Salesman Auditor problem: Which mean and median is correct? What is preferred?

Question

Trip	Days	Expenses	Expenses per day
1	0.5	13.50	27
2	2.0	12.00	6
3	3.5	17.50	5
4	1.0	9.00	9
5	9.0	27.00	3
6	0.5	9.00	18
7	8.5	17.00	2
Total	25.0	105.00	70

Question: The following data represents travel expenses ( other than transportation) for 7 trips made during november by a salesman of small firm. The auditor criticised these expenses as excessive asserting average expense per day is Rs 10 ( 70 divided by 7 ). The salesman replied that average is only Rs 4.20 ( 105 divided by 25 ) and that in any event median is the appropriate measure and is only Rs 3. The auditor rejoined that the arithmetic mean is the appropriate measure, but that the median is 6. a) Explain the proper interpretation of each of the four averages mentioned b) Which average seems to be appropriate to you

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I'm self-studying statistics, I have read that median is an appropriate measure in case of skewed data or if outlier is present. I understand how the auditor and salesman are calculating the mean, and I think 105/25 = 4.20 is the correct mean. However, I do not understand how they are arriving at the median and which approach/statistic is correct.

Answer 1

This is a nice question. The auditor treats every number in the Expenses per day column as a data point (you can verify the auditor's median from this). The salesman treats every day as a data point. Verifying the salesman's median is not quite so intuitive, particularly because the data are not given day by day, so you will need to assume that every day in the same trip is treated the same, like for example in Trip 2, the two days have expenses 6 and 6. This still doesn't make it fully intuitive as there are half days. However you can go to "Expenses per day" keeping an eye on the Days column, and then ask, what is the expense per day number so that half of the days (or more) have larger or equal expenses and half of the days (or more) have smaller or equal expenses (definition of the median). You're given that the number is 3, so you just have to check that this is the case.

Regarding which approach is appropriate (which is a better word than "correct" as it depends on the aims of the people involved), think about which of these is directly related to the money that the salesman has spent overall, as this is what they will probably get refunded, i.e., the cost to the company. In fact only one of these (assuming we know the totals but not further details) will allow to directly reconstruct that amount.