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Testing for normality 2
Duncan Golicher
December 4, 2008
Consequences in practice
Small samples
We have seen that our ability to detect non-normality in a population using the data we collect in a sample willdepend on how far the population deviates from normality and the size of our sample. The process of running aKS test or any other test of normality is no extra effort. The software will do all work for us. We know that withsmall sample sizes we are unlikely to reject the null. So what is the problem? Shouldn’t we just run the tests bydefault, then go ahead and carry out the analysis we planned, safe in the knowledge that we can defend ourselvesagainst any accusation that the assumptions needed for the main analysis were not met by saying that we testedthem formally. No. This is potentially really bad practice.
The problem is this. If we run a KS test on a small sample we are likely to get a type two error on the test “fornormality”. That’s another way of restating what we’ve just seen. As an exercise in reductio ad absurdum think ofa sample size of two. We can conclude nothing about the distribution at all, so no test could ever reject the nullhypothesis that it might be drawn from a normal population. This is clearly absurd and we would never run a testwith such a small sample, but with any sample size below 20 the power of the test is still far too low to be reallyuseful.
By adopting the wrong statistical procedure as a result of an initial, apparently innocuous and merely technicaltype two error we are then more likely to get a type one error when we test the more important hypothesis we werereally interested in. This is much more serious. We are likely to find significance when none exists. For example ifby chance one very large value from a right skewed population is included when we take two smallish samples froma single population a KS test may suggest that there is nothing wrong with the assumption of normality. Howevera test on the hypothesis of no difference between samples may be erroneously significant. This is sometimes quitedangerous1. The result is not replicable and it could even get into the literature. If our desire is to conductrigorous sound science we really should not be doing this. It defeats the object of running any statistical tests atall. Therefore if tests of simple hypotheses are required from very small samples non-parametric procedures aregenerally safer whatever the result of a the KS test.
Large samples
There is a much better case to be made for running tests of normality on large samples, providing we are aware ofthe true meaning of the test. In such situations the aim is not to evaluate whether a non-parametric test should beused. Instead we are interested in finding out how well any transformation of the data will work. Once a large dataset has been assembled it is usually a real waste to use it only for simple non parametric null hypothesis testing. Wewant to know about effect sizes and produce confidence intervals. The most useful procedures nearly always involvesome assumptions regarding distributions (models), although more advanced techniqes such as generalised linearmodels assume distributions other than normal. A well designed transformation effectively draws the populationcloser to a normal form. We can use a test of normality of the data both before and after transformation to helpus evaluate how effective it has been. Inference about every aspect of the population is easier with large samplesthan small samples and this goes for the shape of the distribution too.
What to do about it?
So what are the answers in practice? Different authors will have slightly different takes on the problem, but if wefollow these three steps we shouldn’t go that far wrong.
1The fortunate outcome of including an extreme value is that it increases the variance and thus the standard error but this shouldn’tbe taken as a guarantee that a type one error won’t result.
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1. Think about the properties of the underlying population first
2. Look for example data sets taken under similar circumstances
3. Simulate some data that follows from reasonable assumptions regarding the population and your samplingscheme multiple times.
Of the three steps the third is the least likely to be done in practice, but potentially the most informative.
Think about the population
Planning ahead is vital. We should never just collect data haphazardly assuming that the statistical analysis willadapt itself to whatever we happen to come up with. We can discount normality before we even look at any datain many situations
Samples consisting of counts
The fact that a population consists of discrete counts technically doesn’t allow it to take a perfectly normal form.However that is not necessarily much of a problem. It certainly doesn’t discount inference on the mean in a practicalsetting. In many cases it is the best way to handle the data. We intuitively understand the mean number of goalsper team per season, even though logically no team can score exactly 45.3 goals. Inference on the mean is veryuseful for many reasons. It provides us with confidence intervals and it is easy to communicate.
Statistical text books are full of details about normal approximations. If the mean value for counts is highenough (say above 7 as a pragmatic rule of thumb) we can safely assume that normal inference is OK providing wecan make another assumption. That is that the counts follow a Poisson distribution. In this case the population canbe expected to sufficiently symmetrical as to not interfere greatly with the process of inference. Poisson distributionsoccur when occurrences of an event are assigned randomly to classes and the number per class counted. This isa reasonable model for an underlying process in many cases and underlies statistical tests of association such aslog-linear modelling.
Poisson distribution with lambda=7
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Some textbooks suggest much larger values are needed2, but given that so many other things can go wrong inpractice a minor and fairly technical lack of symmetry should be the least of our worries for an empirical study.This is not to suggest that normal inference on this sort of data is always the best way to treat them, just that itgeneral no real harm will come of it.
On the other hand if the mean value for the counts is low, a population of counts can never be treated as normal.The problem is that zero forms a barrier that automatically skews the data.
2True symetry starts at about lambda=10 and at lambda =30 the normal approximation is really very good.
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In ecology there are often far worse problems. They are so common that they should be expected as the defaultwhen we count most types of organisms. They arise from clustering effects. Animals and plants tend to sticktogether, so the assumption of random allocation of individuals is hardly ever met. Take the example of bats whichroost in house lofts. Most houses will have no individuals at all, but some houses will have a colony. The populationmight then look something like this.
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This is a zero inflated poisson. In this sort of case one solution is to separate data analysis in two phases. Firstwe can look at presence and absence as a binary variable and perhaps analyse the reasons for absence. Then wemight look at the number of individuals in the colonies themselves. For something like bats it is quite logical tothink in terms of mean colony size before starting work, but in other cases it may be less obvious that this approachis going to be needed. Sometimes the zeros get mixed in and analysed along with the rest of the data, even thoughthey are really part of a different process. As the mean value for the poisson part is reduced the distribution becomesa mixture. This can really make a mess of the analysis. In some cases if a mean group size is not going to tell usmuch it might be easier to only measure presence and absence.
It is also very common to find a particularly nasty sort of skewed poisson distribution known as the negativebinomial in ecological data. This occurs in cases when there are a few very large agglomerations of individuals andmany small clusters. Flocking birds, such as starlings, can have this sort of pattern, particularly when viewed overthe whole year. Much of the time they are in pairs or small family groups, but in the winter they come together inthe thousands. This is perhaps the hardest sort of data to handle consistently well because if one large cluster ismissed inference can be way out. Hilborn and Mangel give an example of bycatch of allbatross in the South Pacificas an illustration. Most of the time very few birds are caught, but occasionally a boat kills a lot of birds. If thiswere to be missed we might not take adequate conservation measures.
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In all these cases the way to proceed in the case of a real life study is to anticipate the problem beforehand,think carefully and take advice from ecologists with experience working with the organism in question. There aresolutions in all cases, but many require alterations to the study design. No statistical method has ever managedto deal effectively with a very small sample taken from a highly skewed population. Mess up in the planning stageand there is no way back.
Log normal data
Many phenomena in nature are more likely to produce log-normal distributions, or variations on the theme, thannormal distributions. When many effects act additively a normal distribution is expected due to the mechanism ofthe central limit theorem. However when effects act in a multiplicative fashion a log-normal results for a similarreason. Exponential growth or decline is such an effect. Log normal distributions are right skewed. So for examplethe populations of towns often form a log-normal type pattern as do the populations of species. Wealth accumulationalso tends to be log-normal unless governments intervene. It is not so much that the rich get rich and the poorget poorer, rather that the rich get richer more quickly. Dividing up land produces a log normal for the conversereason. If we first split a parcel of land in two and then split one of the resulting parcels in two, then we get avariety of exponential decay. Thus the area of land holdings are often log normal too.
So we end up with situations where we find that most towns are small, but most people live in large towns.More species are rare, but most individuals belong to a a handful of species. Most people do not have much money,but most of the money is in a few people’s bank accounts. Most people have very small parcels of land, but mostof the land belongs to one owner. And so on.
The good thing about a genuine log normal is that simply taking logarithmns makes it normal. So we have adead easy get out if we know beforehand that a log normal effect is likely to occur. In fact taking logs unecessarilyoften does no great harm. A truly normal population will then become left skewed but in real life the outliers areusually on the right so it will tend to draw them in. we can even take logs twice to correct for extreme skew. Themain problem with this may be communication and interpretation of the results.
A log normal population
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Taking logs of a log−normal
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Concentrations
The third main class of non normal populations consist of measurements of concentrations. Unlike the log-normalcase we have no real theory to guide us in this, we have to use intuition and common sense. The problem is suggestedby the very term concentration. Pollutants and contaminants such as heavy metals are often concentrated inspecific areas or in a few individuals or organs within individuals. Thus we might get many soil “samples”3 withvery low levels followed by one taken right next to a discarded battery. The same effect can occur in water samplesor measurements taken from fish such as salmon.
When looking at contaminants it might often be worth asking whether inference on the mean makes any senseat all. The problem of pollutants is precisely that they are concentrated. In most cases there are regulations andguidelines that set “safe” limits for acceptable concentrations. These may be arbitrary but we will often find thatmost samples are well below them but a few are clearly way above. Thus the data can be converted into a binaryvariable (“safe”,”unsafe”) and analysed as such. Or several categories such as normal, slightly contaminated, highlycontaminated can be extracted from the literature. we might not die from eating a plate of shellfish all with themean amount of toxins, but if we get one really bad one that we’ve had it.
If we do actually want to look at mean values the central limit theorem can perhaps come to our aid. Forexample when soil is collected several “samples” taken from a site are often mixed to form a combined “sample”before analysis. Notice that is conventional to call a soil core a soil sample, but in a statistical sense it is not.It is one unit that makes up the statistical sample. The good thing about this procedure is that adding severalnumbers together from almost any form of distribution tends to produce a normal distribution in the end. Howeverif the population is highly skewed that still is unlikely to really solve the problem unless a very large number ofsub-samples are mixed together.
> concentrations <- c(1, 1, 2, 3, 4, 5, 6, 2, 8, 2, 12, 4, 20,
+ 10, 40)
> hist(replicate(1000, sum(sample(concentrations, 10, replace = T))),
+ main = "", xlab = "Pooled soil samples", col = "grey")
3It is very difficult not to use the word sample in this context even though it referes to a sample unit, not a statistical sample
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Pooled soil samples
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A further point regarding concentrations is also that they will tend to be concentrated in time and space. Itis this effect that we are often interested in understanding. However this can lead to autocorrelation in the datawhich can potentially play havoc on statistical inference.
Use other studies
It is suprising how often studies are planned only with reference to the summarised findings of previous similarwork. This is particularly unfortunate if the summaries are only based on null hypothesis tests with no informationregarding effect size and confidence intervals. It is fortunately now becoming common for authors to provide rawdata in appendices on line. If this is available it should always be used as it can provide vital insight into the sortof distribution to be expected.
The subject of the use of prior experience is a large one and should be expanded upon.
Simulate some data from a condidate population
Our intuitions regarding the way sampling affects the patterns we see when looking at histograms, boxplots, qqplotsand ecdfs are often not reliable. A“hole” in a histogram might be an indication of bimodality, but with small samplesit may arise by chance. Even quite large samples never look perfectly normal. This is the justification for usingtests of normality in the first place. If only they worked the way we really want the world would be a much simplerplace.
I have provided a set of simulated results to give some idea as to the patterns to expect. This exercise is muchmore informative when you conduct it yourself using the same design as will be used for the survey or experiment.Histograms are often the most difficult to interpret well and are susceptible to decisions regarding the number ofbins used for the data. Most software takes the decision for you, but it is usually a good idea to try with variousbin sizes.
The idea behind including the appendix that follows is to allow you to get a quick visual impression of thevariability in the results that might be expected when you conduct a standard exploratory analysis on a sampletaken from a known population. In all cases the statistical inference is aimed at the underlying population (that isknown) not the sample.
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Appendix. Simulated data
Sample size 10
Normal
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78
911
Normal Q−Q Plot
Theoretical Quantiles
●
●
●
●
●
●
●
●
●
●68
1012
14
Normal Q−Q Plot
●
●
●
●
●●
●
●
●
●
−1.5 −0.5 0.5 1.5
910
1112
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●●
●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
910
1112
13
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
89
1113
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●
●●
●
●●
● ●
●
89
1011
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●
●
●
●●
●
●
●
−1.5 −0.5 0.5 1.5
68
1012
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
67
89
1113
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
89
1012
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●
●
●
●
●
●
●
●
●
46
810
12
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●●
●
●●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
910
1214
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●●
●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
68
1012
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
89
1113
15
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●
●
●
●
●
●
●
●
●
67
89
11
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
7
Right skewed log normal
x
Fre
quen
cy
0 20 40 60 80 100 120
050
100
150
200
250
300
Histograms
x
0 10 20 30 40
01
23
45
x
0 5 10 20 30
0.0
1.0
2.0
3.0
x
0 20 40 60 80
01
23
45
60
12
34
5
x
Fre
quen
cy
0 10 20 30 40
0.0
1.0
2.0
3.0
x
Fre
quen
cy
0 5 10 20 30
01
23
45
x
Fre
quen
cy
0 5 10 15 20 25
01
23
4
Fre
quen
cy
01
23
45
6
x
Fre
quen
cy
0 5 10 15 20 25
01
23
4
x
Fre
quen
cy
0 10 30 50
01
23
45
x
Fre
quen
cy
0 10 30 50
01
23
4
Fre
quen
cy
01
23
45
x
Fre
quen
cy
0 5 10 15 20 25
01
23
4
x
Fre
quen
cy
0 10 30 50
01
23
4
x
Fre
quen
cy
0 5 10 15 20 25
01
23
4
Fre
quen
cy
0.0
0.5
1.0
1.5
2.0
x
0 10 20 30 40
01
23
4
x
0 10 30 50
01
23
45
67
x
5 10 15 20 25 30
01
23
45
60.
01.
02.
03.
0
x
Fre
quen
cy
0 5 10 20 300
12
34
x
Fre
quen
cy
0 10 20 30 40 50
01
23
45
x
Fre
quen
cy
0 5 10 20 30
01
23
4
Fre
quen
cy
01
23
4
x
Fre
quen
cy
0 5 10 15 20 25
0.0
1.0
2.0
3.0
x
Fre
quen
cy
5 10 20 300.
01.
02.
03.
0
x
Fre
quen
cy
0 10 20 30 40
01
23
4
Fre
quen
cy
01
23
4
x
Fre
quen
cy
0 10 30 50
01
23
45
6
x
Fre
quen
cy
0 5 10 15 20 25
01
23
45
6
x
Fre
quen
cy
0 10 20 30 40 500
12
34
5
Fre
quen
cy
0.0
1.0
2.0
3.0
x
0 5 10 15 20 25
01
23
45
x
0 5 10 15 20 25
01
23
45
x
0 5 10 20 30
0.0
1.0
2.0
3.0
01
23
4
x
Fre
quen
cy
0 5 10 15 20 25
01
23
45
x
Fre
quen
cy
5 10 15 20 25 30
0.0
1.0
2.0
3.0
x
Fre
quen
cy
0 5 15 25 35
01
23
45
Fre
quen
cy
01
23
4
x
Fre
quen
cy
0 10 20 30 40 50
01
23
45
x
Fre
quen
cy
0 20 40 60
01
23
45
x
Fre
quen
cy
0 5 10 15 20 25
01
23
4
Fre
quen
cy
01
23
4
x
Fre
quen
cy
5 10 20 30
0.0
1.0
2.0
3.0
x
Fre
quen
cy
0 5 10 20 30
0.0
1.0
2.0
3.0
x
Fre
quen
cy
0 5 10 15 20 25
0.0
1.0
2.0
3.0
Fre
quen
cy
01
23
45
Boxplots
●
●
●
510
2030
510
1520
255
1015
20
●
515
2535
510
1520
2530
●
510
1520
25
●
1020
3040
●
510
1520
●
1020
3040
510
1520
25
●
1020
3040
50
●
1020
3040
46
812
165
1015
2025
510
1520
2510
2030
40
1020
3040
46
812
165
1015
2025
306
812
16
510
1520
510
1520
250
1020
3040
50
●
515
25
510
1520
510
1520
255
1015
2025
510
1520
●
●
020
4060
46
812
16
●
1020
3040
●
1020
3040
510
15
●
510
2030
●
510
1520
1020
3040
●
510
1520
68
1216
510
1520
255
1015
20
010
2030
405
1015
20
●
515
2535
46
810
14
510
1520
510
1520
2530
●
510
1520
256
810
1214
Qqplots
●
●
●
●
●
●
●
●
●●
−1.5 −0.5 0.5 1.5
510
1520
Normal Q−Q Plot
Theoretical Quantiles
●
●
●
●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
1020
3040
50
Normal Q−Q Plot
Theoretical Quantiles
●
●
●
●
●
●
●
●●
●
−1.5 −0.5 0.5 1.5
510
1520
25
Normal Q−Q Plot
Theoretical Quantiles
●●
●
●
●
●
●
●
●
●
510
1520
Normal Q−Q Plot
●
●
●
●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
1020
3040
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●●
●
−1.5 −0.5 0.5 1.5
510
15
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●
●
●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
510
1520
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
● ●
●
●
●
●
●
●
510
1520
25
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●●
●
●●
●
●
●
−1.5 −0.5 0.5 1.5
510
1520
2530
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●●●
●
●
−1.5 −0.5 0.5 1.5
1020
3040
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●●
●
●
● ●●●
−1.5 −0.5 0.5 1.5
1030
5070
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●●
●
●
●
510
1520
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
515
2535
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●
●
●
●
●
●
●
●●
−1.5 −0.5 0.5 1.5
1020
3040
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
● ●●
●
●
●
●
●
●●
−1.5 −0.5 0.5 1.5
46
812
16
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●●
●
●
●
●
●
1020
3040
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
510
1520
25
Normal Q−Q Plot
Theoretical Quantiles
●
●
● ●●
●
●
●
●●
−1.5 −0.5 0.5 1.5
510
1520
25
Normal Q−Q Plot
Theoretical Quantiles
●
●●
●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
510
15
Normal Q−Q Plot
Theoretical Quantiles
●
●
●
●
●
●
●
●
●
●
05
1525
Normal Q−Q Plot
●
●
●
●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
46
810
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●●
●
●
● ●
●
−1.5 −0.5 0.5 1.5
515
2535
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
510
2030
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
● ●
●
●
●
●
●
●
●
●
510
1520
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●
●●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
1020
3040
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●●
●
−1.5 −0.5 0.5 1.5
46
810
1214
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
510
1520
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●●
●
●
●
●●
1020
3040
50
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
1030
50
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
● ●
●
● ●
●●
●
●
−1.5 −0.5 0.5 1.5
020
4060
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
515
2535
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●
●
●
●
●
●
●
●
●
1015
2025
30
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
● ●
●
●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
1020
3040
Normal Q−Q Plot
Theoretical Quantiles
●
●
●●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
515
2535
Normal Q−Q Plot
Theoretical Quantiles
●
●
●
●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
510
1520
Normal Q−Q Plot
Theoretical Quantiles
●
●
●
●
●
●
●
●
●
●
510
1520
Normal Q−Q Plot
●
●
●
●
●
●
●
●
● ●
−1.5 −0.5 0.5 1.5
510
1520
25
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●●
●
●
−1.5 −0.5 0.5 1.5
515
2535
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
● ●
●●
●
●
●
●
−1.5 −0.5 0.5 1.5
515
2535
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●
●
●
●
●
●
●
●
●
515
2535
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●
●
●
●●
●●
●
−1.5 −0.5 0.5 1.5
510
1520
25
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
510
1520
25
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●●
●
●
● ●
●
−1.5 −0.5 0.5 1.5
510
1520
25
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●●
●
●
●
●
●
●●
510
1520
25
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●
●
●
●●
●
●
●
−1.5 −0.5 0.5 1.5
68
1216
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●●
●
●
●
●
●
●
−1.5 −0.5 0.5 1.5
515
2535
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●● ●
−1.5 −0.5 0.5 1.5
510
2030
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●
●
●
●
●
●
●
●
●
1015
20
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
8
Sample size 20
Normal
x
Fre
quen
cy
5 10 15
050
010
0015
0020
00
Histograms
x
8 9 10 12 14
01
23
45
67
x
4 6 8 10 12 14
01
23
45
x
4 6 8 10 14
02
46
80
12
34
x
Fre
quen
cy
4 6 8 10 14
02
46
8
x
Fre
quen
cy
6 8 10 12 14 16
02
46
810
x
Fre
quen
cy
6 8 10 12 14 16
01
23
45
6
Fre
quen
cy
02
46
8
x
Fre
quen
cy
8 9 10 12 14
01
23
45
x
Fre
quen
cy
7 8 9 11 13
01
23
45
6
x
Fre
quen
cy
6 8 10 12 14
01
23
45
67
Fre
quen
cy
01
23
45
x
Fre
quen
cy
7 8 9 11 13
01
23
45
x
Fre
quen
cy
7 8 9 11 13
01
23
45
x
Fre
quen
cy
6 8 10 12 14
01
23
45
6
Fre
quen
cy
01
23
45
6
x
6 7 8 9 11 13
01
23
45
6
x
8 10 12 14
01
23
45
6
x
6 8 10 12 14
01
23
45
60
12
34
x
Fre
quen
cy
6 8 10 12 14
01
23
45
67
x
Fre
quen
cy
6 7 8 9 11 13
01
23
45
x
Fre
quen
cy
6 7 8 9 11 13
01
23
45
6
Fre
quen
cy
01
23
45
6
x
Fre
quen
cy
2 4 6 8 10 14
02
46
8
x
Fre
quen
cy
6 8 10 12
01
23
45
x
Fre
quen
cy
6 8 10 12 14
02
46
8
Fre
quen
cy
01
23
45
x
Fre
quen
cy
8 10 12 14
01
23
45
x
Fre
quen
cy
6 7 8 9 11 13
01
23
45
x
Fre
quen
cy
7 8 9 10 12
01
23
45
67
Fre
quen
cy
01
23
45
6
x
7 8 9 10 11 12
01
23
45
67
x
6 8 10 12 14
01
23
45
x
6 8 10 12 14
01
23
45
67
02
46
810
x
Fre
quen
cy
6 8 10 12 14
01
23
45
6
x
Fre
quen
cy
6 8 10 12
01
23
45
x
Fre
quen
cy
6 8 10 12 14
01
23
45
6
Fre
quen
cy
01
23
4
x
Fre
quen
cy
6 8 10 12 14
01
23
45
x
Fre
quen
cy
6 8 10 12 14 16
02
46
8
x
Fre
quen
cy
7 8 9 10 12
01
23
45
Fre
quen
cy
01
23
45
6
x
Fre
quen
cy
7 8 9 11 13
01
23
45
6
x
Fre
quen
cy
6 8 10 12 14
01
23
45
x
Fre
quen
cy
6 8 10 12 14
01
23
45
67
Fre
quen
cy
01
23
45
Boxplots
68
1012
146
810
124
68
1014
67
89
11
●
68
1012
14
●
68
1012
810
1214
810
1214
68
1012
78
911
13
●
911
1315
●
68
1012
●
●68
1012
●
810
1214
810
1214
●
●
68
1012
14
78
911
137
89
1011
810
1214
810
1214
78
911
137
89
1113
78
911
137
89
1113
68
1012
●
89
1113
810
1214
67
89
1113
89
1011
126
810
126
810
128
910
12
89
1012
78
911
138
1012
14
●
●
●
●
89
1113
68
1012
89
1011
128
910
1112
13
●
810
1214
68
1012
78
911
136
810
128
910
12
46
810
126
810
1214
●
●
68
1012
148
1012
14
Qqplots
●
●
●
●●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
−2 −1 0 1 2
78
911
13
Normal Q−Q Plot
Theoretical Quantiles
●
●
●
●●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
46
810
12
Normal Q−Q Plot
Theoretical Quantiles
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
●
●
●●
●
●
●
●
●
●
●
●
●● ●
●
●
●
●●
68
1012
Normal Q−Q Plot
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
78
910
1112
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
−2 −1 0 1 2
78
911
13
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●
●
●
68
1012
14
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
78
911
13
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
−2 −1 0 1 2
78
910
12
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●●
●●
●
●●
●●●
● ●
●
●
●
●
●
●
−2 −1 0 1 2
810
1214
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●●●
●
●
●●
●●
●
●
●
●
68
1012
14
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
68
1012
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
−2 −1 0 1 2
68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
● ●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
89
1011
12
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●●
89
1113
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
78
910
12
Normal Q−Q Plot
Theoretical Quantiles
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
−2 −1 0 1 2
810
1214
Normal Q−Q Plot
Theoretical Quantiles
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
−2 −1 0 1 2
78
910
12
Normal Q−Q Plot
Theoretical Quantiles●
●
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
68
1012
Normal Q−Q Plot
●●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
89
1011
12
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
−2 −1 0 1 2
810
1214
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
−2 −1 0 1 2
89
1011
12
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
68
1012
14
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
−2 −1 0 1 2
89
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
68
1012
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
−2 −1 0 1 2
810
1214
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
89
1012
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
−2 −1 0 1 2
810
1214
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
89
1113
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
−2 −1 0 1 2
78
910
12
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
89
1012
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●
●
●
−2 −1 0 1 2
68
1012
Normal Q−Q Plot
Theoretical Quantiles●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
67
89
1113
Normal Q−Q Plot
Theoretical Quantiles
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
68
1012
Normal Q−Q Plot
Theoretical Quantiles
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
810
1214
Normal Q−Q Plot
●
●
●
●
●●●
●
●
●
●
●●●●
●
●
●
●
●
−2 −1 0 1 2
810
1214
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●●
●
●
●●
●
●
●●
●
●
●●
●
●
●
−2 −1 0 1 2
67
89
1113
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●●
●
●●
●
●
●
●
●●
●
●
●
●
●●
●
−2 −1 0 1 2
68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●●
●●
●
●●
●
●
●
●
●
68
1014
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●●
●
●●
●
●
●●
●
●
●●
●
●●
●●●
−2 −1 0 1 2
68
1014
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●●
●
●
●
●
●●
●●
●●
●
●
●
●
●
−2 −1 0 1 2
68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
● ●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
68
1012
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●
●
●●
●●
●
●
●●
●
●●
●
●
●
●
●
−2 −1 0 1 2
68
1012
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●
−2 −1 0 1 2
78
911
13
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
−2 −1 0 1 2
67
89
11
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
● ●
●●
●
●
●
●
●
●
●
●
●
68
1012
14
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
9
Right skewed log-normal
x
Fre
quen
cy
0 20 40 60 80 100
020
4060
8010
012
014
0
Histograms
x
0 5 15 25 35
02
46
8
x
0 20 40 60
02
46
8
x
0 10 30 50
02
46
80
12
34
5
x
Fre
quen
cy
0 5 10 15 20 25
01
23
45
67
x
Fre
quen
cy
0 5 10 20 30
02
46
8
x
Fre
quen
cy
0 20 40 60 80
02
46
810
Fre
quen
cy
02
46
810
x
Fre
quen
cy
0 20 40 60
01
23
45
67
x
Fre
quen
cy
0 10 20 30 40 50
01
23
45
67
x
Fre
quen
cy
0 5 15 25 35
01
23
45
6
Fre
quen
cy
02
46
810
x
Fre
quen
cy
0 10 20 30 40
01
23
45
6
x
Fre
quen
cy
0 5 15 25 35
01
23
45
6
x
Fre
quen
cy
0 10 30 50
02
46
8
Fre
quen
cy
01
23
45
6
x
0 10 20 30 40
01
23
45
6
x
0 5 15 25 35
02
46
8
x
0 20 40 60
02
46
810
01
23
45
67
x
Fre
quen
cy
5 10 15 200
12
34
5
x
Fre
quen
cy
0 10 20 30 40
01
23
45
6
x
Fre
quen
cy
0 10 20 30 40
01
23
45
6
Fre
quen
cy
02
46
810
x
Fre
quen
cy
0 10 20 30 40
02
46
8
x
Fre
quen
cy
0 5 10 20 300
24
68
x
Fre
quen
cy
0 5 15 25 35
01
23
45
Fre
quen
cy
02
46
8
x
Fre
quen
cy
0 5 15 25 35
01
23
45
6
x
Fre
quen
cy
0 5 15 25 35
01
23
45
67
x
Fre
quen
cy
0 10 20 30 400
24
68
Fre
quen
cy
01
23
45
67
x
0 10 20 30 40
01
23
45
67
x
0 10 20 30 40
01
23
45
6
x
0 5 10 15 20 25
02
46
80
12
34
56
7
x
Fre
quen
cy
0 10 20 30 40
02
46
8
x
Fre
quen
cy
0 5 10 15 20 25
01
23
45
67
x
Fre
quen
cy
0 5 10 20 30
02
46
8
Fre
quen
cy
01
23
45
67
x
Fre
quen
cy
0 5 10 20 30
01
23
45
x
Fre
quen
cy
0 5 15 25 35
02
46
810
x
Fre
quen
cy
0 5 15 25 35
02
46
8
Fre
quen
cy
01
23
45
x
Fre
quen
cy
0 5 15 25 35
01
23
45
x
Fre
quen
cy
0 5 15 25 35
01
23
45
6
x
Fre
quen
cy
0 20 40 60
02
46
8
Fre
quen
cy
01
23
45
6
Boxplots
510
1520
255
1015
2025
30
●
510
1520
515
25
●
515
2535
515
2535
●
510
1520
25
●
1020
3040
510
15
●
●
1020
3040
●
●
510
1520
25
●
510
2030
510
1520
515
25
●
510
1520
●
●
010
3050
●
510
2030
46
812
165
1015
2025
510
1520
25
510
1520
515
2535
●
1020
3040
●
510
1520
25
●
515
2535
●
1020
3040
515
2535
●
510
1520
25
●
510
2030
●
515
2535
●
510
1520
●
510
1520
●
●
●
1020
3040
●
510
1520
25
●
1020
3040
●
515
2535
●
515
2535
●
●
020
4060
80
●
510
2030
●●
●
●
050
100
150
●●
515
25
●
●
010
2030
40
●
1020
3040
24
68
1216
●
1020
3040
50
●
1020
3040
●
●
1020
3040
●
515
2535
Qqplots
●●
●
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●
●●
●
●
●
●
●●
●
●
●
●
●●
●
−2 −1 0 1 2
515
2535
Normal Q−Q Plot
Theoretical Quantiles●
●
●
●
●
●●
●
●●
●
●
●
●
●
●●
●●
●
−2 −1 0 1 2
510
1520
Normal Q−Q Plot
Theoretical Quantiles
●
●
●
●
●
●
● ●●
●
●● ●
●●
●
●
●
●●
−2 −1 0 1 2
1030
50
Normal Q−Q Plot
Theoretical Quantiles
●
●
●
●
●
●
●
●
●
●●
●●
●●
●
●
●●
●
1020
3040
Normal Q−Q Plot
●
●●
●
●
●●
●●
●●
●●
●
●
●
●●
●●
−2 −1 0 1 2
1020
3040
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
● ●●
●●
●
●
●
●
●
●
●
●
●
●●
−2 −1 0 1 2
515
2535
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
510
2030
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●●
●
●
●
●●●
●
●
●
●
●
●
●
●
515
2535
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●●
●
●
●
●
●●
●
●●●
●
●
●
●
●●●
●
−2 −1 0 1 2
010
2030
4050
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
510
2030
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●●
−2 −1 0 1 2
1020
3040
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
●
●●
●
●●
●
●●
●
1020
3040
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
● ●●
●●
●●
●
●●
●
●
●
●
●
●●
●
●
●
−2 −1 0 1 2
1030
50
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
−2 −1 0 1 2
510
1520
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●●
●●
●
●
●
●●
●
●
●●●●
●
●●
●
−2 −1 0 1 2
2040
6080
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
1020
3040
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●
●● ●
●
●
●
●●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
510
2030
Normal Q−Q Plot
Theoretical Quantiles
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
−2 −1 0 1 2
515
2535
Normal Q−Q Plot
Theoretical Quantiles
● ●●
●●
●●
●
●
●
●●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
1030
50
Normal Q−Q Plot
Theoretical Quantiles
●
●●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●●
1020
3040
Normal Q−Q Plot
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●●
●
−2 −1 0 1 2
515
2535
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
−2 −1 0 1 2
05
1020
30
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
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510
1520
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Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
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ntile
s
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Normal Q−Q Plot
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ple
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Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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Normal Q−Q Plot
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ple
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ntile
s
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2030
Normal Q−Q Plot
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ple
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515
2535
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
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ntile
s
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1020
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Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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510
15
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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Normal Q−Q Plot
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ple
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s
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Normal Q−Q Plot
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Normal Q−Q Plot
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ple
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510
2030
Normal Q−Q Plot
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ple
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s
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1030
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Normal Q−Q Plot
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s
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515
2535
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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1020
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Normal Q−Q Plot
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ple
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s
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Normal Q−Q Plot
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ple
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Normal Q−Q Plot
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ple
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1030
50
Normal Q−Q Plot
Theoretical Quantiles
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ple
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s
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Normal Q−Q Plot
Sam
ple
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s
10
Sample size 30
Normal
x
Fre
quen
cy
5 10 15
050
010
0015
0020
00
Histograms
x
6 8 10 12 14
02
46
810
x
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01
23
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46
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4 6 8 10 12 14
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46
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8 10 12 14
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23
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6 8 10 12 14
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6 8 10 12 14
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6 8 10 12 14
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45
60
510
15
x
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quen
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4 6 8 10 14
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46
810
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4 6 8 10 14
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810
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02
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810
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810
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812
Boxplots
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12
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Normal Q−Q Plot
Theoretical Quantiles
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Normal Q−Q Plot
Theoretical Quantiles
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13
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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68
1012
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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78
910
12
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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Normal Q−Q Plot
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ple
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810
1214
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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68
1012
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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Normal Q−Q Plot
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ple
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12
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14
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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46
810
14
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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810
1214
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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68
1012
Normal Q−Q Plot
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ple
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ntile
s
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68
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Normal Q−Q Plot
Theoretical Quantiles
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810
1214
Normal Q−Q Plot
Theoretical Quantiles
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−2 −1 0 1 2
89
1012
Normal Q−Q Plot
Theoretical Quantiles●
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68
1012
Normal Q−Q Plot
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−2 −1 0 1 2
810
1214
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
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ntile
s
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−2 −1 0 1 2
78
911
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
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ntile
s
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68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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810
1214
Normal Q−Q Plot
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ple
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ntile
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68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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−2 −1 0 1 2
78
911
13
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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68
1012
14
Normal Q−Q Plot
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ple
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−2 −1 0 1 2
910
1112
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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−2 −1 0 1 2
89
1012
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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46
810
1214
Normal Q−Q Plot
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ple
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−2 −1 0 1 2
810
1214
16
Normal Q−Q Plot
Theoretical Quantiles
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−2 −1 0 1 2
810
1214
Normal Q−Q Plot
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1113
Normal Q−Q Plot
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Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
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1012
Normal Q−Q Plot
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ple
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ntile
s
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67
89
1113
Normal Q−Q Plot
Theoretical Quantiles
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ple
Qua
ntile
s
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78
911
13
Normal Q−Q Plot
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ple
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ntile
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89
11
Normal Q−Q Plot
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ple
Qua
ntile
s
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1012
14
Normal Q−Q Plot
Theoretical Quantiles
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ple
Qua
ntile
s
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68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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89
1011
12
Normal Q−Q Plot
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ple
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ntile
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810
1214
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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89
1012
Normal Q−Q Plot
Theoretical Quantiles
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ple
Qua
ntile
s
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78
910
12
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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810
1214
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
11
Right skewed log-normal
x
Fre
quen
cy
0 20 40 60 80
020
4060
8010
012
014
0
Histograms
x
0 5 15 25 35
02
46
812
x
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1015
x
0 10 20 30 40 50
02
46
812
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46
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quen
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46
812
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quen
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46
810
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quen
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1015
2025
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1015
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46
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05
1015
2025
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1015
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quen
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02
46
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quen
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46
8
x
0 5 10 20 30
02
46
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x
0 10 30 50
05
1015
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0 5 10 20 30
02
46
80
24
68
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quen
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24
68
10
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1015
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46
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quen
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46
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02
46
812
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quen
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0 5 15 25 350
24
68
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quen
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46
8
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quen
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02
46
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quen
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0 20 40 60
05
1015
x
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quen
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0 5 10 20 30
02
46
810
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quen
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24
68
10
Fre
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02
46
812
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0 20 40 60 80
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1015
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x
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15
x
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quen
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1015
20
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46
810
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1015
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quen
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0 10 20 30 40
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46
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1015
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1520
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2535
Normal Q−Q Plot
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2535
Normal Q−Q Plot
Theoretical Quantiles
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ple
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s
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010
3050
Normal Q−Q Plot
Theoretical Quantiles
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ple
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020
4060
Normal Q−Q Plot
Theoretical Quantiles
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ple
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2535
Normal Q−Q Plot
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05
1525
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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010
2030
40
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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010
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4050
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
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Normal Q−Q Plot
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1020
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Normal Q−Q Plot
Theoretical Quantiles
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010
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40
Normal Q−Q Plot
Theoretical Quantiles
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2040
6080
Normal Q−Q Plot
Theoretical Quantiles
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05
1525
35
Normal Q−Q Plot
Theoretical Quantiles
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ple
Qua
ntile
s
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−2 −1 0 1 2
010
3050
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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−2 −1 0 1 2
1030
5070
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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515
2535
Normal Q−Q Plot
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ple
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ntile
s
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1030
5070
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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515
2535
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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−2 −1 0 1 2
510
2030
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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510
2030
Normal Q−Q Plot
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ple
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ntile
s
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1020
3040
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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510
1520
25
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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515
2535
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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Normal Q−Q Plot
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ple
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ntile
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Normal Q−Q Plot
Theoretical Quantiles
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510
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Normal Q−Q Plot
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Normal Q−Q Plot
Theoretical Quantiles
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Normal Q−Q Plot
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010
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4050
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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1030
50
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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1020
3040
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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Normal Q−Q Plot
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ple
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ntile
s
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1020
3040
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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1020
3040
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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020
4060
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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4050
Normal Q−Q Plot
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ple
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ntile
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1020
3040
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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−2 −1 0 1 2
1020
3040
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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−2 −1 0 1 2
010
3050
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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2535
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
12
Sample size 50
Normal
x
Fre
quen
cy
5 10 15
050
010
0015
0020
00
Histograms
x
6 8 10 12 14
02
46
810
x
6 8 10 12 14
05
1015
x
6 8 10 14
02
46
812
05
1015
x
Fre
quen
cy
4 6 8 12 16
05
1015
20
x
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quen
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4 6 8 10 14
02
46
812
x
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quen
cy
6 8 10 12 14
02
46
810
Fre
quen
cy
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1015
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6 8 10 12 14 16
02
46
812
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1015
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4 6 8 10 14
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1015
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6 8 10 12 14
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46
810
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4 6 8 10 14
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1015
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6 8 10 12 14
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810
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4 6 8 10 12 14
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810
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4 6 8 10 14
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1015
02
46
810
x
Fre
quen
cy
6 8 10 12 14
02
46
810
x
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quen
cy
8 10 12 14
02
46
812
x
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quen
cy
6 8 10 12 14
02
46
812
Fre
quen
cy
02
46
810
x
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quen
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6 8 10 12 14
02
46
812
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quen
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6 8 10 12 14
02
46
810
x
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quen
cy
4 6 8 12 16
02
46
812
Fre
quen
cy
05
1015
x
Fre
quen
cy
6 8 10 12 14
02
46
810
x
Fre
quen
cy
4 6 8 10 14
05
1015
x
Fre
quen
cy
6 8 10 12 14
02
46
810
Fre
quen
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05
1015
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812
x
6 8 10 12 14
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46
810
x
8 10 12 14
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812
02
46
8
x
Fre
quen
cy
6 8 10 12 14
02
46
812
x
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quen
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6 8 10 12 14
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46
810
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quen
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2 4 6 8 10 14
05
1015
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quen
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1015
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quen
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6 8 10 12 14
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1015
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quen
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6 8 10 14
02
46
810
x
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4 6 8 10 14
05
1015
20
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05
1015
x
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quen
cy
8 10 12 14 16
02
46
812
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quen
cy
4 6 8 10 14
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1015
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6 8 10 12 14
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810
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810
1214
Normal Q−Q Plot
Theoretical Quantiles
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Normal Q−Q Plot
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68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
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911
13
Normal Q−Q Plot
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−2 −1 0 1 2
810
12
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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1012
14
Normal Q−Q Plot
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ple
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ntile
s
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−2 −1 0 1 2
68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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−2 −1 0 1 2
68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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68
1012
14
Normal Q−Q Plot
Sam
ple
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ntile
s
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−2 −1 0 1 2
68
1014
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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−2 −1 0 1 2
46
810
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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46
810
1214
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
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−2 −1 0 1 2
68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
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−2 −1 0 1 2
46
810
1214
Normal Q−Q Plot
Theoretical Quantiles
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−2 −1 0 1 2
810
1214
16
Normal Q−Q Plot
Theoretical Quantiles
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68
1012
14
Normal Q−Q Plot
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68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
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ple
Qua
ntile
s
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46
810
12
Normal Q−Q Plot
Theoretical Quantiles
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ple
Qua
ntile
s
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−2 −1 0 1 2
810
1214
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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68
1012
14
Normal Q−Q Plot
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ple
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ntile
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68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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−2 −1 0 1 2
68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
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ple
Qua
ntile
s
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−2 −1 0 1 2
68
1012
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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46
810
14
Normal Q−Q Plot
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ple
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ntile
s
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−2 −1 0 1 2
46
810
1214
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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−2 −1 0 1 2
810
1214
Normal Q−Q Plot
Theoretical Quantiles
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ple
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ntile
s
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−2 −1 0 1 2
810
1214
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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68
1012
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
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−2 −1 0 1 2
68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
●
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−2 −1 0 1 2
810
1214
Normal Q−Q Plot
Theoretical Quantiles
●
●
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●
−2 −1 0 1 2
810
1214
Normal Q−Q Plot
Theoretical Quantiles
●
●
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●
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68
1012
1416
Normal Q−Q Plot
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−2 −1 0 1 2
68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●●
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●
−2 −1 0 1 2
810
1214
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
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−2 −1 0 1 2
68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●
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68
1012
14
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
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−2 −1 0 1 2
46
810
1214
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●●
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● ●
●
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−2 −1 0 1 2
67
89
1113
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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−2 −1 0 1 2
68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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68
1012
14
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
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−2 −1 0 1 2
46
810
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
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−2 −1 0 1 2
68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
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−2 −1 0 1 2
68
1012
14
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
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68
1014
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
13
Right skewed log-normal
x
Fre
quen
cy
0 20 40 60 80
050
100
150
Histograms
x
0 10 30 50
05
1015
x
0 10 20 30 40
05
1015
x
0 10 20 30 40
05
1015
05
1015
2025
x
Fre
quen
cy
0 10 20 30 40 50
02
46
812
x
Fre
quen
cy
0 10 30 50
05
1015
20
x
Fre
quen
cy
0 10 30 50
05
1020
30
Fre
quen
cy
05
1015
x
Fre
quen
cy
0 10 30 50
05
1015
20
x
Fre
quen
cy
0 10 30 50
05
1015
x
Fre
quen
cy
0 10 20 30 40 50
05
1015
Fre
quen
cy
05
1020
x
Fre
quen
cy
0 20 40 60
05
1015
20
x
Fre
quen
cy
0 20 40 60 80
05
1020
x
Fre
quen
cy
0 10 20 30 40
05
1015
2025
Fre
quen
cy
05
1015
20
x
0 10 30 50
05
1015
20
x
0 20 40 60 80
05
1020
x
0 10 20 30 40
05
1015
05
1015
20
x
Fre
quen
cy
0 10 30 500
510
1520
x
Fre
quen
cy
0 10 20 30 40
05
1015
x
Fre
quen
cy
0 20 40 60 80
05
1015
2025
Fre
quen
cy
05
1015
x
Fre
quen
cy
0 20 40 60 80
05
1015
2025
x
Fre
quen
cy
0 10 30 500
510
1520
x
Fre
quen
cy
0 20 40 60 80
05
1015
20
Fre
quen
cy
05
1015
x
Fre
quen
cy
0 10 20 30 40 50
05
1015
20
x
Fre
quen
cy
0 10 20 30 40
02
46
812
x
Fre
quen
cy
0 20 40 60 800
510
1520
25
Fre
quen
cy
05
1020
30
x
0 10 30 50
05
1015
2025
x
0 10 20 30 40 50
05
1015
x
0 20 40 60
05
1015
2025
05
1015
x
Fre
quen
cy
0 10 30 50
05
1015
2025
x
Fre
quen
cy
0 5 15 25 35
05
1015
x
Fre
quen
cy
0 10 20 30 40
05
1015
Fre
quen
cy
05
1020
30
x
Fre
quen
cy
0 20 40 60 80
05
1015
2025
x
Fre
quen
cy
0 10 30 50
05
1015
2025
x
Fre
quen
cy
0 10 20 30 40
05
1015
Fre
quen
cy
05
1015
x
Fre
quen
cy
0 10 20 30 40
05
1015
20
x
Fre
quen
cy
0 10 20 30 40
05
1015
x
Fre
quen
cy
0 20 40 60 80
05
1020
Fre
quen
cy
05
1015
20
Boxplots●
010
2030
4050
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1030
50
●
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50
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3040
50
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4060
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4060
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4060
80
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50
515
2535
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3040
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4060
80
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4060
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2030
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4060
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50
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4060
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50
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4060
80
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515
2535
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2535
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2030
Qqplots
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−2 −1 0 1 2
1020
3040
50
Normal Q−Q Plot
Theoretical Quantiles
●●●●
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●
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−2 −1 0 1 2
1030
50
Normal Q−Q Plot
Theoretical Quantiles
●
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−2 −1 0 1 2
010
2030
40
Normal Q−Q Plot
Theoretical Quantiles
●
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●
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010
3050
Normal Q−Q Plot
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●
●●●
●
●
●
●
−2 −1 0 1 2
515
2535
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●
●●
●●●
●
●●
●
●
●
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●
●●●●
●●
●
●
●
●
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●●
●
●
●
−2 −1 0 1 2
1030
50
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●● ●●
●●
●
●●●●
●
●
●
●●
●
●●
●
●●●
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●
●●●●
●
●
●
●
●
●
●
●
●●●●
●●
●
●
●●
●●
−2 −1 0 1 2
020
4060
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
●
●
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●
●
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●
●
●●●
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●
●
●
●
●
●
●
●●
●
●●
●●
●
●
515
2535
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●●● ●
●
●
●
●
●
●
●
●●●
●●●
●●
●●●
●
●
●●●● ●
●
●
●
−2 −1 0 1 2
020
4060
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●● ●
●
●
●
●●
●
●● ● ●
●●
●
●
●
●
●●
●●●
●
●
●
● ●● ●
●●
●
●●
●
●
●
●●●
●
●●
●●
●
●●
−2 −1 0 1 2
1030
50
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●●
●●
●
●
●
●
●●
●
●●
●
●
●
●
●●●
●
●
●
●●● ●
●●
●●
●
●
●
●●
●●
●
●
●
●
●●
−2 −1 0 1 2
1020
3040
50
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●●●
●
●
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●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
515
2535
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●●
●●●
●
●
●●
●
●
●●
●●
●●●
●●
●●
●●
●
●
●
●● ●
●
●
●
●
●●●
●
●
●●
●
●●
●
●
●
●
−2 −1 0 1 2
020
4060
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●●
●
●●
●●
●
●●
●
●
●
●
●
●
●
●
●
●●
●●
−2 −1 0 1 2
1020
3040
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●●
●●
●
●
●●
●●
●●
●
●
●●
●● ●
●
●
●●
●
●
●●
●
●●●●
●
●
●
●●
●●
●
●
●
●
●
●
●
−2 −1 0 1 2
1020
3040
50
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
● ●●
●
●
●●
●●
●
●
●●
●●●
●
●
●
●
●
●
●
●
●●
●
●●
●
●
●
●
●●
●
●
●
●●
●●●
●
●
●●
●
●
●1020
3040
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●●
●
●●
●
●
●●
●●
●●
●●
●●
●●●
●
●
● ●●● ●
●
●●
●
●●● ●
●
●
● ●
●●●
●
●● ●
●●
●●
−2 −1 0 1 2
020
6010
0
Normal Q−Q Plot
Theoretical Quantiles
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●●
●●
●
●
●
●
● ●●
●●
●
●
●
●●
●
●●●●
●●
●
●
●●
●
●
●
−2 −1 0 1 2
010
2030
4050
Normal Q−Q Plot
Theoretical Quantiles
●
●
●●
●
●●
●●
●
●●
●●
●●
●●
●
●
●
●
●
●●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●●
●●●
●●
●●
−2 −1 0 1 2
1030
5070
Normal Q−Q Plot
Theoretical Quantiles
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●●
●●
●
●
●
●
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●
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●
●
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●
●
●
●
●
●
●
● ●
010
2030
4050
Normal Q−Q Plot
●●
●●
● ●
●
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●
●
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●
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●●●●
● ●
−2 −1 0 1 2
2040
6080
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●
●
●
●●●
●
●
●
●
●
●
●
●
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●
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●
●
●
●
●
●
●
●
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●
●
●
−2 −1 0 1 2
1020
3040
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●
●
●
●
●●●
●
●●
●
●
●
●●
●
●
●
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● ●●●
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●
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●
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●
●
●
●●
●
●
●●
●
●●
●
●
●
●●
−2 −1 0 1 2
1030
50
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●●●
●
●●● ●
●
●●
●
●
●
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●
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●
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●
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●
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●
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●
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●
●
●
●
●
●●
●1020
3040
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●●
●
●
●
●●
●
●●
●
●●●
●
●
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● ●●
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●
●●
●●
●
●
●
●●●
●
●●
●
●
●●
●
−2 −1 0 1 2
515
2535
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●
●●
●●●●
●●
●
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●
●
●●
●
●
●●
●●●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
1020
30
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●●●
●
●
●●●●
●
●
●
●
●●
● ●
●
●
●
●
●●
●●
●●●
●
●
●
●●
●
●
●●
●
●
●●
●
●●●
●●
●
−2 −1 0 1 2
1020
3040
50
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●
●
●●
●
●
●
●
●
●
●
●
●
●
●●
●●
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●●
●●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
1020
3040
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●
●
●
●●
●●
●●
●
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●
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●
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●
●●
●
●
●●
●
●
●
●
●●
●●
●
●
●
●
●
−2 −1 0 1 2
010
2030
4050
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●●
●
●
●●●
●
●●
●
●
●
●
●
●
●
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●
●
●●
●●
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●
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●
●
●
●
●
●●●
●
●
●
●
●
●
●
●●
●
−2 −1 0 1 2
1020
3040
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●●
●
●
●●
●
●●
●
●
●●
●
●
●●
●
●●
●●●
●
●
●
●●●
●
●
●●●
●
●●
●●●●
●
●
●
●
●●
●
−2 −1 0 1 2
1020
3040
50
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●
●●
●●
●
●
●●●●
●
●
●
●
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●
●●
●●
●
●
●
● ●●
●
●
●●
●
●
●
●
●●●
●
●
●●
●
1020
3040
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●●
●●
●
●
●
●●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●●
●
●
●
●●
●
●●●
●
●
●
●
●
●
●
●
●●
●
●
−2 −1 0 1 2
1020
3040
Normal Q−Q Plot
Theoretical Quantiles
●
●●
●●●
●
●
●
●
●
●
●
●
●
●
●●
●●●
●●
●
●●
●
●
●●
●
●
●
●● ●
●●
●●
●●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
010
3050
Normal Q−Q Plot
Theoretical Quantiles
●
●
●
●
●●●●●
●●
●●
●●●
●●●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●●
●
●
●
●
●●●
●
●●
●
●
●
−2 −1 0 1 2
515
25
Normal Q−Q Plot
Theoretical Quantiles
●
●●
●
●●●●
●
●
●●
●
●
●●
●●●
●
●
●●
●●●
●
●●
●
●
●
●
●●●
●
●
●
●
●
●
●
●
●
●●●
●
●
010
3050
Normal Q−Q Plot
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●●
●
●●
●●
●
●●
●
●
●
●
●●
●
●●
●●
●●
●
●●
●
●●
●
●
●
●●
●
−2 −1 0 1 2
1020
3040
50
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
● ●
●
●●
●
●
●●
●●
●
●●
●●
●●
●
●
●
●●
●
●●
●
●
●●
●●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
−2 −1 0 1 2
515
2535
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●
●●
●●●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
1020
3040
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●●
●
●●
●
●
●
●●
●
●●●
●●
●
●●
●
●●
●
●●
●
●
●
●
●●●●
●
●
●
●
●●●
●
●●
●
● ●
●
● ●
●
020
4060
80
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●●
●●
●●
●●
●
●●●
●
●
●
●●
●
●
●●
−2 −1 0 1 2
515
2535
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
● ●
●
●
●
●
●●
●
●●
●
●●●
●●
●
●●
●
●
●
●
●
●
●●●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●
●
●
−2 −1 0 1 2
010
3050
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●●
●
●
●●
●
●
●
●●
●
●
●
●
●
● ●●
●
●
●
●
●
●
●
●●●
●
●
●
●●
●●
●
●
●
●●
●
●
●
●
●
−2 −1 0 1 2
1020
3040
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●●
●
●●
●
●
●●
●●
●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●
●●
●
●●
●
●
●●
●
●
●
●
05
1020
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
●
●
●
●
●
●●●
●
●
●
●
●
●
●●
●
●●
●
●
●●
●
●
●
●
●●
●
●
●
●
●
●
●
●
●●●●●
●●●
●
●
●●●
−2 −1 0 1 2
1020
3040
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●
●●
●
●
●●
●
●
●
●
●
●
●
●
●●
●
●
●
●
●●
●●
●
●
●
●●
●●●
●●●
●
●
●
●
●
●
●
●
●
●
●
−2 −1 0 1 2
010
3050
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●
●●
●
●
●
●●
●
●●●●
●●
●●
●
●
●
●●
●
●●
●●●●
●●●
●
●●
●●
●
●●
●
●●
●
●
●
●●
−2 −1 0 1 2
010
2030
4050
Normal Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
●
●
●●●
●
●
●
●●●●●
●●
●●
●
●
●
●
●
●
●●●
●
●
●
●
●●
●
●●
●
●●
●●
●
●
●
●
●
●
●
●
●
●
1020
3040
50
Normal Q−Q Plot
Sam
ple
Qua
ntile
s
References
Hilborn, R. and Mangel, M. (1997). The Ecological Detective: Confronting Models with Data.
14
