math 141 - quantile-quantile plots - reed collegepeople.reed.edu/~jones/courses/p14.pdforder...
TRANSCRIPT
![Page 1: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/1.jpg)
Math 141Quantile-Quantile Plots
Albyn Jones1
1Library [email protected]
www.people.reed.edu/∼jones/courses/141
Albyn Jones Math 141
![Page 2: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/2.jpg)
Outline: Quantile-Quantile Plots
Quantiles and Order Statistics
Quantile-Quantile Plots
Normal Quantile Plots
Albyn Jones Math 141
![Page 3: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/3.jpg)
Quantiles and Order Statistics
Definition: p-th quantile qpEarlier we defined the p-th quantile of the distribution of a RV Xas any number qp satisfying
P(X ≤ qp) ≥ p
andP(X ≥ qp) ≥ (1− p).
Albyn Jones Math 141
![Page 4: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/4.jpg)
Order Statistics
Sample quantiles are based on Order Statistics:Let X1,X2, . . . ,Xn be a sample of size n. The order statisticsX(1),X(2), . . . ,X(n) are just the observations sorted intoascending order.
# The data> x[1] 76 92 83 105 102 109 106 91 110 89
# The order statistics> sort(x)[1] 76 83 89 91 92 102 105 106 109 110
Albyn Jones Math 141
![Page 5: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/5.jpg)
Order Statistics and Sample Quantiles
There are numerous definitions of sample quantiles chosen toperform well under various conditions. All involve interpolationbetween neighboring order statistics.
Suppose that we want the pth quantile, where p liesbetween k/n and (k + 1)/n. There are variations, but allare, for some choice of a ∈ [0,1]:
q̂p = aX(k) + (1− a)X(k+1)
Example, when n is even, the sample median q.5 is usuallytaken to be the average of the two middle observations.
Albyn Jones Math 141
![Page 6: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/6.jpg)
Order Statistics and Sample Quantiles
There are numerous definitions of sample quantiles chosen toperform well under various conditions. All involve interpolationbetween neighboring order statistics.
Suppose that we want the pth quantile, where p liesbetween k/n and (k + 1)/n. There are variations, but allare, for some choice of a ∈ [0,1]:
q̂p = aX(k) + (1− a)X(k+1)
Example, when n is even, the sample median q.5 is usuallytaken to be the average of the two middle observations.
Albyn Jones Math 141
![Page 7: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/7.jpg)
Order Statistics and Sample Quantiles
There are numerous definitions of sample quantiles chosen toperform well under various conditions. All involve interpolationbetween neighboring order statistics.
Suppose that we want the pth quantile, where p liesbetween k/n and (k + 1)/n. There are variations, but allare, for some choice of a ∈ [0,1]:
q̂p = aX(k) + (1− a)X(k+1)
Example, when n is even, the sample median q.5 is usuallytaken to be the average of the two middle observations.
Albyn Jones Math 141
![Page 8: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/8.jpg)
Order Statistics as Sample Quantiles
Let’s turn it around, and ask what sample quantiles correspondto the order statistics?
Consider 4 observations from a population. On the average, weexpect them to at least approximately divide the population intoequal chunks corresponding to equally spaced percentiles:
15,25,35,45
In other words, they correspond to the sample quantiles
q.2,q.4,q.6,q.8
Albyn Jones Math 141
![Page 9: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/9.jpg)
Comparing Two Samples
Suppose we have two samples of size n, X1,X2, . . .Xn andY1,Y2, . . .Yn.
If they were samples from the same distribution, then theorder statistics X(1),X(2), . . .X(n) and Y(1),Y(2), . . .Y(n)would be estimates of the same quantiles.Thus we expect that X(1) ≈ Y(1), X(2) ≈ Y(2), etc.
Albyn Jones Math 141
![Page 10: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/10.jpg)
Comparing Two Samples
Suppose we have two samples of size n, X1,X2, . . .Xn andY1,Y2, . . .Yn.
If they were samples from the same distribution, then theorder statistics X(1),X(2), . . .X(n) and Y(1),Y(2), . . .Y(n)would be estimates of the same quantiles.
Thus we expect that X(1) ≈ Y(1), X(2) ≈ Y(2), etc.
Albyn Jones Math 141
![Page 11: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/11.jpg)
Comparing Two Samples
Suppose we have two samples of size n, X1,X2, . . .Xn andY1,Y2, . . .Yn.
If they were samples from the same distribution, then theorder statistics X(1),X(2), . . .X(n) and Y(1),Y(2), . . .Y(n)would be estimates of the same quantiles.Thus we expect that X(1) ≈ Y(1), X(2) ≈ Y(2), etc.
Albyn Jones Math 141
![Page 12: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/12.jpg)
The QQ plot
The quantile–quantile plot, or QQplot, is a simple graphicalmethod for comparing two sets of sample quantiles. Plot thepairs of order statistics
(X(k),Y(k)).
If the two datasets come from the same distribution, the pointsshould lie roughly on a line through the origin with slope 1.
Albyn Jones Math 141
![Page 13: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/13.jpg)
QQ plot example
● ●
●●
●●
●●●
●●●●●●●●
●●●●●●●●●●
●●●●●●●●●● ●●
●●●●●●●●●●●●●●
●●●●●●●●
●●●●●●●●●●●●●
●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●
●●●●●●●●●●
●●●●●●●●●●●
●●●●●●●●●●
●●●●●●●●●●●●●●●●●
●●●●●●●●●●
●● ●
●●
●
●
−3 −2 −1 0 1 2
−3
−2
−1
01
23
X
Y
QQplot of two N(0,1) samples of size 200
Albyn Jones Math 141
![Page 14: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/14.jpg)
QQ plot example, small sample!
Alert!! With small samples, expect variation!
●
●
●
●
●
●●
●
●
●
● ●● ●
●
●
●
●
●
●
−3 −2 −1 0 1
−2.
0−
1.5
−1.
0−
0.5
0.0
0.5
1.0
X
Y
QQplot of two N(0,1) samples of size 20
Albyn Jones Math 141
![Page 15: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/15.jpg)
QQ plot example: location shift
Two samples from similar distributions which differ only inlocation: the green reference line is y = x .
●
●
●
● ●●●●
●●●●●
●●●●●
●●●●●●●●●●●●●●
●●●●●●●●●
●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●
●●●●●●●●●●●
●●●●●●●●●●●
●●●●● ●●●●●●
●●●●●
●●● ● ● ●
●●
−2 −1 0 1 2
01
23
45
6
X
Y
QQplot of two samples of size 200
Albyn Jones Math 141
![Page 16: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/16.jpg)
QQ plot example: different spread
Two samples from similar distributions which differ only inspread, with reference line.
●
●
●
● ●●●●
●●●●●
●●●●●
●●●●●●●●●●●●●●
●●●●●●●●●
●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●
●●●●●●●●●●●
●●●●●●●●●●●
●●●●● ●●●●●●
●●●●●
●●● ● ● ●
●●
−2 −1 0 1 2
−6
−4
−2
02
46
X
Y
QQplot of two samples of size 200
Albyn Jones Math 141
![Page 17: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/17.jpg)
QQ plot example: different shape
Two samples from distributions which differ in shape, as well aslocation and spread, with reference line.
●
●●●●● ●
●●●●●
●●●● ●●●●●●
●●●●●●●
●●●●●●●●●●●
●●● ●●●●●●●●●
●●●●●●●●●●●●●
●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●
●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●
●●●●●●●●●●●●●
●●●●●●●●●●●●●●●
●●●●●●●
●●●●●●
●●●●●
●●●●●●●●
●●●●●●●●
●
●
●
●
●
●
0.0 0.2 0.4 0.6 0.8 1.0
−2
−1
01
23
X
Y
QQplot of two samples of size 200
Albyn Jones Math 141
![Page 18: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/18.jpg)
QQ plot example: Anorexia data
The Family Therapy group had 17 subjects, the Control Therapy26. qqplot() uses estimated quantiles for the larger dataset.
●
●
●
●
●
● ●
●
●●
●●
●
●● ●
●
−10 −5 0 5 10 15
−5
05
1015
20
Control
Fam
ily
QQplot of Family Therapy vs Control
Albyn Jones Math 141
![Page 19: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/19.jpg)
Normal Quantile Plots
Often we wish to compare a dataset to the Normal distribution,a theoretical population, rather than to a second dataset. R hasa function that plots the order statistics of a sample against thecorresponding quaintiles of the standard normal distribution:
qqnorm(X )
If the plot is roughly linear, then our data are approximatelynormally distributed.
Albyn Jones Math 141
![Page 20: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/20.jpg)
Normal Quantile Plot: Normal data
Don’t expect a perfectly straight plot even with normal data!
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
−2 −1 0 1 2
05
1015
20
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
Albyn Jones Math 141
![Page 21: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/21.jpg)
Normal Quantile Plot: Mean and SD
We can estimate the mean and SD from a Normal quantile plot:the mean is roughly equal to the median (plotted above 0), andthe slope is roughly the SD.
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
−2 −1 0 1 2
05
1015
20
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
rise: 4.8
Albyn Jones Math 141
![Page 22: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/22.jpg)
Normal Quantile Plot: Short Tails
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
● ●
●
●
●
−2 −1 0 1 2
−5
05
1015
2025
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
Albyn Jones Math 141
![Page 23: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/23.jpg)
Normal Quantile Plot: Short Tails
Short tails are hard to diagnose from a density plot!
−5 0 5 10 15 20 25
0.00
0.01
0.02
0.03
0.04
0.05
0.06
Short Tails
N = 50 Bandwidth = 2.436
Den
sity
Albyn Jones Math 141
![Page 24: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/24.jpg)
Normal Quantile Plot: Long Tails
● ●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
−2
−1
01
2
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
Albyn Jones Math 141
![Page 25: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/25.jpg)
Density Plot: Long Tails
−2 0 2 4
0.0
0.1
0.2
0.3
0.4
0.5
Long Tails
N = 50 Bandwidth = 0.3177
Den
sity
Albyn Jones Math 141
![Page 26: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/26.jpg)
Normal Quantile Plot: Positive Skewness
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
02
46
810
12
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
Albyn Jones Math 141
![Page 27: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/27.jpg)
Density Plot: Positive Skewness
0 5 10 15
0.00
0.05
0.10
0.15
Positive Skewness
N = 50 Bandwidth = 0.9877
Den
sity
Albyn Jones Math 141
![Page 28: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/28.jpg)
Normal Quantile Plot: Negative Skewness
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
−2 −1 0 1 2
68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
Albyn Jones Math 141
![Page 29: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/29.jpg)
Normal Quantile Plot: Bimodal Data
●
●●
●●
●
●
●
●●
●
●
●
●
●● ●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●●
−2 −1 0 1 2
24
68
1012
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
Albyn Jones Math 141
![Page 30: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/30.jpg)
Density Plot: Bimodal Data
0 5 10 15
0.00
0.05
0.10
0.15
Bimodal data
N = 50 Bandwidth = 1.372
Den
sity
Albyn Jones Math 141
![Page 31: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/31.jpg)
Other distributions
Suppose we would like to make a theoretical quantile plot for adataset X to compare to some other distribution, say aChisquared distribution with 5 degrees of freedom. Easy:
Sort your dataset:SampleQuantiles <- sort(X)
Compute the theoretical quantiles:ChiSqQuantiles <- qchisq(ppoints(X),5)
Plot:plot(ChiSqQuantiles,SampleQuantiles,
pch=19)
Albyn Jones Math 141
![Page 32: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/32.jpg)
Other distributions
Suppose we would like to make a theoretical quantile plot for adataset X to compare to some other distribution, say aChisquared distribution with 5 degrees of freedom. Easy:
Sort your dataset:SampleQuantiles <- sort(X)
Compute the theoretical quantiles:ChiSqQuantiles <- qchisq(ppoints(X),5)
Plot:plot(ChiSqQuantiles,SampleQuantiles,
pch=19)
Albyn Jones Math 141
![Page 33: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/33.jpg)
Other distributions
Suppose we would like to make a theoretical quantile plot for adataset X to compare to some other distribution, say aChisquared distribution with 5 degrees of freedom. Easy:
Sort your dataset:SampleQuantiles <- sort(X)
Compute the theoretical quantiles:ChiSqQuantiles <- qchisq(ppoints(X),5)
Plot:plot(ChiSqQuantiles,SampleQuantiles,
pch=19)
Albyn Jones Math 141
![Page 34: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/34.jpg)
Other distributions
Suppose we would like to make a theoretical quantile plot for adataset X to compare to some other distribution, say aChisquared distribution with 5 degrees of freedom. Easy:
Sort your dataset:SampleQuantiles <- sort(X)
Compute the theoretical quantiles:ChiSqQuantiles <- qchisq(ppoints(X),5)
Plot:plot(ChiSqQuantiles,SampleQuantiles,
pch=19)
Albyn Jones Math 141
![Page 35: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/35.jpg)
Gamma Quantile Plot
●
● ●
● ●
● ●
●
● ● ●
●●
● ●●
● ●●
●
●
●
●
●
●
2 4 6 8 10
24
68
10
Gq
sort
(X)
Gamma Quantile Plot
Albyn Jones Math 141
![Page 36: Math 141 - Quantile-Quantile Plots - Reed Collegepeople.reed.edu/~jones/Courses/P14.pdfOrder Statistics and Sample Quantiles There are numerous definitions of sample quantiles chosen](https://reader031.vdocuments.site/reader031/viewer/2022030500/5aacab577f8b9aa9488d4a5f/html5/thumbnails/36.jpg)
Summary
QQplots are an excellent graphical tool for comparing twosamples to each other, or one sample to a theoreticaldistribution like the Normal. They reveal differences in location,spread and shape more clearly than do density plots orhistograms.
Albyn Jones Math 141