The Placement Report: Two-Thirds Placed?
A business school reports on its 300-student graduating batch. Of these, 180 registered for campus placements; the other 120 opted out to join family businesses or pursue higher studies. Among those who registered, 120 students received at least one offer. The placement office counted 165 offers in total, because 45 students received two offers each and no student received more than two. The offers fell into three salary bands. The 90 offers in the ₹6–8 lakh band averaged ₹6.8 lakh; the 55 offers in the ₹8–12 lakh band averaged ₹9.6 lakh; the 20 offers above ₹12 lakh averaged ₹14.5 lakh. Last year the average salary across all offers was ₹7.9 lakh. The dean's press note states: "Placements have improved: the average offer is now ₹8.67 lakh, up from ₹7.9 lakh last year, and two-thirds of the batch is placed."
1. Verify the dean's two figures and state precisely what each one measures.
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Average offer: total salary value = (90 × 6.8) + (55 × 9.6) + (20 × 14.5) = 612 + 528 + 290 = ₹1,430 lakh. Dividing by 165 offers gives 1,430 ÷ 165 = ₹8.67 lakh. The figure is arithmetically correct.
But note the trap: a simple average of the three band averages, (6.8 + 9.6 + 14.5)/3 = ₹10.3 lakh, would be wrong, because the bands contain very different numbers of offers. The ₹8.67 lakh is a weighted average, and it is an average per offer, not per student.
Two-thirds placed: 120 ÷ 180 = 66.7 per cent, which is two-thirds of the students who registered. Measured against the full batch of 300, the figure is 120 ÷ 300 = 40 per cent. So the dean's number is true only for the self-selected group that opted into placements, a group likely to be more placement-oriented than the batch as a whole. The press note's wording, "two-thirds of the batch", is therefore misleading.
2. Can the average salary per placed student be computed from the given data? Explain, and state what would be needed.
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No. The data give the number of offers in each band and the average of each band, but not which students hold which offers. The 45 students with two offers could have both offers in the ₹6–8 lakh band, or one low and one high, or both high; the average per student changes substantially across these possibilities.
To see the size of the problem, consider the extremes. If every double-offer student held two offers from the ₹6–8 lakh band, removing one offer from each would raise the per-student average; if every double-offer student held one high-band and one low-band offer, the per-student average would fall. Per-offer data alone cannot settle this.
What is needed is a student-level table showing each student's offers and band. Failing that, the placement office could report the median salary, or the average of each student's highest offer, both of which are standard and avoid double counting. Reporting offers rather than students is the single biggest source of distortion in this report.
3. Is the comparison with last year's ₹7.9 lakh valid evidence that placements improved? Give two reasons and one piece of data that would make the comparison fair.
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The comparison is weak for at least two reasons.
First, mix effects. The average can rise because the composition of offers changed, not because any individual did better. If this year's batch had more offers in the above-₹12 lakh band, or if the number of offers grew faster than the number of students, the average rises mechanically. Two years with identical individual outcomes but different band counts will produce different averages.
Second, definitional drift. Last year's ₹7.9 lakh may have been computed per student, per offer, or over a different set of recruiters; the note does not say. Comparing an average per offer with a figure of unknown definition is not a like-for-like comparison.
A fair comparison needs band-wise offer counts and band-wise averages for both years, or better, the median of each student's best offer in both years. With those, the school could separate a genuine rise in pay from a change in the shape of the offer pool.