from my data set, is this sample significant or random

Question

I apologize for how much of a "beginner" question this must be.

This is a general question, but I will spell out the actual numbers here as much as possible just so you can see what it looks like.

I have 6,680 items, in a particular sequence, ordered by a score. The score ranges from 0 to 529, and the median score is 3.86.

Added -- and the average is 12.03 and the standard deviation is 26.356. I don't know how to further describe the distribution in a useful way but the scores are all here, there are no unknowns about that.

I have been handed a sample of four items from this group. I need to say quantify how likely it is that this sample is random (that's the null hypothesis). Of course I am hoping that the sample is not random. It would be great to say "the chance of this sample being random is less than .001" or something like that.

Here are the scores from this sample. But I'm not asking you to tell me the answer, I need to learn how to do this myself, for future samples, whose size might be different. The current sample in question has scores 0.32, 0.74, 1.02, and 4.95.

(And is the same kind of calculation possible if I ignored the individual scores but could still rely on the sequence of items in my group of 6,680? I suspect that the individual scores are not too important so long as the sequence of items is preserved.)

Thanks in advance if you can point me in the right direction.

Answer 1

Comment: Here is one way to make sense of the the question.

You have a large population with median $\eta = 3.84.$ You sample $n = 4$ observations from this population obtaining the stated observations, three of which are below $\eta.$ Is the sample consistent with $H_0: \eta = 3.84$ or should we reject $H_0$ in favor of the alternative $H_a: \eta < 3.84?$

We do not know the distribution of the population, and we are asked to compare the sample to the population median (not mean). Those are two clues that the author of the question might expect you to use a Wilcoxon one-sample ("signed-rank") test. Here are results from R statistical software:

wilcox.test(x, mu = 3.86, alt="less")

        Wilcoxon signed rank test

data:  x 
V = 1, p-value = 0.125
alternative hypothesis: true location is less than 3.86 

You may be 'hoping' for evidence that the sample (with sample median $H = 0.88$) is 'centered' significantly below $\eta = 3.86,$ but there is there is not evidence of that. If the P-value were below 0.05, then we could reject $H_0$ at the 5% significance level.

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Notes: (1) Even if the last of the four observations were 1.94 (below $\eta$), there still be be insufficient evidence to reject $H_0.$ Four observations are not enough to draw interesting conclusions.

(2) Because you have given absolutely no context for this question, it is possible that that you have never heard of a Wilcoxon signed-rank test, and that the author had entirely something else in mind.