# Does a sample have the same statistics as another sample taken from the same population?

Let's say I take a sample from a population and find the statistics of it like the mean, variance, minimum and maximum. Then, I take another sample from the same population and find its statistics. Would the statistics be the same for both samples? I'm guessing the minimum and maximum has a chance of being different, but I'm not sure for the mean and variance.

• When you say you "take a sample," how is the sampling done? Is it a random sample you're taking, or something else? – Mico Sep 11 '16 at 6:09

## 4 Answers

In general the sample statistics will tend to be different. With continuous random variables this should always be the case (up to rounding, which brings us back to "actually that's only notionally continuous") and with discrete random variables it will often be the case with some statistics and perhaps more often not with some others (with how often depending on the pattern of the distribution, the sample size, and the particular statistics you're looking at).

You can answer your own question by direct experiment, in simple cases.

For example, consider rolling a particular six-sided die (a well-manufactured one that is very close to fair). You can draw two samples of some desired size ($n_1=20$ and $n_2=20$ say) and calculate their sample statistics. I suggest you try it!

Actually, not being one to ask you to try something I wouldn't do myself, here's my attempts, first with one die (two samples each of size 20) and then a repeat with a different die:

Die A     Outcome:   1    2    3    4    5    6
Sample 1 (Counts)    2    3    3    2    4    6
Sample 2             6    2    3    4    3    2

Die B     Outcome:   1    2    3    4    5    6
Sample 1 (Counts)    3    7    3    0    3    4
Sample 2             1    4    1    5    4    5


And so here are some summary statistics:

Die A      Range  median mean  sd
Sample 1     5      4.5  4.05 1.791
Sample 2     5       3   3.10 1.774

Die B      Range  median mean  sd
Sample 1     5      2.5  3.25 1.860
Sample 2     5       4   4.10 1.619


If you do it you will probably get the same maximum and minimum both times (you'd expect both 1 and 6 to show up in a sample of 20 about 95% of the time) but the means and standard deviations would be different.

The medians might well be the same (about a 25% chance of that, with the usual definition of sample median for even $n$), but easily might not.

There's some chance to get the same mean for two of these (because we're sampling a discrete distribution with only a few outcomes) but there's a low chance of seeing it (about 3.7%); you can also get the same standard deviation, but the chance is much lower still ... about 2/3 of a percent.

At larger or smaller sample sizes those chances change; and they change again if you draw from other distributions than that of an (approximately) fair die roll.

That all of those statistics I mentioned would be the same would be very unlikely.

What other kinds of samples might you easily make an experiment with? What do you expect to find?

Let's try it out, using Python to take two samples of size 100 from a standard normal distribution:

>>> import numpy as np
>>> samp1 = np.random.randn(100)
>>> samp2 = np.random.randn(100)
>>> np.mean(samp1), np.mean(samp2)
(-0.021265142109962453, 0.10432818350501703)
>>> np.var(samp1), np.var(samp2)
(0.97512014413601544, 0.97388659947212219)
>>> np.min(samp1), np.min(samp2)
(-2.5134700890849775, -2.0177700510108623)
>>> np.max(samp1), np.max(samp2)
(2.9687061563574924, 2.8165937970247885)


If you're sampling from continuous distributions, then the values of the statistics you get from two different samples are going to be different with probability 1.

Sampling from a discrete distribution (which may be what you mean by "a population"), there's going to be some chance that you get the same values, but it's definitely not guaranteed. For example, think about taking a sample of size 1 from a uniform distribution over $\{1, 2, 3, 4\}$. Taking larger samples makes it more likely that they're closer, but they're not going to be exactly the same.

That said, if the samples are the same size the distribution of the statistic that you get is the same. (Without looking at samp1 or samp2, I have no reason to think that one is going to have a larger mean than the other, for example.)

Consider a population consisting of 5 people with ages 10, 20, 30, 40 and 50. Assume we are interested in statistics of their ages. If sample size = 4, these are 5 possible samples:

1. 10, 20, 30, 40
2. 10, 20, 30, 50
3. 10, 20, 40, 50
4. 10, 30, 40, 50
5. 20, 30, 40, 50

It is clear that almost all of the standard statistics are different across samples.

If sample size = population size, all statistics would be the same across all samples (obviously, because each of the samples would be the same ). As you decrease the sample size, the statistics you observe vary across samples, and the variability is modelled using sampling distributions.

Assuming the samples are taken from the same population; as sample size increases the sample statistics should approach the population statistics.

All that said:

The mean, median, min, max, or any other statistic can be different from the population for any given sample. This is where things like t-test come in to determine if there is a statistically significant difference between the sample and the population. On the predicate that you don't know if you are sampling from your population of course.

• Can you clarify how you get the statement in the first paragraph from the central limit theorem? The last sentence sounds like a misinterpretation of hypothesis tests, though I'm not sure what "the likelihood" means in this context. – Juho Kokkala Sep 11 '16 at 10:00
• @JuhoKokkala It works by taking a sample of size n calculate your statistic, do this N times. Take the average of all the N samples and it should approach the population statistic. As n and/or N increase in size it will take less iterations to get a value that is close to the population statistic. I used mean here, but can be done for almost any statistic mean, sd, median. By likelihood I mean a sample did not pass a statistical significance test, suggesting, it did not come from the population. – Matt L. Sep 12 '16 at 21:53
• The central limit theorems are about $\sqrt{n}\,(\bar{x}-\mu)$ converging in distribution to a certain normal distribution. I don't see how your statement is related - do you have a reference or a proof? Also, if only $N$ increases and $n$ is constant, I don't believe your statement holds except if the statistic is an unbiased estimator of the population statistic. Take, for example, samples of size $n=3$ (with or without replacement) from $\{-2,-1,0,1000,2000\}$ and compute medians -- the expected value of the sample median is not equal to the population median. – Juho Kokkala Sep 13 '16 at 4:35
• @JuhoKokkala Ah I see, yes, I was overzealous in my statements. There are several cases where this won't work. That being said it does work for many distributions. As can be seen here. Thanks for keeping me on my toes. The sampled statistic has to have a normal distribution. I'll try to go back and clarify when I get a chance. - I just removed it, rather not confuse people :P And it was way above what was being asked. – Matt L. Sep 13 '16 at 12:51