What This Episode Covers
Statistics shows up on nearly every GMAT® you'll take, and the good news is that a handful of core properties do most of the heavy lifting. The time you invest in this episode is the kind that pays back on test day. We work through three retired Official Guide problems focused on averages, medians, and statistical constraints, building from a clean warm-up to a legitimately challenging optimization question.
The first problem introduces (or refreshes) the foundational definitions of average and median, then reveals a shortcut that converts what looks like a two-minute computation into a near-instant answer: in any evenly spaced set, the mean always equals the median. Having said that, this only works for evenly spaced sets — blindly applying it elsewhere will cost you points, so we explain exactly when it holds and when it doesn't.
The second problem involves five pieces of wood with a given average length and a given median length, asking for the maximum length of the shortest piece. This is a super useful medium-difficulty constraint problem that rewards careful visual organization. We walk through how to set up the constraints on paper, why the median anchors one piece of the solution, and how to systematically solve for the target value without guessing.
The third problem is the hardest: 30 students, a known distribution of how many books each group borrowed, a given class average, and the goal of maximizing one student's book count. It's a multi-step optimization that integrates the average formula, subtraction, and strategic reasoning — and it's a great example of how the same core skills compound when you stack problem complexity.
Problems Covered
Problem 1 — Average vs. Median of Multiples of 5 (Easy): If M is the average and capital-M is the median of the first 10 positive multiples of 5, what is capital-M minus M? Answer: 0. Key lesson: in an evenly spaced set, mean = median — know this shortcut cold.
Problem 2 — Five Pieces of Wood (Medium): Five pieces have an average length of 124 cm and a median of 140 cm. What's the maximum possible length of the shortest piece? Answer: 100 cm. Key lesson: write out the constraints clearly, anchor on the median, and use algebra to solve for the extreme.
Problem 3 — Library Books (Hard): 30 students, with specific numbers borrowing 0, 1, and 2 books each, a class average of 2 books per student, and a remaining group borrowing 3 or more. What's the maximum any single student could have borrowed? Answer: 13 books. Key lesson: use the average formula to unlock the total, minimize all but one value, and let the algebra do the rest.
Key Takeaways
- Average = median in evenly spaced sets. This is one of the most valuable shortcuts in GMAT® statistics. Consecutive integers, consecutive multiples, arithmetic sequences — if the spacing is uniform, mean equals median, full stop.
- Define average as sum ÷ count. This formula is what makes "average questions" solvable when you need to find a total or an unknown group's stats. Master it as an equation, not just a concept.
- Median requires an ordered list. For an odd number of values, it's the middle term. For an even number, it's the average of the two middle terms. Don't skip writing out the list under time pressure.
- Write what's given before you solve anything. On multi-constraint problems, capturing all the given information first — before touching the math — is what prevents you from finding the right answer to the wrong question.
- Maximizing one value requires minimizing the others. This is the core logic of the hard problem. When you want to push one variable as high as possible within a fixed total, set everything else to its minimum allowed value.
- Brute force is fine when nothing else comes to mind. If you can execute it in under three minutes and there's no faster approach, do the math. Just know that when a shortcut exists, it's usually worth learning.
Statistics problems can feel abstract at first, but the handful of properties we covered here go a long way. Build the habit of writing what's given before you touch the math, and these questions get a lot more manageable — if that makes sense. Stay positive and stay consistent.