Donation Drive: Choosing a Summary and Checking a Probability Claim
The student council ran a donation drive. The 15 council members reported these totals in dollars: 5, 8, 8, 10, 12, 12, 12, 15, 15, 18, 20, 20, 25, 40, 250. The $250 was a single gift routed through one member by a grandparent. The principal's newsletter draft reads: "Our council members raised an average of $31.33 each - a great showing of student generosity." A council member objects and wants a different summary. She also wants to include the statement: "If you pick one of us at random, there is a 1-in-4 chance that person raised under $10."
1. Compute the mean, median, and mode of the data. Which one should the newsletter use to describe a typical member, and why?
Read the worked answer
Totals: the 15 values sum to $470.
- Mean: 470 ÷ 15 = $31.33
- Median: with 15 values ordered, the median is the 8th value = $15
- Mode: $12 appears three times, more than any other value = $12
Recommendation: the median, $15. The $250 gift is a legitimate data point but is far outside the cluster (the next highest is $40). It pulls the mean to $31.33, a figure no member actually raised and which 13 of the 15 members fell below. The median is resistant to that single extreme value and better represents a typical member.
The mode ($12) is a defensible secondary figure - it is the most common amount - but it understates the group because it ignores the larger contributions entirely. Reporting "median $15" alongside "total $470 from 15 members" is honest and complete.
2. Evaluate the principal's sentence as a piece of statistical communication. What exactly is wrong with it, and what would a corrected version say?
Read the worked answer
The arithmetic is right but the inference is not.
- The mean is not a description of members; it is a total divided by a count. Saying members raised "$31.33 each" implies a typical member, which is false: 13 of 15 raised less than that, and 10 of 15 raised $20 or less.
- A single outlier drives the figure. Remove the $250 gift and the mean falls to (470 − 250) ÷ 14 = $15.71, essentially the median. The headline number is an artifact of one donation.
- "Great showing of generosity" is being supported by a statistic that mostly measures one donor.
Corrected version: "Fifteen council members raised $470 in total, a median of $15 per member, including one extraordinary gift of $250." This reports total, typical value, and the outlier's influence explicitly, so the reader can judge for themselves.
3. Check the council member's probability claim. Is it correct as stated, and what assumption about the random selection does it depend on?
Read the worked answer
Count the qualifying members: amounts under $10 are 5, 8, and 8 → 3 members out of 15.
P(under $10) = 3/15 = 1/5 = 20%, not 1/4 (25%). The claim overstates the probability. She may have divided 3 by 12, or counted the two $8 gifts but handled the $5 incorrectly; either way the denominator must be all 15 members.
The assumption: the claim requires each member to be equally likely to be chosen - a uniform draw over the 15 people. Note what changes under a different sampling scheme: if you instead drew a dollar at random from the $470 total, the probability of landing on a dollar given by someone who gave under $10 would be only 21/470 ≈ 4.5%. Both numbers are correct for their own sampling scheme, so the wording "pick one of us at random" must be preserved exactly; rewording it as "a random dollar donated" would silently change the answer by a factor of more than four.