I am doing a Weibull and normal distribution analysis for a set of my data which are :

336256 620316 958846 1007830 1080401

So to avoid putting the whole code here, I refer you directly to the post I followed :


The PDF and CDF plots I get are so small and get this form :



Just as info , I put the Weibull distribution results here , to show the values that are small as well :

enter image description here

update :

as per request I put the code to plot pdf here :

my data are as indicated 336256 620316 958846 1007830 1080401

so to reproduce the code, you need to save it as csv and run this code, on your directory. My problem is that I can change and scale the graphs on

xs <- seq(0, 5, len=500)

by changing to :

xs <- seq(10, 1650000, len=5000)

I get :


The problem is when I change the second and third argument , I mean :

1650000 and len=5000 to for example 9650000, len=5000 , the peaks position also displace and don't remain in the same place so it's not only a re scaling the graph.

# 5. Bootstrapping the pointwise confidence intervals


rw.small <- c(336256,620316,958846,1007830,1080401)

xs <- seq(0, 5, len=500)

boot.pdf <- sapply(1:1000, function(i) {
  xi <- sample(rw.small, size=length(rw.small), replace=TRUE)


boot.cdf <- sapply(1:1000, function(i) {
  xi <- sample(rw.small, size=length(rw.small), replace=TRUE)


# Plot PDF

par(bg="white", las=1, cex=1.2)
plot(xs, boot.pdf[, 1], type="l", col=rgb(.6, .6, .6, .1), ylim=range(boot.pdf),
     xlab="x", ylab="Probability density")
for(i in 2:ncol(boot.pdf)) lines(xs, boot.pdf[, i], col=rgb(.6, .6, .6, .1))

# Add pointwise confidence bands

quants <- apply(boot.pdf, 1, quantile, c(0.025, 0.5, 0.975))
min.point <- apply(boot.pdf, 1, min, na.rm=TRUE)
max.point <- apply(boot.pdf, 1, max, na.rm=TRUE)
lines(xs, quants[1, ], col="red", lwd=1.5, lty=2)
lines(xs, quants[3, ], col="red", lwd=1.5, lty=2)
lines(xs, quants[2, ], col="darkred", lwd=2)
#lines(xs, min.point, col="purple")
#lines(xs, max.point, col="purple")
  • 3
    $\begingroup$ "I still get" that is because you still didn't scale it properly enough. This still seems to be just a matter of making the graphs. Scale the graphs differently and you will be able to align them with the left graphs in your Weibull results overview (these have more on the x-scale, showing more of the entire curve, and less on the, taller, y-scale making the curve less flat). If after correct scaling you still do not get a good result, then show your (reproducible) code or otherwise it will be difficult to see what else could have gone wrong (it isn't clear how the code in the link is used). $\endgroup$ Aug 5, 2018 at 21:58
  • 1
    $\begingroup$ Note that small values for a pdf are perfectly fine. It is a probability density function, which will be lower if the total probability (the integrated total/mass/area will be 1) is spread out over a longer range events. $\endgroup$ Aug 5, 2018 at 22:10
  • $\begingroup$ @MartijnWeterings Thank you very much, I updated my question please check it $\endgroup$ Aug 6, 2018 at 7:10
  • 1
    $\begingroup$ "The problem is when I change the second and third argument , I mean : 1650000 and len=5000 to for example 9650000, len=5000 , the peaks position also displace and don't remain in the same place so it's not only a re scaling the graph" I don't see the problem. Can you explain it more. (also, is this question answered or not?, you have accepted the answer below, but you have made additional questions in the comments afterwards) $\endgroup$ Aug 6, 2018 at 11:18
  • 1
    $\begingroup$ Please read stats.stackexchange.com/questions/4220 $\endgroup$
    – whuber
    Aug 6, 2018 at 12:24

2 Answers 2


"the peaks position also displace"

Could you say what the coordinates of the peaks are in the two different cases?

I am not sure what you are all doing. The bootstrapping part is very vague (and I doubt it is correct to do it like that) so I took it out. The code below is creating fine graphs (peaks at the same coordinates every time, but of course when you scale the x-axis the position on the screen/plot shifts). Can you look into this and comment/explain again what your question about the graphs is?

code output


# settings
df <- c(336256,620316,958846,1007830,1080401)
xs <- seq(0, 2*10^6, len=500)

# estimation
MLE.est <- suppressWarnings(fitdistr(df, densfun="weibull", lower = 0))  
boot.pdf <- dweibull(xs, shape=as.numeric(MLE.est[[1]][1]), scale=as.numeric(MLE.est[[1]][2]))
boot.cdf <- pweibull(xs, shape=as.numeric(MLE.est[[1]][1]), scale=as.numeric(MLE.est[[1]][2]))

# plotting

plot(xs, boot.pdf, type="l", col=1, ylim=range(boot.pdf),
     xlab="x", ylab="")
points(df,dweibull(df, shape=as.numeric(MLE.est[[1]][1]), scale=as.numeric(MLE.est[[1]][2])),pch=21,col=1,bg=2)

plot(xs, boot.cdf, type="l", col=1, ylim=range(boot.cdf),
     xlab="x", ylab="")
points(df,pweibull(df, shape=as.numeric(MLE.est[[1]][1]), scale=as.numeric(MLE.est[[1]][2])),pch=21,col=1,bg=2)

legend(0.9*10^6,0.12,c("fit","data points"),col=1,lty=c(1,NA),pch=c(NA,21),pt.bg=c(0,2),cex=0.7)

  • $\begingroup$ @FabioSpaghetti it is unclear to me why you have accepted my answer? I have only cleaned up your code and was still trying to figure out what your question really was. Was it about the low values due to your imagine only showing the tail of the distribution (as alexis' answer points out), or was is about the low values of the pdf in general (and which is explained in the other question which Whuber links to, but then it is about high values instead of low values, but the principle is the same)? $\endgroup$ Aug 6, 2018 at 13:02
  • 1
    $\begingroup$ For future reference (helping visitors that may land on this page) we should make this question and answer more clear. Can you still clarify your question, even though you seem to have gotten something of an answer. $\endgroup$ Aug 6, 2018 at 13:04
  • $\begingroup$ Sure Martijin, I will do that now $\endgroup$ Aug 8, 2018 at 7:13

The scale of $x$ on your graphs is nowhere near the five observed values (which have a mean value of 800,729.8, a minimum value of 336,256, and maximum of 1,080,401), and therefore you would expect the very small probabilities on your $y$ axis.

  • $\begingroup$ Thank you , I modified the code , but as you see it remains the same $\endgroup$ Aug 5, 2018 at 21:29
  • 1
    $\begingroup$ It remains the same because the graph is the same graph with different axis labels. I agree with @MartijnWeterings first comment above. $\endgroup$
    – Alexis
    Aug 6, 2018 at 0:34
  • $\begingroup$ Thank you very much, so far, I thought I can only scale x, and the function pdf and cdf are scales as a consequence $\endgroup$ Aug 6, 2018 at 6:19
  • $\begingroup$ @Martijin Weterings $\endgroup$ Aug 6, 2018 at 7:09
  • $\begingroup$ I updated my question, please review $\endgroup$ Aug 6, 2018 at 7:09

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