen size and underestimates large prostates15,16. However other investigators using MRI EVF volumetrics report overestimation compared to specimen volume (corrected for density of 1.05 gm/cc)9.

There are scant studies of inter-observer reliability for the measurement of total prostatic volume using standard methods10,11,17. TRUS intra-rater reliability using the ellipsoid formula was examined in one study and was 0.9312. Bangma, et al.18 compared TRUS step-section planigraphy with several geometric multiplane models and found the prolate ellipsoid to be comparable (r = 0.89). Early application of MRI using the EVF showed acceptable precision and accuracy in one small study6.

The purpose of this study is three-fold. We review detailed anatomy required to more confidently find measurement boundaries. We introduce a new multiplane EVF approach to make these measurements using MRI, and report results of inter-rater and intra-rater reliability with this technique.

## Materials and Methods

### Patient selection

This study was approved by our institutional Internal Review Board and Ethics Committee (protocol number 00003941) with a waiver of informed consent. 140 patients were selected from a population of men undergoing MRI because of abnormal PSA, digital rectal examination or both. Ages ranged from 41–76 (mean, 64 years). They were selected by the senior radiologist (NFW) with over 40 years of experience who acted as administrator/expert. All MRI lobar classification types of BPH were represented in the selection4.

### Study design and statistics

Two body imagers with 10 years of experience reading prostate MRIs were assigned the task of measuring total prostatic volume (EN, BS) using T2-weighted MRI on the 140 patient cohort. They were provided with a spreadsheet showing only the patient hospital identification numbers. After determining the total prostate volumes, these spreadsheets were returned to the administrator. Intra-rater reliability was tested only on the expert’s data. Re-measurement of prostate volume was done between 3 and 9 months after primary measures. Inter-rater reliability (precision) for continuous variables for volumes was analyzed for actual paired case-to-case correlation using the Pearson product moment and Lin’s Coefficient of Concordance19. Linear regression was calculated to find r2 values, Y intercepts, p values and 95% confidence intervals (CI). These were calculated from the open access statistical programs at the National Institute of Water and Atmospheric Research (NIAWA.com) and QI Macro Statistics® (KnowWare International, Inc., Denver CO, USA). Measurements of the administrator were used as a proxy for gross pathological specimen weight for the determination of rater “accuracy”. Correlations were analyzed by linear regression and graphed. A comparison of means was also done after the data was available to establish data normality. Significance was defined as p ≤ 0.05.

### MRI acquisition techniques

Prostate MRI was performed at 3 T with a pelvic phased array coil in 27 with and 113 without endorectal coil (Magnetom Skyra or TrioTim; Siemens Healthcare; Erlangen Germany). T2W FSE imaging was performed in the axial plane (TR/TE 3700/80 msec; NEX 3; 3 mm slice thickness; no interslice gap; flip angle 160 deg; FOV 140 mm, matrix 320 × 256) and coronal plane (TR/TE 4030/100 msec; NEX 2; 3 mm slice thickness; no interslice gap; flip angle 122 deg; FOV 180 mm; matrix 320 × 256). T2W 3D SPACE images were obtained (TR/TE 1400/101 msec; flip angle 135 deg; 256 × 256 × 205 matrix, 180 mm FOV). In addition to 1 mm axial images, T2W 3D SPACE images were reconstructed at 3 mm in the axial, sagittal, and coronal planes. Additionally, dynamic contrast enhanced T1-weighted (3 mm slice thickness; TR/TE 4.9/1.8 msec; 224 × 156 matrix, 250 mm FOV, temporal resolution <10 sec), and diffusion-weighted images at b values of 50, 800, and 2000 s/mm3 were performed.

### Anatomy

Anatomic boundaries and landmarks for measurement of total prostate volume were chosen after a thorough review of the anatomic literature, including the work of Salvadore Gil-Vernet (salvadoregilvernet.com)20, Robert Meyers21,22 and others23,24,25. The Prostate is not only a glandular but also a muscular organ23,26. Its external boundary, that may be referred to as the external prostatic capsule (EPC) extends posterior, lateral, and antero-lateral. The EPC represents the peripheral compressed stromal (fibro-muscular) component of the latticework supporting the glandular elements23. This is the outer boundary that is visible on imaging. Walz, et al.27 refers to this as the “true capsule”, although it is not what some anatomists would define as a true “capsule” that can be “peeled off” and separated from the gland. It will be referred to as a “capsule” in this presentation due to its common use in clinical practice and because it is the outer boundary we see on imaging.

The controversy surrounding the naming of this “pseudocapsule” is discussed in several anatomic studies including those that refer to the periprostatic fascia as the “capsule”25,27. This fascial plane is not visible on imaging. The anterior external limit of the prostate is less distinct, comprising anterolateral elements of the EPC, the surgical capsule (SC) surrounding enlarged transition zones (TZ) when BPH is present, and anterior fibro-muscular stroma (AFMS). The multilayered loose fascia surrounding the prostate is disrupted with surgical or post-mortem removal making correlation between the in vivo gland to the ex vivo specimen more difficult22.

The “surgical” capsule, (SC) is so-called because it encases the enlarged transition zone, which on open surgery can be enucleated from surrounding tissues. It represents compressed circular smooth muscle fibers of the pre-prostatic urethral wall (above the verumontanum) and prostatic urethral wall (below the veru)23. The anterior SC is indistinct to absent unless there is more advanced nodular hyperplasia of the (TZ). The AFMS is thickest proximally where it is formed by smooth muscle fibers from the bladder wall. This boundary is critical to define when performing maximal AP prostate measurement. The above features are delineated in Fig. (1A–C). The EPC is incomplete at the apical peripheral zone (PZ)22,23,24 (Fig. 1D).

### MRI image analysis & selection of boundaries

All images were reviewed on high-resolution monitors using Philips Intellispace (Koninklijke Philips N.V.) software. Because of new interest in the urological literature pertaining to herniated portions of enlarged prostate into the bladder base, an alternative scheme of volumetrics was devised to include measurement of intravesical prostatic protrusion (IPP). A new vocabulary is needed to describe these boundaries e.g. vesico-prostatic angle (VPA), vesico-prostatic line (VPL), and apical line (AL) discussed below. Our approach is calibrated to provide the most reproducibility for intra-and interrater reliability.

Transverse measurements were made in the axial plane estimated to show maximal diameter. It allows best visualization of the lateral EPC boundaries for transverse measurement. The axial plane is also the only plane that allows for selection of the anterior boundary by interpolation of anterior ends of the right and left EPC for accurate antero-posterior (AP) measurement. The mid-sagittal plane is unsatisfactory for that determination. (The coronal plane, though reasonable for transverse measurement, is suboptimal for AP diameter measurement).

Pericapsular veins and levator ani muscles are confounding anatomical structures that, depending on axial level, lie adjacent to the EPC and make its outer boundary difficult to define (Fig. 2A). Therefore, for consistency, when measuring the transverse diameter, we arbitrarily chose to measure from the inner margins of the EPC.

The maximal AP diameters were made at or near the same axial plane as the transverse taking care to not include the inferior extension of the anterior bladder detrusor. Measurement was made from the inner margin of the mid-posterior EPC to the mid-anterior AFMS. If there was a midline posterior indentation of the EPC (posterior commissure), the inner point was selected, even though the PZ projected slightly more posterior on either side. The anterior landmark chosen varied depending on definable anatomy.

There are three general configurations of the anterior boundary. In some patients, there is a discrete continuation of the merged SC and bilateral anterolateral EPC across the midline allowing for selection of its outer boundary (Fig. 3A). In the second, the EPC is interrupted and invisible in the midline, in which case we drew an interpolated arc-like line connecting the ends of these structures (Fig. 3B). (This is the rationale for using the axial rather that sagittal plane for AP landmarks). The third configuration is where there is apparent complete fusion of the AFMS, SC and EPC, completely obliterating visualization of the anterior EPC. This configuration was seen in less than 5% of our cohort. An interpolated arc-like line was dawn if the two antero-lateral ends of the EPC were close enough to connect. Otherwise, the axial AP anterior landmark was taken as the inner boundary of the AFMS (Fig. 3C). The appearance may be caused by the “salami effect” of a trans-axial plane that is oblique with respect to the longitudinal axis of the prostate (Fig. 4).

Length measurement was made in the mid-sagittal plane. The mid-sagittal plane is defined as that which best shows the membranous and bulbous urethra. Landmarks for measuring mid-sagittal length started by finding the anterior and posterior vesico-prostatic angles (VPA) as defined as the points where the detrusor connects with the prostate and connecting these angles by a line called the vesico-prostatic line (VPL) (Fig. 5). The VPL arbitrarily defines the level of bladder “outlet.” The inferior landmark was the caudal margin of the PZ defining the apex where an apical line (AL) was drawn approximately perpendicular to the long axis of the prostate drawn from the bladder outlet to the membranous urethra (Fig. 5). This PZ inferior boundary may occur posteriorly or in a parasagittal plane, therefore, it is essential to use a localizer tool in the coronal view to verify the level of the AL in the mid-sagittal plane (Fig. 6). If the prostate was entirely below the VPL, the mid-sagittal length was defined as the length of a line drawn from the mid-urethra at the level of the AL extending perpendicular to the VPL. If the prostate projected above the VPL, the midsagittal length was defined as the sum of the distance below the VPL plus the distance of a line from the from the superior margin of this tissue drawn perpendicular to the VPL (Fig. 5). This segment superior to the VPL represents intravesical prostatic protrusion (IPP).

Total prostatic volume was calculated according to the formula of the geometric model for a prolate ellipsoid:

$$\mathrm{Volume}=\mathrm{AP}\times \mathrm{Transverse}\times \mathrm{Length}\times \pi /6$$

The size of the retrourethral lobe (RU) was arbitrarily evaluated by measuring the maximal AP diameter as visualized on the mid-sagittal view. This involved three steps: Step 1: The RU long axis was defined by a line from the cranial margin of the RU to the mid-verumontanum (conjunction of the ejaculatory ducts with urethra). Step 2: We traced the approximate course of the urethra from the bladder neck through the level of the verumontanum using the “freehand” tool. Step 3: we estimated the thickest AP level of the RU as measured from the inner margin of its posterior capsule perpendicular to its long-axis ending at the posterior wall of the urethra (Fig. 7A,B).

### Ethics statement

The Internal Review Board for research and Ethics Committee are combined at the University of Minnesota into a single on-line submission called Ethosirb for which there is a single identifying protocol number (00003941). Informed consent was waved for this retrospective study carried out in accordance with the relevant guidelines and regulations of our institutional Internal Review Board and Ethics Committee, meeting all international standards for human research.

## Results

Analysis of the alternative method of determining total prostatic volume resulted in an interobserver correlation (precision) of 0.95 for the two interpreting radiologists (Fig. 8A). Intra-observer correlation (precision) was 0.98 for the senior radiologist (administrator) (Fig. 8B). Interobserver correlation between the two interpreting radiologists and the administrator (accuracy) was 0.94 and 0.97 respectively (Fig. 9A,B). All correlation p values < 0.000. Since the data was non-normal in distribution, a parametric test of total prostate volume means (Wilcoxan Sign Rank) was applied and means for interrater reliability (precision) and accuracy compared to the administrator were all the same p = 0.001 (Fig. 10).

## Discussion

The purpose of this study was to determine the inter- and intra-rater reliability of an alternative method of measuring total prostatic volume based on well-defined anatomic landmarks in order to improve precision among observers. After review of the anatomy and correlation with MRI, the resultant measurements show excellent statistical agreement.

The conventional transverse measurement method attempts to find the outside EPC margins. Because of the adjacent position of pericapsular veins and in the mid- and lower levels the levator ani muscles, these structures are often included in the transverse measurement. MRI is superior to ultrasound in selecting the outer boundary of the EPC. We chose to select the inner boundary of the EPC for axial plane landmarks to improve interobserver consistency (precision). In the everyday clinical setting, measurement of the outer boundary of the EPC, when visible in the maximal transverse view, is acceptable. We also describe better criteria for identifying a more consistent selection of the anterior and posterior landmarks for measurement in the axial view. Some might take issue with our technique of not including the entire AFMS in some cases wherein the maximal transverse axial plane falls at an extreme proximal level. In these situations, the detrusor (primarily smooth muscle) component of the AFMS is focally thickest and not representative of the lower two-thirds of the prostate. In addition to the “salami” effect mentioned earlier, this may explain why EVF calculations in most studies indicate that volume in glands <35 cc tend to be overestimated12.

The single most common cause for variation of total prostatic volume between examiners is likely selection of precise lower and upper landmarks in the mid-sagittal plane while measuring length5,28. Common errors include incorrect identification of the inferior boundary of the apex as the caudal landmark, inconsistent identification of the cranial landmark, making measurements in a parasagittal (rather than mid-sagittal) plane, and measuring from the anterior or posterior prostatic base to apex. Our understanding of the location of the inferior boundary of the PZ is critical. Experience has shown that relying only on the apparent lower margin of the PZ in the mid-sagittal plane can be misleading and that confirmation in the coronal view using a localizer function is required for accuracy. Collins, et al.29 pointed out two methods of longitudinal methods of measurement, one using the bladder base as the cranial landmark and the other the superior margin of the prostatic tissue including any “median” lobe (Fig. 11). We refer to our proposed method of length measurement as “biproximate” as it conveniently embraces both.

While measurement of the AP diameter of the RU may be useful for lobar classification of BPH4, it is non-contributory in determining total prostatic volume. However, the proposed alternative method of volumetrics includes measurement of IPP that may be of use to urologists for patient management30,31,32,33,34,35,36,37,38,39. Some studies indicate that IPP may be an independent variable effecting surgical or medical outcome40. IPP of greater than 5 mm as measured on transabdominal ultrasound, has been associated with clinical signs and symptoms of bladder outlet obstruction30,31 and responsiveness to medical management31,32,38.

Landmarks presented on MRI in our study are much more accurately visualized than on TRUS and could be expected to produce better intra- and interobserver concordance than those selected on TRUS. Another advantage of MRI, in contrast to transabdominal and TRUS, is that MRI does not require complete bladder distention for adequate visualization of landmarks.

The choice of an expert as proxy for “ground truth” measure of total prostatic volume rather than the pathological specimen may be considered a limitation or advantage of our study. Other clinical research studies have accepted the concept of expert consensus as a “gold standard”41,42,43. One limitation of our study is that we used a single expert rather than a panel of experts due to the relative paucity of those with experience using the measurement techniques under examination. Expert opinion is also the current standard for evaluation of MRI prostate segmentation44.

One could argue that since the objective should be to make in vivo measurements under normal physiological conditions, the volumes obtained were more realistic than under post-surgical or post-mortem conditions. Ex vivo measurements in the pathology department have disadvantages that reduce accuracy compared to those made in the living subject. The prostate is no longer infused with active circulation. This decreases specimen weight (volume)15. Processing of the specimen has been variable in previous studies. Comparison is usually made between the imaging measured volume and the “wet weight” of the fresh specimen on a scale or by water displacement in a graduated cylinder45. More accurate weights can be derived from electronic scales14.

Since the specific gravity of prostate tissue and water are nearly identical (1.05 gm to 1.0 ml)13, weight and volume have traditionally been used interchangeably in the literature, introducing mild inaccuracies when comparing image volumes to pathologic weights in larger glands. Some investigators use this number as a correction factor for comparing weight to volume9,46. This assumption fails to take into account that it is likely that there is a spectrum of density values in larger prostates with different proportions of stromal, glandular, and malignant tissues13. There is no evidence that the presence of carcinoma changes the accuracy between TRUS and cadaver volumes47. The specimen may include the attached seminal vesicles, vas deferens and surrounding soft tissues in many studies47. In some, the seminal vesicles and vas deferens are physically removed before weighing9,14,48, and in others the estimated mean volume of the seminal vesicles, as determined from the medical literature, is subtracted from the total weight of the prostate and accessory structures15,46. Changes in prostate size due to drying or fixation complicate comparisons in cadaver studies48,49.

MRI volume imaging eliminates these variables, and it could be argued that MRI volumetrics in the living patient should replace post-mortem measurement as the “gold standard”6. Hass, et al.50 has suggested that MRI-measured EVF might be considered the new standard for total volume estimate. In a small early small study using semi-automatic planigraphy, MRI showed improved results compared with TRUS6,9.

The application of artificial intelligence algorithms and the use of segmentation programs to establish prostate volume with MRI were reviewed by Turkbey, et al.51. These include innovative semi-automatic and automatic computer-based methods to “find” elusive prostate boundaries (edge detection). The software used is only available for research and not clinical use. These technologies share with geometric orthogonal models, such as the prolate ellipsoid formula (EVF) the fact that they are based on mathematical models and not direct measurement e.g. weight. The former requires significant boundary adjustments by an expert reader who still must ultimately understand the anatomy and landmarks we have described. At present, these techniques are thought by most to be too labor intensive for speedy clinical use9,46,50. Fully automatic unsupervised computerized assisted programs are under study, but performance compared with simpler methods is less well understood44. The great advantage of a fully automatic method would be precision since there is no user variation. Benxinque, et al.52 recently compared prolate ellipsoid and other prostate volume measures to a commercially available fully automated segmentation program. Although fully automatic measurement was not reliable compared to manually adjusted MRI segmentation, they demonstrated strong correlation of manually adjusted MRI 3D segmentation by an experienced radiologist relative to a medical student (ICC 0.91) and relative to MRI EVF (ICC 0.90). They also found that in the hands of a radiologist, the manually adjustment time was acceptably short.

It should be remembered that prostate imaging is not necessary in the initial work-up of lower urinary tract symptoms, and only is considered when surgical or minimally invasive procedures are under discussion53. TRUS, because of its cost effectiveness and availability will likely continue to be the basic clinical volumetric tool. But MRI will be useful for research studies of treatments requiring follow-up monitoring6 and in the clinical assessment of larger prostates and could replace ultrasound if limited protocols can be found and costs reduced.

It may be time to accept MRI as the new “gold standard” for assessing prostate volume, especially in research protocols6,12,50.

## Conclusions

We present a succinct review of the scientific foundations for identifying the anatomic boundaries necessary for reproducible multiplane volume measurement of the prostate using MRI. Detailed landmarks for these measures are sufficiently shown to allow for reliable transfer of method to the general community of radiologists and urologists. An alternative method of measurement (biproximate) is introduced to enhance measurement consistency and to answer the needs of operating urologic surgeons interested in the effects of IPP. The measurements tested showed excellent inter- and intra-observer reliability and is suitable especially for multi-observer research.


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# point estimate of the population mean calculator

FAQ. A little bird, a Mocking Jay perhaps, tells you that you can end the game by shooting an arrow into the sky and hitting some unknown point that will disable the power source of the city that put you there in the first place. A point estimate of the mean of a population is determined by calculating the mean of a sample drawn from the population. Table of contents. Let’s input the values we already have into the formula: Therefore, the error is 13.4 and the point estimate for the population mean is 139.2. Note that a Finite Population Correction has been applied to the sample size formula. After calculating all of the four values manually, you can check the accuracy of your calculation using the online calculator. The population mean is the average of the data of a group characteristics of a population, where the group characteristic can be anything such as people, things, animals etc. For example, in the data set survey, the survey is performed on a sample of the student population. Also, it provides you with the results for all the formulas. Find the point estimate of students who believe in BigFoot. computing the point estimate of a population mean example: estimating a population proportion calculator: how to find sample size needed for confidence interval: the point estimate of the population mean is: 68 confidence interval calculator: critical value confidence level calculator: finding confidence interval without standard deviation Point Estimate Calculator – How to Find Point Estimate? To calculate a confidence interval, you will first need the point estimate and, in some cases, its standard deviation. Population variance (σ 2) indicates how data points in a given population are distributed. If the value is 0.9 < MLE, select the smaller value between the Laplace and Jeffrey Estimations as this is the most accurate. surveyed was 1.28 hours. In more formal terms, the estimate occurs as a result of point estimation applied to a set of sample data. Expected value of the estimator. To illustrate this, let’s work with an example. Another way to express the confidence interval is as the point estimate plus or minus a margin of error; in this case, it is 198 ± 6 pounds. How to Calculate Minkowski Distance in R (With Examples). Thus, the variance itself is the mean of the random variable Y = (X − μ)2. Learn more. Consider these examples: When you look at this in a more formal perspective, the occurrence of the estimate is a result of the application of the point estimate to a sample data set. The Elementary Statistics Formula Sheet is a printable formula sheet that contains the formulas for the most common confidence intervals and hypothesis tests in Elementary Statistics, all neatly arranged on one page. About Population Variance Calculator . You want to go back 20 years and calculate monthly return but that will become very hectic. The confidence interval gives us a range of reasonable values for the difference in population means μ 1 − μ 2. Although most often it is not the actual population mean, it does serve as a good point estimate. Statology is a site that makes learning statistics easy. The calculation of the mean is the sum of all sample values divided by the number of values. Arithmetic Mean For Samples And Populations. Get the formula sheet here: Statistics in Excel Made Easy is a collection of 16 Excel spreadsheets that contain built-in formulas to perform the most commonly used statistical tests. If you have an unbiased coin and you toss it several times, you should get a result of about 50% tails and 50% heads. In the example we used, this refers to the number of tails you got after tossing the coin for a certain number of times. These are the Maximum Likelihood Estimation formula or MLE, the Wilson Estimation formula, the Laplace Estimation formula, and the Jeffrey Estimation formula. A tutorial on computing the point estimate of population mean from a simple random sample. How to use the point estimate calculator? But what happens if your coin has a slight bias. Going back to calculating the point estimate manually, follow these steps: In some cases, you need to determine the point estimate of a population mean. Here is the simple online best point estimation calculator to find the best guess along with the Laplace, Jeffrey, Wilson, Maximum Likelihood Estimation. Find a 95% confidence interval for a population mean, given the following information: sample mean x = 12; sample size n = 19 As you learn how to find point estimate, there are different point estimate formulas for you to use. The above discussion suggests the sample mean, ¯ X, is often a reasonable point estimator for the mean. Finally, you should also have the value for the confidence interval. This point estimate calculator is very useful, especially in finding point estimate statistics. Just enter the values for the number of success, a number of trials and confidence interval to find the estimation. Let say you want to invest in IBM and very keen to look at its past performance and returns. Show Video Lesson. Obtaining a point estimate of a population parameter is rather easy: just use the corresponding sample statistic. This point estimate calculator can help you quickly and easily determine the most suitable point estimate according to the size of the sample, number of successes, and required confidence level. Similarly, a sample proportion can be used as a point estimate of a population proportion. Point Estimate and Margin of Error Calculator The best thing about this online tool is that it’s very easy to use. For example, a sample mean can be used as a point estimate of a population mean. Each time you toss the coin, take note of the result. For example, a sample mean can be used as a point estimate of a population mean. In our example, this refers to the number of times you tossed the coin. It makes use of the different point estimate formulas to provide you with the most precise possible value. If you instead know the population standard deviation, you should use our Confidence Interval Calculator for the Mean with known Population Standard Deviation. You also need to have the value of the z-score which comes from when you’ve calculated the confidence interval. This lecture presents some examples of point estimation problems, focusing on mean estimation, that is, on using a sample to produce a point estimate of the mean of an unknown distribution. How To Calculate A Point Estimate For A Population Mean? Biostatistics: A Foundation for Analysis in the Health Sciences. Confidence Interval Calculator for the Population Mean This calculator will compute the 99%, 95%, and 90% confidence intervals for the mean of a normal population, given the sample mean, the sample size, and the sample standard deviation. ̅=7.35, =1.28 2 = 1−.98 2 First, you need to have the value for the number of successes. Unbiased estimation of standard deviation however, is highly involved and varies depending on distribution. This shows that the slight bias changes the probability of getting either of the results for that specific coin. After you have tossed your biased coin for a certain number of times and you’ve collected enough data pertaining to the “behavior” of the coin, you can use that data when using the point estimate calculator.eval(ez_write_tag([[300,250],'calculators_io-box-4','ezslot_8',104,'0','0']));eval(ez_write_tag([[300,250],'calculators_io-box-4','ezslot_9',104,'0','1']));eval(ez_write_tag([[300,250],'calculators_io-box-4','ezslot_10',104,'0','2'])); Of course, you can also perform the calculations manually then check the results with the calculator. Aside from determining the point estimate of the population mean, you also have to determine the margin of error when the lower bound of your confidence interval is 125.8 and the upper bound of your confidence interval is 152.6.eval(ez_write_tag([[970,90],'calculators_io-banner-1','ezslot_11',105,'0','0'])); In the example, we’re given [125.8<=mu<=152.6]. More formally, it is the application of a point estimator to the data. Variance of the estimator. On the other hand, interval estimation uses sample data to calcul… A study aims to estimate the mean systolic blood pressure of British adults by randomly sampling and measuring the blood pressure of 100 adults from the population. Normal IID samples. Distribution of the estimator. Point Estimate for the Population Variance & Standard Deviation. σ (Greek letter sigma) is the symbol for the population standard deviation. If the value is 0.5 < MLE < 0.9, select the Maximum Likelihood Estimation as this is the most accurate. for a population mean; σ unknown. Using your TI-83/84/89 Calculator: Estimating a Population Mean (σ Unknown)Dr. Laura Schultz Statistics I When the population standard deviation (σ) is not known (as is generally the case), a confidence interval estimate for a population mean (μ) is constructed using a critical value from the Student’s tdistribution. FAQ. If you need to find the most accurate point estimates, follow these steps: eval(ez_write_tag([[728,90],'calculators_io-medrectangle-3','ezslot_7',110,'0','0'])); eval(ez_write_tag([[300,250],'calculators_io-medrectangle-4','ezslot_4',103,'0','0']));eval(ez_write_tag([[300,250],'calculators_io-medrectangle-4','ezslot_5',103,'0','1']));eval(ez_write_tag([[300,250],'calculators_io-medrectangle-4','ezslot_6',103,'0','2']));For you to understand the concept of point estimate better, let’s use a simple example. You are 90 percent certain that the true population mean of football player weights is between 192 and 204 pounds. Point Estimate of Population Mean A tutorial on computing the point estimate of population mean from a simple random sample. The great thing about this calculator is that it selects the most relevant result for you automatically. Once you press ENTER, the 95% confidence interval for the population mean will be displayed: The 95% confidence interval for the population mean is (12.675, 15.325). Once you’ve verified that they’re all correct, you can select the most accurate value. Apart from learning how to use this point estimate calculator, we’ll also learn how to find point estimate manually, and other relevant information. The estimator . Similarly, a sample proportion can be used as a point estimate of a population proportion. by Marco Taboga, PhD. Once you have all of the required values, you can use the formulas to calculate the point estimate. The expectation of the observed values of many samples (“average observation value”) equals the corresponding population parameter. Point Estimate Calculator This point estimate calculator can help you quickly and easily determine the most suitable point estimate according to the size of the sample, number of … Population variance (σ 2) indicates how data points in a given population are distributed.This is the average of the distances from each data point in the population to the mean square. How do you find the point estimate of the population mean? The Population Standard Deviation Calculator is used to calculate the population standard deviation of a set of numbers. This calculator will compute the 99%, 95%, and 90% confidence intervals for the mean of a normal population, given the sample mean, the sample size, and the sample standard deviation. Show Instructions. After entering all of the required values, the calculator will generate a number of results including the Best Point Estimation, the Maximum Likelihood Estimation, the Laplace Estimation, Jeffrey’s Estimation, and the Wilson Estimation. Using the simplest definition, any statistic can also be a point estimate. There are a lot of point estimators with different properties. Population Mean = (14+61+83+92+2+8+48+25+71+12) / 10; Population Mean = 416 / 10; Population Mean = 41.6; Example #2. Please enter the necessary parameter values, and then click 'Calculate'. We can't directly calculate the population average because we only have a sample, so we have to estimate it. A population parameter is assumed to be fixed or take only one value. It is usually an unknown constant. Sample statistics or statistics are observable because we calculate them from the data (or sample) ... An example, would be to use the sample mean as a point estimate of the population mean, here the population mean is the population parameter we are interested in finding out about. Find the point estimate A sample was taken from students to ask them how many believe in BigFoot. It is desirable for a point estimate to be: (1) Consistent. This online population mean calculator helps you to find the mean for the given set of data of a group characteristics. Point Estimate for Population Proportion n x pˆ = 1. It uses sample data when calculating a single statistic that will be the best estimate of the unknown parameter of the population. A sample standard deviation “s” is the point estimate of a population standard deviation “σ.”, A sample mean “x” is a point estimate of a population mean “μ.”, A sample variance “s2” is a point estimate of a population variance “σ2.”. Confidence Interval Calculator for the Population Mean. When the sample mean is used as a point estimate of the population mean, some error can be expected owing to the fact that a sample, or subset of the population, is used to compute the point estimate. In such a case, you might discover after tossing the coin a lot, either the head or the tail appears more frequently. Your email address will not be published. The population standard deviation measures the variability of data in a population. Find the population mean using this formula:eval(ez_write_tag([[250,250],'calculators_io-large-leaderboard-2','ezslot_12',111,'0','0'])); [bar(x)-“error”<=mu<=bar(x)+”error”] where [bar(x)] is the point estimate for the mean. Try to imagine that you’re tossing a coin. Similar to this example, you might … The other important piece of information is the confidence level required, which is the probability that the confidence interval contains the true point estimate. In cases where you’ve collected a lot of data about a specific population and you want to calculate the “best guess” parameter, this point estimate calculator proves extremely useful. The calculator uses four estimation approaches to compute the most suitable point estimate: the maximum likelihood, Wilson, Laplace, and Jeffrey's methods. The points are individual values compared to the interval estimates which are a set of values. Instructions: Use this Confidence Interval Calculator to compute a confidence interval for the population mean $$\mu$$, in the case that the population standard deviation $$\sigma$$ is known. Confidence interval (limits) calculator, formulas & workout with steps to measure or estimate confidence limits for the mean or proportion of finite (known) or infinite (unknown) population by using standard deviation or p value in statistical surveys or experiments. It is usually an unknown constant. The population standard deviation measures the variability of data in a population. Point Estimate. It produces a single value while the latter produces a range of values. Unbiased estimation of standard deviation however, is highly involved and varies depending on distribution. What is Population Variance? Given a lower bound of 17, upper bound of 27, and sample size of , calculate the point estimate and margin of error (2) Unbiased. However, there are several ways to calculate the point estimate of a population proportion, including: To find the best point estimate, simply enter in the values for the number of successes, number of trials, and confidence level in the boxes below and then click the “Calculate” button. Data collected from a simple random sample can be used to compute the sample mean, x̄, where the value of x̄ provides a point estimate of μ. In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). Sample/Population Variance Calculator. Example 2: C.I. Point estimation is the opposite of interval estimation. The following reference explains how the FPC is used to adjust a variance estimate when sampling without replacement (see pages 141-142). Therefore, you must use these formulas in different situations too. This calculator uses the following logic to determine which point estimate is best to use: Your email address will not be published. Of the 300 surveyed, 25 believe in BigFoot. The two main types of estimators in statistics are point estimators and interval estimators. For example, the sample mean X̄ is the point estimate of the population mean μ. The arithmetic mean is a single value meant to “sum up” a data set. Now, suppose that we would like to estimate the variance of a distribution σ2. Get the spreadsheets here: Try out our free online statistics calculators if you’re looking for some help finding probabilities, p-values, critical values, sample sizes, expected values, summary statistics, or correlation coefficients. Daniel WW (1999). In practice, when the sample mean difference is statistically significant, our next step is often to calculate a confidence interval to estimate the size of the population mean difference. This is similar to what researchers and statisticians do: out of an arena of possibl… Assuming 0 < σ2 < ∞, by definition σ2 = E[(X − μ)2]. σ (Greek letter sigma) is the symbol for the population standard deviation. This is the average of the distances from each data point in the population to the mean square. Please type the sample mean, the population standard deviation, the sample size and the confidence level, and the confidence interval will be computed for you: Upgrade to Math Mastery. Because we are estimating the population mean, we need to find the point estimate, and we will use the t-table in our confidence interval for the critical value. For the given set of values, the calculator will find their variance (either sample or population), with steps shown. A point estimate gives statisticians a single value as the estimate of a given population parameter. Calculating a confidence interval involves determining the sample mean, X̄, and the population standard deviation, σ, if possible. A point estimator is a statistic used to estimate the value of an unknown parameter of a population. From their sample, they estimate the sample mean to be 70mmHg and the sample standard deviation to be 8mmHg. A point estimate is the value of a statistic that estimates the value of a parameter. Best Point Estimation Calculator. This refers to the probability that you’ve made the best point estimate correctly and it’s within the margin of error. Point estimation involves the use of sample data to calculate a single value (known as a statistic) which is to serve as a "best guess" or "best estimate" of an unknown (fixed or random) population parameter. Point Estimate. As such, the "corrected sample standard deviation" is the most commonly used estimator for population standard deviation, and is generally referred to as simply the "sample standard deviation." You should also have the value for the number of trials. The Population Variance Calculator is used to calculate the population variance of a set of numbers. This is because a statistic serves as an estimator of a given parameter in a population. Similarly, the sample proportion p is a point estimate of the population proportion p when binomial modeling is involved. These are the only values needed by this calculator to give you the point estimate statistics. What would happen to the confidence interval if you wanted to be 95 percent certain of it? Base your selection on these rules: If the value of the MLE ≤ 0.5, select the Wilson Estimation as this is the most accurate. Points are single values, in comparison to interval estimates, which are a Using each of these formulas will provide you with results that differ slightly. The point estimate refers to the probability of getting one of the results. Point estimation of the mean. What Is Population Standard Deviation? As such, the "corrected sample standard deviation" is the most commonly used estimator for population standard deviation, and is generally referred to as simply the "sample standard deviation." The larger the sample size, the more accurate the estimate. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. Calculating confidence intervals: Calculating a confidence interval involves determining the sample mean, X̄, and the population standard deviation, σ, if possible. For instance, what if it’s bent slightly? For example, the sample mean X̄ is the point estimate of the population mean μ. A sample mean “x” is a point estimate of a population mean “μ.” A sample variance “s2” is a point estimate of a population variance “σ2.” When you look at this in a more formal perspective, the occurrence of the estimate is a result of the application of the point estimate to a sample data set. For example, the sample mean is a point estimate of the population mean. There are other confidence intervals you can use such as the confidence interval for the sample variance, the confidence interval for slope coefficients, or confidence intervals and prediction intervals for regression estimate . A point estimate gives statisticians a single value as the estimate of a given population parameter. If the population standard deviation cannot be used, then the sample standard deviation, s, can be used when the sample size is greater than 30. We can compute the sample mean and use it as an estimate of the corresponding population parameter. Best Guess Estimation Calculation: Feel free to use this online best point estimation calculator to calculate the point estimation which involves the utilization of sample data to determine a single value which is to serve as a 'best guess' or 'best estimate' of an unknown population parameter. The expected value is the average value in the target population - the population mean. Here are the equations for the different formulas: for the Maximum Likelihood Estimation, the equation is MLE = S / T, for the Laplace Estimation, the equation is Laplace = (S + 1) / (T + 2), for the Jeffrey Estimation, the equation is Jeffrey = (S + 0.5) / (T + 1), for the Wilson Estimation, the equation is Wilson = (S + z²/2) / (T + z²). Please enter the necessary parameter values, and then click 'Calculate'. Point Estimate Calculator. Calculate and interpret a 98% confidence interval for the mean number of hours practiced by the dancers at the competition. Imagine you are trapped inside a dangerous dome with 20 game contestants who can only win the game by being the last person left alive. Point Estimate Calculator A point estimate represents our “best guess” of a population parameter. Using this information the 95% confidence interval is calculated as between 68.43 and 71.57mmHg.