Two Rice Mills, Equal Averages: Consistency, Contract Limits and a ₹2,500 Saving
A wholesale grocer in Nagaon packs rice in 1 kg bags and is choosing between two mills. Both quote on declared net weight, and both deliver bags whose average is very close to 1,000 g, so the price per kilogram alone does not settle the matter. Ten bags were drawn at random from each mill's last dispatch and weighed. Mill P gave 995, 1002, 998, 1005, 993, 1000, 1008, 997, 1004, 998 g. Mill Q gave 1010, 990, 1005, 985, 1015, 995, 1000, 1010, 990, 1000 g. Mill P quotes ₹48.50 per kg of declared weight and Mill Q quotes ₹48.00. The grocer's supply contract with a local hotel chain allows no more than 10% of bags to fall below 990 g, and short-weight complaints have already cost the grocer one customer this year. An order of 5,000 bags is being finalised, and the grocer wants a defensible basis for the decision rather than a comparison of averages.
1. Compute the mean, standard deviation and coefficient of variation of bag weight for each mill, and state which mill fills more consistently.
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Mill P. Values: 995, 1002, 998, 1005, 993, 1000, 1008, 997, 1004, 998. Sum = 10,000, so mean = 1,000 g. Deviations from 1,000: −5, 2, −2, 5, −7, 0, 8, −3, 4, −2. Squared: 25, 4, 4, 25, 49, 0, 64, 9, 16, 4 → Σd² = 200. Variance = 200/10 = 20, SD = 4.47 g, CV = 4.47/1000 × 100 = 0.45%.
Mill Q. Values: 1010, 990, 1005, 985, 1015, 995, 1000, 1010, 990, 1000. Sum = 10,000, mean = 1,000 g. Deviations: 10, −10, 5, −15, 15, −5, 0, 10, −10, 0. Squared: 100, 100, 25, 225, 225, 25, 0, 100, 100, 0 → Σd² = 900. Variance = 90, SD = 9.49 g, CV = 0.95%.
The means are identical, so the average conceals the whole difference. Mill P's spread is roughly half of Mill Q's on both SD and CV; the CV is the right comparison here because it expresses risk relative to the 1,000 g target. Using the n − 1 divisor instead gives SDs of 4.71 and 10.00, which leaves the conclusion unchanged.
2. Applying the hotel chain's limit of no more than 10% of bags below 990 g, estimate how many underweight bags per 1,000 each mill would produce, and comment on how much this estimate can be trusted.
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Mill P. The lowest of the ten bags is 993 g, so no sampled bag falls below 990 g. Estimated rate: 0 per 1,000 bags. Even the worst bag is 7 g clear of the limit.
Mill Q. Only the 985 g bag is strictly below 990 g (the two 990 g bags sit exactly on the line and are not below it). That is 1 in 10, i.e. an estimated 100 bags per 1,000.
Trustworthiness. With only ten observations the estimate is very fragile. One underweight bag out of ten is compatible with a true rate anywhere from under 1% to over 40%; the sample cannot distinguish a mill that is genuinely at 10% from one that is at 3% or 25%. Mill P's zero is equally weak - a defect rate of 5% would often show no defect in ten bags.
So the safe statement is directional, not numerical: P is clearly the tighter filler, and Q's process is centred on 1,000 g in a way that puts a visible share of bags on the wrong side of a contractual line. Before any penalty clause is invoked, both mills should be re-sampled at 50–100 bags.
3. Mill P costs ₹48.50 per kg and Mill Q ₹48.00 per kg on a 5,000 bag order. Give a recommendation with the arithmetic behind it, and say what would change your mind.
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Price gap. 5,000 bags at 1 kg declared weight: Q saves ₹0.50 × 5,000 = ₹2,500 on the order.
Weight risk. If Q's 1-in-10 sample rate holds, about 500 of the 5,000 bags would fall below 990 g. The ₹2,500 saving therefore works out to roughly ₹5 per underweight bag - a very thin cushion against one hotel chain audit, one return consignment, or the loss of a customer, which has already happened once this year.
Contract risk. Q's estimated 10% sits exactly on the contractual ceiling. Because the estimate comes from ten bags, Q is as likely to breach the limit as to meet it on any given dispatch; P gives a margin of 7 g on its worst bag, which is genuine protection.
Recommendation: buy from Mill P, and treat the ₹2,500 as the price of the weight guarantee, not as a loss. Reopen the comparison only if Q (a) brings its declared weight to at least 1,000 g with a floor of 995 g, (b) offers a written replacement clause for underweight bags, or (c) drops to ₹47.50 so the saving becomes ₹5,000 - still only ₹10 per suspect bag, so a re-sample would be needed first.