In the country I live in, by law, every employer (with some exceptions) must disclose this data about how much is paid for their employees every month (the data set): arithmetic mean, the three quartiles and standard deviation. Also, the number of employees is known. I think that it is interesting to think what kind of things you can deduce about any company when this kind of information is available publicly. To make it more real, here is an example of some real-ish data: $$n = 320$$ $$\bar{x}= 2160$$ $$Q_1 = 1460$$ $$Q_2 = 2150$$ $$Q_3 = 2710$$ $$\sigma = 930$$ Does it make sense to assume that such data set follows normal distribution and would this help to get some more information? I guess that it depends on the concrete data set but is there some "test" to see if such data fits some commonly used distribution? Also, is it possible to say something interesting about how a specific employee compares to the rest? For example $x_i=2500$.

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    $\begingroup$ In most countries the salary structure in companies is hugely non-normal: upper level management is paid many times what most employees are paid. It sounds like your question requires knowledge of companies in your country in order to be answerable. $\endgroup$ – whuber Sep 20 '18 at 20:00
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    $\begingroup$ In addition, many companies have very different roles and remunerations. A retailer may have many low-paid store personnel (many of who may be part time workers), and a few comparatively well paid IT workers. A given IT worker may earn in the top 10% of the entire company workforce and still be underpaid compared to his peers. $\endgroup$ – S. Kolassa - Reinstate Monica Sep 20 '18 at 20:22
  • $\begingroup$ In the particular example you give, it might be reasonable to assume normal dist'n: mean approx equal median and $Q_2 - Q_1 \approx Q_3 - Q_2. \approx .67S.$ (I assume you mean $S,$ not $\sigma.)$ // Astonishingly similar to normal. Is that really a typical example? What industry? // At least you can say, for this employer, that salary 2500 is above mean and median, but below 75th percentile. // You're really vague about what inferences you might like to draw. $\endgroup$ – BruceET Sep 20 '18 at 21:26
  • $\begingroup$ "Does it make sense to assume that such data set follows normal distribution" -- the normal would be a very poor approximation in most cases. $\endgroup$ – Glen_b -Reinstate Monica Sep 21 '18 at 9:48
  • $\begingroup$ From just these empirical summary statistics, it would be a long shot to extrapolate or estimate a distribution. You need the full data set for this, plot their empirical PMF or PDF (for different binnings), pick a theoretical distribution model, then employ goodness-of-fit tests, Q-Q plots, etc. to evaluate your chosen theoretical model PDF. I would rather suggest you look into nonparameteric models. As already pointed out, salary data are unlikely to be Gaussian in typical cases. $\endgroup$ – Lucozade Sep 21 '18 at 17:03

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