Why don't we automatically have outliers when mean and median differ strongly? Assume you have a data set with information on income of all students in the lecture. The mean value is 1500\$. The median value is however only 800\$. Which of the following conclusions is wrong?
The answer is: The difference between median and mean can be explained by outliers.
Why is this wrong? When Mean and Median deviate in such a manner, there have to be outliers or am I completely wrong? I can check outliers with |z| > 2.5 or a boxplot with IQR*1.5
But why do there not have to be outliers in such a case? I am completely lost and cannot think of a data set that fulfils such a case. Could you please help me?

Outliers from what? You must be assuming a normal (Gaussian) distribution. But with lognormal distributions, some high values are expected as the distribution can be quite skewed. The mean is almost always larger than the median with data sampled from lognormal distributions.  To test this idea, take the logarithm of all the values, and see if that distribution is closer to normal (Gaussian).
