Which "average day" should go to the bank
A shopkeeper records her daily takings for twelve working days, in rupees: 4,200; 3,800; 5,000; 4,500; 4,100; 3,900; 4,600; 4,300; 120,000; 4,000; 4,400; 3,700. The 120,000 day was a single bulk order from a school for a sports day, the third such order in two years. She applies for a small business loan and the form asks for "average daily sales". She calculates the mean of all twelve days, rounds it to Rs 13,875, and writes that in the box. Her cousin says she is exaggerating and should use the middle value instead. The bank will use the figure to judge whether she can manage monthly repayments of Rs 95,000.
1. Advise her on what to put on the form. Your answer must deal with both accuracy and the purpose of the form.
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The number she wrote is arithmetically correct but presents a false picture of her trading. A bank needs a sustainable figure, because the repayments must be met every month from ordinary sales, not from a bulk order that arrives twice a year.
Better options, in order of preference:
- Report the median, Rs 4,250, and label it as the median. This is honest, is a standard statistical measure, and reflects a typical day.
- Report both: "median daily sales Rs 4,250; mean Rs 13,875, inflated by one bulk order of Rs 120,000 that occurs about twice a year." The disclosure lets the bank judge for itself.
Either is defensible; what is not defensible is a bare "average: Rs 13,875" with no note, because it invites the bank to expect Rs 400,000 a month where the shop turns over about Rs 100,000. Note the consequence on the loan: repayments of Rs 95,000 a month are roughly equal to her entire normal monthly takings, so on the realistic figure the loan is probably unaffordable and she should ask for a smaller amount or a longer term.
2. The shopkeeper argues that the mean is "the true average" and that using the median hides the bulk order. Assess this argument and state what would change your advice if the bulk order were regular.
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She is half right and half wrong.
The mean is the total divided by the number of days, and it does not ignore the bulk order - but that is exactly the problem. A mean is a good summary of a typical day only when the data are reasonably symmetric. Here one value sits 28 times above the rest, so the mean describes a day that has never happened and never will; every ordinary day falls below Rs 5,000, yet the mean is Rs 13,875. The median does not "hide" the bulk order - it survives it, and it is the measure that answers the question actually asked: what does a normal day look like?
What would change the advice: if bulk orders arrived, say, every fortnight and were similar in size, the distribution would no longer be dominated by a single freak value; the mean and median would converge, and the mean would then be the better basis for forecasting monthly income. The right approach is to look at the pattern of the data first and choose the measure to match, not to pick a measure and defend it.