What is going on?
This experiment draws independent observations from an exponential population with mean 1 and standard deviation 1. Individual values are right-skewed. Each sample is averaged, and the histogram records those sample means. Increasing the observations per sample usually produces a narrower, more symmetric shape.
Work through it
- Choose how many observations go into each average.
- Repeat that sampling many times to form a histogram of averages.
- Increase the sample size and compare the spread with 1 / √n.
Put numbers to the idea
For this population, samples of size 1 have standard deviation 1. Averages of 25 observations have standard deviation 1 / √25 = 0.2. Averages of 100 observations have standard deviation 0.1. Their center remains 1.
A little more clarity
Does the original population become normal?
No. The original distribution stays exponential. The distribution of sample averages becomes more nearly normal as the sample size grows.
Is a sample size of 30 always enough?
No. Required size depends on the population’s shape and the accuracy needed. Very skewed or heavy-tailed populations may need much larger samples.
Why does the histogram change between runs?
Each histogram uses a finite random collection of samples. A new run changes the observed bars even though the underlying population and theoretical spread stay the same.