Which Average Should a Ration Shop Inspector Report?
A ration shop inspector in a block of Madurai district has to recommend how the monthly rice quota of the block's 63 ration shops should be fixed. She collects the average daily sales of eleven shops, selected because they were the first eleven listed in the register: 40, 42, 45, 45, 48, 50, 52, 55, 60, 65 and 480 kilograms. The last figure comes from a shop that also runs a wholesale godown. The inspector's assistant calculates the mean and writes in the draft note that "the average shop sells about 89 kg a day", on which basis the block office plans to release stock. The inspector is uneasy: most of the shops she visited look far smaller than that. She has also been told that the eleven shops were not chosen at random, and that the block has shops in both the town and the surrounding villages. She must submit a note with a corrected average, a justification for the measure she chooses, and a caution about the data.
1. Compute the mean, median and mode of the eleven observations, showing your working, and comment on the shape of the distribution.
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Data in ascending order: 40, 42, 45, 45, 48, 50, 52, 55, 60, 65, 480 (n = 11)
Mean = ΣX ÷ n = 982 ÷ 11 = 89.27 kg
Check the sum: 40 + 42 + 45 + 45 + 48 + 50 + 52 + 55 + 60 + 65 + 480 = 982.
Median = value of the ((n + 1) ÷ 2)th item = the 6th item = 50 kg
(The sixth value in the ordered list is 50: 40, 42, 45, 45, 48, 50.)
Mode = the most frequently occurring value = 45 kg (it appears twice; every other value appears once, so the distribution is unimodal).
Shape. Mean (89.27) > Median (50) > Mode (45). This ordering identifies a positively skewed distribution with a long right tail, produced entirely by the single 480 kg observation. In a symmetric distribution the three averages coincide; the greater the gap between mean and median, the greater the skewness. This comparison of the three averages is itself a standard way of describing the shape of a distribution without drawing it.
2. Which average should the inspector use for fixing the quota, and what happens to the mean and the median if the 480 kg shop is excluded?
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Choice: the median.
The median is a positional average - it depends only on the order of the observations, not on the magnitude of the extreme ones - so the godown-cum-shop cannot pull it up. The mean uses the value of every observation and is therefore dragged upward by any outlier. Here the mean of 89.27 kg describes almost none of the shops: only one of the eleven sells more than 89 kg, and ten sell 65 kg or less. The mode (45 kg) is also unaffected by the outlier but is a poor choice, because it picks the commonest size rather than the middle of the distribution.
Effect of excluding 480 kg
- New sum = 982 − 480 = 502; n = 10 → mean = 50.2 kg
- New ordered list: 40, 42, 45, 45, 48, 50, 52, 55, 60, 65 → median = average of the 5th and 6th items = (48 + 50) ÷ 2 = 49 kg
| Measure | With 480 kg | Without 480 kg | Change |
|---|---|---|---|
| Mean | 89.27 | 50.20 | −39.07 (about 44%) |
| Median | 50.00 | 49.00 | −1.00 |
The mean moves by about 39 kg while the median moves by 1 kg. That is the practical meaning of robustness: the median resists extreme values, the mean is sensitive to every observation.
Practical consequence. Quotas based on the mean of 89 kg would over-supply the single large shop many times over and under-supply the ten small shops. Quota should be fixed shop by shop from card-holder numbers, with the median (about 50 kg) taken as the typical value for the block.
3. The eleven shops were the first eleven in the register. What cautions should the inspector record about using this average for all 63 shops?
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1. Selection bias / representativeness. The first eleven entries in a register are almost never a random sample: registers are commonly ordered by ward, by date of licensing or by size of shop, so the eleven may over-represent one locality, one size class or one period. A statistic is only as good as the sample behind it, and here the sample was chosen for convenience, not for representativeness.
2. Small sample. Eleven out of 63 shops is about 17 per cent of the population. With so few observations a single unusual unit - the godown shop - shifts the mean by nearly 40 kg and can even shift the median, because the median of an even-sized set rests on just two middle values. Conclusions drawn from such a small base carry a large margin of uncertainty.
3. Measurement problems. "Average daily sales" over what period? Sales on festival days, at the start of the month when cards are credited, and wholesale sales from the godown are not comparable with ordinary retail sales of a small shop. The 480 kg figure almost certainly mixes wholesale turnover with ration-card sales. All shops should be measured over the same days, and wholesale should be recorded separately.
4. What the note should state. The measure used (median), the period and method of collection, the fact that the eleven shops were not randomly selected, and that the figure is an estimate for the sampled shops, not a census of the block. If a block-level average is really required, the inspector should take a stratified random sample - separate draws from town shops and village shops in proportion to their numbers - and weight each shop's sales by its number of card holders.