Polygon

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Polygon

Steven Yen
Can someone help me with polygon. The following codes are self-runnable
and mark a shaded area under the standard normal curve in the x-range
(-3,-1).
Is there a way to also mark the area in (1,3), at the same time.
That is, I want shaded areas in both tails. Thank you...

===
# Create data for the area to shade
cord.x <- c(-3,seq(-3,-1,0.01),-1)
cord.y <- c(0,dnorm(seq(-3,-1,0.01)),0)

# Make a curve
curve(dnorm(x,0,1), xlim=c(-3,3), main='Standard Normal')

# Add the shaded area.
polygon(cord.x,cord.y,col='skyblue')

--
[hidden email] (S.T. Yen)


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Re: Polygon

Rui Barradas
Hello,

Continue with


polygon(-rev(cord.x), rev(cord.y), col = 'skyblue')


Hope this helps,

Rui Barradas

Às 17:25 de 02/10/2018, Steven Yen escreveu:

> Can someone help me with polygon. The following codes are self-runnable
> and mark a shaded area under the standard normal curve in the x-range
> (-3,-1).
> Is there a way to also mark the area in (1,3), at the same time.
> That is, I want shaded areas in both tails. Thank you...
>
> ===
> # Create data for the area to shade
> cord.x <- c(-3,seq(-3,-1,0.01),-1)
> cord.y <- c(0,dnorm(seq(-3,-1,0.01)),0)
>
> # Make a curve
> curve(dnorm(x,0,1), xlim=c(-3,3), main='Standard Normal')
>
> # Add the shaded area.
> polygon(cord.x,cord.y,col='skyblue')
>

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Re: Polygon

Steven Yen
Great. Thanks! It did wonders.
1. Is there a way to suppress the obvious tick stops (-3,-2,-1,0,1,2,3)
and mark only the ticks -1.96 and 1.96.
2. Better yet, to draw vertical lines at x = -1.96 and x = 1.96.
Thanks.
Steven

On 10/3/2018 12:51 AM, Rui Barradas wrote:

> Hello,
>
> Continue with
>
>
> polygon(-rev(cord.x), rev(cord.y), col = 'skyblue')
>
>
> Hope this helps,
>
> Rui Barradas
>
> Às 17:25 de 02/10/2018, Steven Yen escreveu:
>> Can someone help me with polygon. The following codes are self-runnable
>> and mark a shaded area under the standard normal curve in the x-range
>> (-3,-1).
>> Is there a way to also mark the area in (1,3), at the same time.
>> That is, I want shaded areas in both tails. Thank you...
>>
>> ===
>> # Create data for the area to shade
>> cord.x <- c(-3,seq(-3,-1,0.01),-1)
>> cord.y <- c(0,dnorm(seq(-3,-1,0.01)),0)
>>
>> # Make a curve
>> curve(dnorm(x,0,1), xlim=c(-3,3), main='Standard Normal')
>>
>> # Add the shaded area.
>> polygon(cord.x,cord.y,col='skyblue')
>>
>

--
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Re: Polygon

David Winsemius

> On Oct 2, 2018, at 5:50 PM, Steven Yen <[hidden email]> wrote:
>
> Great. Thanks! It did wonders.
> 1. Is there a way to suppress the obvious tick stops (-3,-2,-1,0,1,2,3)
> and mark only the ticks -1.96 and 1.96.
> 2. Better yet, to draw vertical lines at x = -1.96 and x = 1.96.
> Thanks.

Read ?plot.default

Then add xaxt="n" to the arguments to curve, ... and afterwards:

axis( 1, at=c(-1.96, 1.96),lab=c(-1.96, 1.96) )
abline( v= c(-1.96, 1.96) )

--

> Steven
>
> On 10/3/2018 12:51 AM, Rui Barradas wrote:
>> Hello,
>>
>> Continue with
>>
>>
>> polygon(-rev(cord.x), rev(cord.y), col = 'skyblue')
>>
>>
>> Hope this helps,
>>
>> Rui Barradas
>>
>> Às 17:25 de 02/10/2018, Steven Yen escreveu:
>>> Can someone help me with polygon. The following codes are self-runnable
>>> and mark a shaded area under the standard normal curve in the x-range
>>> (-3,-1).
>>> Is there a way to also mark the area in (1,3), at the same time.
>>> That is, I want shaded areas in both tails. Thank you...
>>>
>>> ===
>>> # Create data for the area to shade
>>> cord.x <- c(-3,seq(-3,-1,0.01),-1)
>>> cord.y <- c(0,dnorm(seq(-3,-1,0.01)),0)
>>>
>>> # Make a curve
>>> curve(dnorm(x,0,1), xlim=c(-3,3), main='Standard Normal')
>>>
>>> # Add the shaded area.
>>> polygon(cord.x,cord.y,col='skyblue')
>>>
>>
>
> --
> [hidden email] (S.T. Yen)
>
>
> [[alternative HTML version deleted]]
>
> ______________________________________________
> [hidden email] mailing list -- To UNSUBSCRIBE and more, see
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.

David Winsemius
Alameda, CA, USA

'Any technology distinguishable from magic is insufficiently advanced.'   -Gehm's Corollary to Clarke's Third Law

______________________________________________
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Re: Polygon

Steven Yen
Thanks!!! It did wonders.
Steven

On 10/3/2018 9:39 AM, David Winsemius wrote:

>> On Oct 2, 2018, at 5:50 PM, Steven Yen <[hidden email]> wrote:
>>
>> Great. Thanks! It did wonders.
>> 1. Is there a way to suppress the obvious tick stops (-3,-2,-1,0,1,2,3)
>> and mark only the ticks -1.96 and 1.96.
>> 2. Better yet, to draw vertical lines at x = -1.96 and x = 1.96.
>> Thanks.
> Read ?plot.default
>
> Then add xaxt="n" to the arguments to curve, ... and afterwards:
>
> axis( 1, at=c(-1.96, 1.96),lab=c(-1.96, 1.96) )
> abline( v= c(-1.96, 1.96) )
>
> --
>> Steven
>>
>> On 10/3/2018 12:51 AM, Rui Barradas wrote:
>>> Hello,
>>>
>>> Continue with
>>>
>>>
>>> polygon(-rev(cord.x), rev(cord.y), col = 'skyblue')
>>>
>>>
>>> Hope this helps,
>>>
>>> Rui Barradas
>>>
>>> Às 17:25 de 02/10/2018, Steven Yen escreveu:
>>>> Can someone help me with polygon. The following codes are self-runnable
>>>> and mark a shaded area under the standard normal curve in the x-range
>>>> (-3,-1).
>>>> Is there a way to also mark the area in (1,3), at the same time.
>>>> That is, I want shaded areas in both tails. Thank you...
>>>>
>>>> ===
>>>> # Create data for the area to shade
>>>> cord.x <- c(-3,seq(-3,-1,0.01),-1)
>>>> cord.y <- c(0,dnorm(seq(-3,-1,0.01)),0)
>>>>
>>>> # Make a curve
>>>> curve(dnorm(x,0,1), xlim=c(-3,3), main='Standard Normal')
>>>>
>>>> # Add the shaded area.
>>>> polygon(cord.x,cord.y,col='skyblue')
>>>>
>> --
>> [hidden email] (S.T. Yen)
>>
>>
>> [[alternative HTML version deleted]]
>>
>> ______________________________________________
>> [hidden email] mailing list -- To UNSUBSCRIBE and more, see
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
>> and provide commented, minimal, self-contained, reproducible code.
> David Winsemius
> Alameda, CA, USA
>
> 'Any technology distinguishable from magic is insufficiently advanced.'   -Gehm's Corollary to Clarke's Third Law
>
>
>
>
>
>

--
[hidden email] (S.T. Yen)


        [[alternative HTML version deleted]]

______________________________________________
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Re: Polygon

Rui Barradas
Hello,

If you change cord.x and cord.y to


cord.x <- c(-3, seq(-3, -1.96, 0.01), -1.96)
cord.y <- c(0, dnorm(seq(-3, -1.96, 0.01)), 0)


then you will not need the call to abline.

Hope this helps,

Rui Barradas

Às 02:50 de 03/10/2018, Steven Yen escreveu:

> Thanks!!! It did wonders.
> Steven
>
> On 10/3/2018 9:39 AM, David Winsemius wrote:
>>> On Oct 2, 2018, at 5:50 PM, Steven Yen<[hidden email]>  wrote:
>>>
>>> Great. Thanks! It did wonders.
>>> 1. Is there a way to suppress the obvious tick stops (-3,-2,-1,0,1,2,3)
>>> and mark only the ticks -1.96 and 1.96.
>>> 2. Better yet, to draw vertical lines at x = -1.96 and x = 1.96.
>>> Thanks.
>> Read ?plot.default
>>
>> Then add xaxt="n" to the arguments to curve, ... and afterwards:
>>
>> axis( 1, at=c(-1.96, 1.96),lab=c(-1.96, 1.96) )
>> abline( v= c(-1.96, 1.96) )
>>
>> --
>>> Steven
>>>
>>> On 10/3/2018 12:51 AM, Rui Barradas wrote:
>>>> Hello,
>>>>
>>>> Continue with
>>>>
>>>>
>>>> polygon(-rev(cord.x), rev(cord.y), col = 'skyblue')
>>>>
>>>>
>>>> Hope this helps,
>>>>
>>>> Rui Barradas
>>>>
>>>> Às 17:25 de 02/10/2018, Steven Yen escreveu:
>>>>> Can someone help me with polygon. The following codes are self-runnable
>>>>> and mark a shaded area under the standard normal curve in the x-range
>>>>> (-3,-1).
>>>>> Is there a way to also mark the area in (1,3), at the same time.
>>>>> That is, I want shaded areas in both tails. Thank you...
>>>>>
>>>>> ===
>>>>> # Create data for the area to shade
>>>>> cord.x <- c(-3,seq(-3,-1,0.01),-1)
>>>>> cord.y <- c(0,dnorm(seq(-3,-1,0.01)),0)
>>>>>
>>>>> # Make a curve
>>>>> curve(dnorm(x,0,1), xlim=c(-3,3), main='Standard Normal')
>>>>>
>>>>> # Add the shaded area.
>>>>> polygon(cord.x,cord.y,col='skyblue')
>>>>>
>>> --
>>> [hidden email]  (S.T. Yen)
>>>
>>>
>>> [[alternative HTML version deleted]]
>>>
>>> ______________________________________________
>>> [hidden email]  mailing list -- To UNSUBSCRIBE and more, see
>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>> PLEASE do read the posting guidehttp://www.R-project.org/posting-guide.html
>>> and provide commented, minimal, self-contained, reproducible code.
>> David Winsemius
>> Alameda, CA, USA
>>
>> 'Any technology distinguishable from magic is insufficiently advanced.'   -Gehm's Corollary to Clarke's Third Law
>>
>>
>>
>>
>>
>>
>
> --
> [hidden email]  (S.T. Yen)
>

______________________________________________
[hidden email] mailing list -- To UNSUBSCRIBE and more, see
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PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
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Re: Polygon

Steven Yen
In reply to this post by David Winsemius
David, Rui, and All:
Greetings.
1. I need a helping hand with the polygon statement below so that I can
have the area under the curve highlighted, between (z1,z2).
2. Is it possible to label the X-axis with in two scale, in the current
z-scale and another, say x = (z+5)*2?
Thank you.

z1<- -1
z2<-  2
curve(dnorm(x,0,1),xlim=c(-4,4),main="Standard
Normal",xaxt="n",frame=F,xlab="z")
jj<-seq(z1,z2,0.01)
cord.x<-c(jj)
cord.y<-c(dnorm(jj))
#polygon(cord.x,cord.y,col="skyblue")
axis(1,at=c(-5,z1,0,z2,5),lab=c(-5,z1,0,z2,5))
abline(v=c(z1,z2))

--
[hidden email] (S.T. Yen)


        [[alternative HTML version deleted]]

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Re: Polygon

David Carlson
I think this does what you want.

z1<- -1
z2<-  2
oldp <- par(mgp=c(3, .5, 0))
curve(dnorm(x,0,1), xlim=c(-5, 5),main="Standard Normal",
    xaxt="n", frame=F, xlab="")
jj<-seq(z1,z2,0.01)
cord.x<- jj
cord.y<- dnorm(jj)
polygon(c(-1, cord.x, 2), c(0, cord.y, 0), col="skyblue")
axis(1,at=c(-5,z1,0,z2,5),lab=c(-5,z1,0,z2,5))
abline(v=c(z1,z2))
axis(1, at=c(-5, z1, 0, z2, 5), lab=(c(-5,z1,0,z2,5) + 5)*2, line=2)
par(oldp)

When you assign a vector to an object, you do not need to use c(), e.g. cord.x <- jj. I expanded xlim= since you were labeling the axis at -5, and 5, but only drawing the plot from -4 to 4.

To get the polygon to work, you need to add points to the baseline at at either end.

The line= argument in axis() lets you put an axis line into the margin of the plot. I used the mgp= argument in par() to close the space between the tickmarks and their values.
--------------------------------------
David L. Carlson
Department of Anthropology
Texas A&M University

-----Original Message-----
From: R-help [mailto:[hidden email]] On Behalf Of Steven Yen
Sent: Sunday, October 21, 2018 1:47 PM
To: David Winsemius <[hidden email]>
Cc: [hidden email]
Subject: Re: [R] Polygon

David, Rui, and All:
Greetings.
1. I need a helping hand with the polygon statement below so that I can
have the area under the curve highlighted, between (z1,z2).
2. Is it possible to label the X-axis with in two scale, in the current
z-scale and another, say x = (z+5)*2?
Thank you.

z1<- -1
z2<-  2
curve(dnorm(x,0,1),xlim=c(-4,4),main="Standard
Normal",xaxt="n",frame=F,xlab="z")
jj<-seq(z1,z2,0.01)
cord.x<-c(jj)
cord.y<-c(dnorm(jj))
#polygon(cord.x,cord.y,col="skyblue")
axis(1,at=c(-5,z1,0,z2,5),lab=c(-5,z1,0,z2,5))
abline(v=c(z1,z2))

--
[hidden email] (S.T. Yen)


        [[alternative HTML version deleted]]

______________________________________________
[hidden email] mailing list -- To UNSUBSCRIBE and more, see
https://stat.ethz.ch/mailman/listinfo/r-help
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and provide commented, minimal, self-contained, reproducible code.
______________________________________________
[hidden email] mailing list -- To UNSUBSCRIBE and more, see
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Re: Polygon

David Carlson
Yes, par(oldp) resets the parameter to its original value.

The simplest way to get the area in this case would be to use pnorm():

> pnorm(2) - pnorm(-1)
[1] 0.8185946

If you have an arbitrary polygon, you can use polyarea() in the pracma package:

> library(pracma)
> polyarea(rev(c(-1, cord.x, 2)), rev(c(0, cord.y, 0)))
[1] 0.8185917

The second differs slightly from the first because it is an approximation based on 101 points (the default for the curve() function). The polyarea() function wants the polygon to be defined counter-clockwise so we use rev() to reverse the coordinates. Without this you will get a negative area, but you can always use abs() to fix that:

> polyarea(c(-1, cord.x, 2), c(0, cord.y, 0))
[1] -0.8185917
> abs(polyarea(c(-1, cord.x, 2), c(0, cord.y, 0)))
[1] 0.8185917

One more point. Don’t use abbreviations for TRUE and FALSE (e.g. frame=F). FALSE cannot be redefined, but F can be. If you used a variable called F in earlier code, it could change the behavior of curve():

> curve(dnorm(x,0,1), xlim=c(-5, 5), frame=F)
> FALSE <- 1
Error in FALSE <- 1 : invalid (do_set) left-hand side to assignment
> F <- 1
> curve(dnorm(x,0,1), xlim=c(-5, 5), frame=F)

----------------------------------------
David L Carlson
Department of Anthropology
Texas A&M University
College Station, TX 77843-4352



From: Steven Yen <[hidden email]>
Sent: Sunday, October 21, 2018 9:54 PM
To: David L Carlson <[hidden email]>
Subject: Re: [R] Polygon

Thank you David. This is great!!
I can follow all you do, except one. What does the last statement do: par(oldp). I would think that it reset the parameters, needed only if there is a next job.

I am a little embarrassed to push this further, but let me see if I can bounce another thing out of this.

Is it possible to have a "call-out", somehow indicating that the probability associated witht he shaded area is, say, 0.45. Is this possible? Am I asking too much of R? Fine if not. Thanks again!!!
On 10/22/2018 4:24 AM, David L Carlson wrote:
I think this does what you want.

z1<- -1
z2<-  2
oldp <- par(mgp=c(3, .5, 0))
curve(dnorm(x,0,1), xlim=c(-5, 5),main="Standard Normal",
    xaxt="n", frame=F, xlab="")
jj<-seq(z1,z2,0.01)
cord.x<- jj
cord.y<- dnorm(jj)
polygon(c(-1, cord.x, 2), c(0, cord.y, 0), col="skyblue")
axis(1,at=c(-5,z1,0,z2,5),lab=c(-5,z1,0,z2,5))
abline(v=c(z1,z2))
axis(1, at=c(-5, z1, 0, z2, 5), lab=(c(-5,z1,0,z2,5) + 5)*2, line=2)
par(oldp)

When you assign a vector to an object, you do not need to use c(), e.g. cord.x <- jj. I expanded xlim= since you were labeling the axis at -5, and 5, but only drawing the plot from -4 to 4.

To get the polygon to work, you need to add points to the baseline at at either end.

The line= argument in axis() lets you put an axis line into the margin of the plot. I used the mgp= argument in par() to close the space between the tickmarks and their values.
--------------------------------------
David L. Carlson
Department of Anthropology
Texas A&M University

-----Original Message-----
From: R-help [mailto:[hidden email]] On Behalf Of Steven Yen
Sent: Sunday, October 21, 2018 1:47 PM
To: David Winsemius mailto:[hidden email]
Cc: mailto:[hidden email]
Subject: Re: [R] Polygon

David, Rui, and All:
Greetings.
1. I need a helping hand with the polygon statement below so that I can
have the area under the curve highlighted, between (z1,z2).
2. Is it possible to label the X-axis with in two scale, in the current
z-scale and another, say x = (z+5)*2?
Thank you.

z1<- -1
z2<-  2
curve(dnorm(x,0,1),xlim=c(-4,4),main="Standard
Normal",xaxt="n",frame=F,xlab="z")
jj<-seq(z1,z2,0.01)
cord.x<-c(jj)
cord.y<-c(dnorm(jj))
#polygon(cord.x,cord.y,col="skyblue")
axis(1,at=c(-5,z1,0,z2,5),lab=c(-5,z1,0,z2,5))
abline(v=c(z1,z2))



--
mailto:[hidden email] (S.T. Yen)
______________________________________________
[hidden email] mailing list -- To UNSUBSCRIBE and more, see
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and provide commented, minimal, self-contained, reproducible code.
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Re: Polygon

David Carlson
Putting it all together:

z1<- -1
z2<-  2
oldp <- par(mgp=c(3, .5, 0))
curve(dnorm(x,0,1), xlim=c(-5, 5),main="Standard Normal",
    xaxt="n", frame=F, xlab="")
jj<-seq(z1,z2,0.01)
cord.x<- jj
cord.y<- dnorm(jj)
polygon(c(-1, cord.x, 2), c(0, cord.y, 0), col="skyblue")
axis(1,at=c(-5,z1,0,z2,5),lab=c(-5,z1,0,z2,5))
abline(v=c(z1,z2))
axis(1, at=c(-5, z1, 0, z2, 5), lab=(c(-5,z1,0,z2,5) + 5)*2, line=2)
segments(.5, .1, 3, .2, lwd=2)
lbl <- paste0("Area = ", round(pnorm(2) - pnorm(-1), 4))
text(3, .2, lbl, pos=4, offset=.25)
par(oldp)


----------------------------------------
David L Carlson
Department of Anthropology
Texas A&M University
College Station, TX 77843-4352





From: Steven Yen <[hidden email]>
Sent: Monday, October 22, 2018 11:53 AM
To: David L Carlson <[hidden email]>
Subject: Re: [R] Polygon

Thanks David. I do know what to do with pnorm.
By call-out I was hoping to get indication/annotation of probability like the attached.
Maybe I am asking too much of the data, as I suspect R is not meant to do that.
I appreciate your help. I am just trying to make these graphs for stat teaching without graphic software like correl draw.
On 10/22/2018 9:34 PM, David L Carlson wrote:
Yes, par(oldp) resets the parameter to its original value.

The simplest way to get the area in this case would be to use pnorm():

pnorm(2) - pnorm(-1)
[1] 0.8185946

If you have an arbitrary polygon, you can use polyarea() in the pracma package:

library(pracma)
polyarea(rev(c(-1, cord.x, 2)), rev(c(0, cord.y, 0)))
[1] 0.8185917

The second differs slightly from the first because it is an approximation based on 101 points (the default for the curve() function). The polyarea() function wants the polygon to be defined counter-clockwise so we use rev() to reverse the coordinates. Without this you will get a negative area, but you can always use abs() to fix that:

polyarea(c(-1, cord.x, 2), c(0, cord.y, 0))
[1] -0.8185917
abs(polyarea(c(-1, cord.x, 2), c(0, cord.y, 0)))
[1] 0.8185917

One more point. Don’t use abbreviations for TRUE and FALSE (e.g. frame=F). FALSE cannot be redefined, but F can be. If you used a variable called F in earlier code, it could change the behavior of curve():

curve(dnorm(x,0,1), xlim=c(-5, 5), frame=F)
FALSE <- 1
Error in FALSE <- 1 : invalid (do_set) left-hand side to assignment
F <- 1
curve(dnorm(x,0,1), xlim=c(-5, 5), frame=F)

----------------------------------------
David L Carlson
Department of Anthropology
Texas A&M University
College Station, TX 77843-4352



From: Steven Yen mailto:[hidden email]
Sent: Sunday, October 21, 2018 9:54 PM
To: David L Carlson mailto:[hidden email]
Subject: Re: [R] Polygon

Thank you David. This is great!!
I can follow all you do, except one. What does the last statement do: par(oldp). I would think that it reset the parameters, needed only if there is a next job.

I am a little embarrassed to push this further, but let me see if I can bounce another thing out of this.

Is it possible to have a "call-out", somehow indicating that the probability associated witht he shaded area is, say, 0.45. Is this possible? Am I asking too much of R? Fine if not. Thanks again!!!
On 10/22/2018 4:24 AM, David L Carlson wrote:
I think this does what you want.

z1<- -1
z2<-  2
oldp <- par(mgp=c(3, .5, 0))
curve(dnorm(x,0,1), xlim=c(-5, 5),main="Standard Normal",
    xaxt="n", frame=F, xlab="")
jj<-seq(z1,z2,0.01)
cord.x<- jj
cord.y<- dnorm(jj)
polygon(c(-1, cord.x, 2), c(0, cord.y, 0), col="skyblue")
axis(1,at=c(-5,z1,0,z2,5),lab=c(-5,z1,0,z2,5))
abline(v=c(z1,z2))
axis(1, at=c(-5, z1, 0, z2, 5), lab=(c(-5,z1,0,z2,5) + 5)*2, line=2)
par(oldp)

When you assign a vector to an object, you do not need to use c(), e.g. cord.x <- jj. I expanded xlim= since you were labeling the axis at -5, and 5, but only drawing the plot from -4 to 4.

To get the polygon to work, you need to add points to the baseline at at either end.

The line= argument in axis() lets you put an axis line into the margin of the plot. I used the mgp= argument in par() to close the space between the tickmarks and their values.
--------------------------------------
David L. Carlson
Department of Anthropology
Texas A&M University

-----Original Message-----
From: R-help [mailto:[hidden email]] On Behalf Of Steven Yen
Sent: Sunday, October 21, 2018 1:47 PM
To: David Winsemius mailto:[hidden email]
Cc: mailto:[hidden email]
Subject: Re: [R] Polygon

David, Rui, and All:
Greetings.
1. I need a helping hand with the polygon statement below so that I can
have the area under the curve highlighted, between (z1,z2).
2. Is it possible to label the X-axis with in two scale, in the current
z-scale and another, say x = (z+5)*2?
Thank you.

z1<- -1
z2<-  2
curve(dnorm(x,0,1),xlim=c(-4,4),main="Standard
Normal",xaxt="n",frame=F,xlab="z")
jj<-seq(z1,z2,0.01)
cord.x<-c(jj)
cord.y<-c(dnorm(jj))
#polygon(cord.x,cord.y,col="skyblue")
axis(1,at=c(-5,z1,0,z2,5),lab=c(-5,z1,0,z2,5))
abline(v=c(z1,z2))





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