Correcting a data-entry outlier and deciding what to publish in a district earnings profile
A district economics and statistics office is preparing a profile of self-employed tailors in a block. An enumerator surveyed 40 tailors and recorded monthly earnings. The office computed the mean as ₹12,400 and the median as ₹9,800. During verification, a tailor's earning of ₹8,500 was found to have been entered as ₹85,000. The erroneous value is the highest in the dataset. The district officer wants to publish 'average monthly earnings' in a one-page brief. A second issue: the survey covered only tailors in the main market and the bus stand, while tailors in four outlying villages were not surveyed. The officer must decide which measure to report, whether to correct the brief or re-survey, and how to describe the limitation. The brief is due tomorrow; a full re-survey would take three weeks. The officer also has the original 40 returns, so the error can be corrected immediately.
1. Compute the corrected mean. Can the office assume the median is unchanged because the erroneous value was an outlier? Explain.
Read the worked answer
Original sum = 40 × 12,400 = ₹4,96,000. Corrected sum = 4,96,000 − 85,000 + 8,500 = ₹4,19,500. Corrected mean = 4,19,500 / 40 = ₹10,487.50. The original mean overstated the corrected mean by ₹1,912.50, or 18.24% of the corrected mean.
The median cannot be assumed unchanged. The erroneous ₹85,000 was the maximum, so removing it does not disturb the middle ranks from the top side. But the corrected value ₹8,500 is below the reported median of ₹9,800. Inserting it into the lower half shifts the 20th and 21st observations: if the insertion falls at or below rank 20, the new median becomes the average of the old 19th and old 20th values, which is at most ₹9,800 and will generally be lower. The exact corrected median requires the original 40 returns; the office must recompute it, not carry forward ₹9,800.
2. Which measure should be the headline in the one-page brief: mean, median, or both? Draft a one-sentence disclosure that is statistically honest and suitable for a district profile.
Read the worked answer
Use the median as the headline for earnings because earnings distributions are typically right-skewed and the median is less influenced by very high earners. Report the mean alongside as supporting information, but only after correction. Do not publish the old mean of ₹12,400.
A defensible sentence is: 'In the surveyed main-market and bus-stand area, the median monthly earning of the 40 self-employed tailors was ₹[recomputed value], and the mean was ₹10,487.50; earnings were skewed, so the median is the better summary of a typical tailor's earning.' If the corrected median has not yet been computed, do not insert ₹9,800 by default; either delay the headline number or publish the mean alone with a clear 'mean' label. State the survey period, the sample size of 40, and that the figures are nominal.
3. How should the officer handle the sampling limitation in tomorrow's brief, and what should the full survey design be?
Read the worked answer
Do not label the result 'block-level' or 'all tailors'. Label it 'main-market and bus-stand tailors' and add a limitation: 'Tailors in four outlying villages were not covered; their earnings may differ, so the figures are not representative of the whole block.' With one day, do not fabricate a block estimate or weight the 40 cases as if they were representative.
For the full survey, use a stratified sample: strata = main market, bus stand, and each of the four villages; allocate the sample by the number of tailors in each stratum, then select shops randomly within each stratum. If a full list is unavailable, use a two-stage cluster design: villages as clusters, then a random sample of shops within selected villages. Pre-test the schedule, train enumerators, and record non-response. This is more defensible than extending the convenience sample.