Differences in abstract Poisson population samplings

Question

Given some abstract population, we take samples of sizes $n=15$, $n=25$, and $n=50$, and then do 10000 trials with each sample with Poisson distribution. As a result, we obtain the following plots:

I'm trying to understand the following:

• Why does increasing the sample size also increases the range of the $y$-axis?

• Why are there fewer (thinner) columns in each histogram when samples increase?

• A sample of 10000 will plot a rectangle (just one wide column). What does this imply?

Would appreciate some clarification.

Answer 1

Some answers:

Increasing range: The variance of $\mathsf{Binom}(n,p)$ is $\sigma^2 = np(1-p),$ which increases with the number $n$ of trials. The greater variability causes the range of the sampled values to increase. [If the values are in a vector x, you can use var(x) to see the actual variance.]

Number of bars: I believe you are making your plots with R statistical software. The algorithm that plots histograms provides a target number of bars for the histogram (based on sample size) and then modifies the actual number used so that the break-points between bins will fall at convenient points (integers, multiples of 5 or 10, and so on). If you don't like the number of bins the algorithm picks, you can use the br (for breaks) parameter. You can give a single number as the target number (which the algorithm may modify) or you can give a vector with the exact, usually equally spaced, breakpoints (which it will not modify). [I suggest you read the documentation on br at ? hist and experiment.]

Single bar: I have no clue. Something is wrong; maybe there's a programming error and you actually have only a few values instead of 10,000. I have seen this happen, but only when I made some sort of mistake.