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Let us say i have salary information of 100 male employees and 100 female employees. I compute the averages and the average male salary is higher than the average female salary. Based on this, can we say that males earn more than females?

Or do i need to form a null and alternative hypothesis and carry out hypothesis testing?

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    $\begingroup$ It depends on what you're trying to make a statement about. In relation to just your observations, you can say so, but usually people are interested in saying something about some broader population from which those values came. You haven't mentioned anything about random sampling, in which case it may be you can't say much of anything. $\endgroup$ – Glen_b -Reinstate Monica Feb 11 '18 at 7:58
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I would say you need to lay out more details to decide in this case.

If you were looking within one company with only 100 male and 100 female employees (i.e., 200 employees total) that all had exactly the same job and all other relevant samples were the same and you were only concerned whether male employees had a higher average salary than female employees, then I would say calculating the means itself would be sufficient information to make a decision, at least in the first order. The 100 male and 100 female employees are your total population and so each group's mean represents the actual average salary. Hence, you can say whether in actuality your male employees have a higher average salary, and then you could make a decision what to do with the information. In reality, in any one company it is likely that not all male and female employees are equal in all relevant variables and there may be other biases in getting the data which would require controlling for these factors and possible hypothesis testing.

Now, if the you had a sample of 100 male and 100 female employees from across all industries (effectively an infinite population size), within one industry (maybe also effectively infinite population size), within one company with thousands of employees, etc., then hypothesis testing is needed. You want to know whether there is a difference in the average salaries of men and women across the population, not just the sample, in order to make a decision. This is what the hypothesis testing is trying to do by using the samples to see if there is enough statistical evidence to reject the null hypothesis or not.

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If these are all the employees of that company, then without something like a hypothesis test you can of course descriptively say that males earn more than females in this company. (If it's a representative sample, you would still need to account for the sampling.)

However you would not be able to

  1. conclude that this is a difference that is not explainable by chance and
  2. conclude that this is not related to the characteristics of the jobs people are doing or the characteristics of the people (skills, experience etc.).

A hypothesis test tries to address the first point under the assumption that the second one is taken care of, which one would typically do with a suitable statistical model (e.g. analysis of covariance on the log-transformed salaries, perhaps with some regularization due to the small population).

Of course, you might also take into account prior knowledge (i.e. use some Bayesian method), given that likely no company is truly that different from others.

What decision you take based on what you see should likely again be based on something else than a hypothesis test, because the cost of wrong decisions either way may be different (e.g. cost of increasing salaries of women if there might be discrimination vs. cost of potential lawsuits and reputational damage). That is something one should rather do with decision analysis than a hypothesis test.

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Yes, you need to form a null and alternative hypothesis. This is the Frequentist method (a Bayesian would do it differently). The average male salary may have been higher by an amount that can be explained purely by chance.

As a first try, you can use an independent samples T (Student t) test. This is well described on the web or in a standard statistical text book. This can also be done easily with any Stats package or with R or by using the functionality in Excel (t statistic under Statistical functions).

Your null hypothesis, H0, is that the salaries levels are the same.

There are some complications that you should consider, namely that your salary data is likely to be skewed. You can use more sophisticated techniques to handle this, but the T test is reasonable start.

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