Daily takings and one bulk order: which average should go to the bank?
A hardware shop in Francistown recorded its daily takings, in pula, for the 12 trading days before applying for a bank loan: 1 850, 1 900, 1 950, 2 050, 2 100, 2 150, 2 250, 2 300, 2 400, 2 600, 2 750, 18 000. The last figure was a single bulk order from a contractor; nothing like it had happened before and the manager does not expect it to happen again. In the loan application the manager writes: 'Average daily takings are P3 525.' The bank asks for the median as well, and for a forecast of takings for a normal 26-day month. The manager argues that the mean is the fairest summary because it uses every day's figures.
1. Verify the manager's figure of P3 525 and calculate the median, showing your working.
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Sum: 1 850 + 1 900 + 1 950 + 2 050 + 2 100 + 2 150 + 2 250 + 2 300 + 2 400 + 2 600 + 2 750 + 18 000 = 42 300
Mean: 42 300 / 12 = P3 525, so the manager's arithmetic is correct.
Median: the data is already in order. With 12 values the median is the mean of the 6th and 7th values, which are 2 150 and 2 250:
(2 150 + 2 250) / 2 = P2 200
2. Comment on whether the statement 'Average daily takings are P3 525' is a fair description of a typical day. Support your answer with both averages and with a recalculation that removes the bulk order.
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The statement is arithmetically true but misleading as a description of a normal day.
- Eleven of the twelve days took less than P2 750, and every one of those eleven is below P3 525. Only the single bulk order lifts the mean.
- Median = P2 200.
- Mean without the bulk order: (42 300 - 18 000) / 11 = 24 300 / 11 ≈ P2 209.
The median and the trimmed mean agree closely (P2 200 and P2 209), which shows they describe the same ordinary trading pattern. The overall mean is about 60% higher than either and describes a pattern that occurred on one day in twelve. The manager is right that the mean uses every value, but that is exactly the weakness here: the mean is not resistant to a single extreme value. Presenting P3 525 without mentioning the bulk order would give the bank a false impression of the shop's normal income, and a misleading loan application is a serious matter.
3. Give a forecast of takings for a normal 26-day month, stating which measure you used and why, and give one limitation of the forecast.
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Forecast using the median: 26 x 2 200 = P57 200.
Using the trimmed mean instead: 26 x 2 209 ≈ P57 434 (about P57 400).
The two agree to within a few hundred pula, so a forecast of about P57 000 to P57 500 is safe.
Why not the overall mean: 26 x 3 525 = P91 650, which is about 60% higher and rests entirely on a one-off order the manager says will not repeat. A forecast should be built from the pattern expected in the future, not from a single past event.
Limitation: the forecast assumes 26 trading days and no further bulk orders, and it ignores seasonal effects such as month-end paydays or the rainy season, when building-supply sales may rise or fall. It is a description of past ordinary days, not a guarantee.