# Is there a relation between sample size and variable range?

I have done simulations with different sample sizes of a normal distribution (mean=0, sd=10) and plotted the number of samples against statistical parameters. I understand that the relation of sample size and mean, sd and the relative frequency of samples ouside of 2*sd is due to the law of great numbers. (Variance reduces with sample size). I do not understand the relation of sample size and variable range. Is there an intuitive explanation for this relation? • By "range", do you literally mean the maximum value in your sample - the min? Also, it may be of interest to some readers if you posted your code. – gung Aug 17 '16 at 15:41
• See some of the discussion here, including a plot showing the asymptotic growth in mean range like $\sqrt{\log n}$ – Glen_b Aug 18 '16 at 3:51

## 1 Answer

If by range you mean the difference between the maximum and minimum value, then the answer is quite simple.

You're simulating data from a Normal distribution with mean 0 and standard deviation of 10. Range is directly related to the most extreme(minimum and maximum) values of your data. The probability of getting very extreme data is small, hence the "extreme" part, but as your sample size goes to infinity, your probability of getting an extreme data point will go to 1. (Where you've defined extreme as greater than 5 standard deviations or something else you feel is "extreme")

Intuitively as your sample from a distribution grows you are more likely to experience extreme data, which has a direct positive impact on the range. As an example I just simulated 100 million data points from a N(0,10) distribution in R and got a range of (-56.36, to 55.69), so the difference here would be about 111. I got values 5 standard deviations away from my mean, because I simulated so much data! Sure if I sat around all day tomorrow and simulated small data set after small data set of N(0,10) data I might see a range this big, but I probably wouldn't because I probably wouldn't simulate 100 million data points worth of small data sets.

As an aside, this is not really a terribly interesting phenomenon since it is exactly what we would expect to happen.

• Yes, by range I mean the difference between maximum and minimum value. Thank you for your explanation, solid and understandable. I guess, if the distribution is known, one can forecast the mean range depending on sample size (the link of glen_b in the comment above holds a good discussion about this). More general, it would be interesting, if the distances between percentiles (eg. between 10 and 90) show a similar relation with sample size. – user128020 Aug 18 '16 at 13:34