How to Solve: Statistics Problems (GRE/GMAT) – Complete Guide

How to Solve: Statistics Problems (GRE/GMAT) – Complete Guide

Score Impact: Statistics questions appear 4-6 times per GRE Quant section and 3-5 times per GMAT Quant section—mastering them can boost your score by 30-50 points, moving you from the 60th to the 80th+ percentile.

WHAT THIS QUESTION TYPE IS ACTUALLY TESTING

The GRE/GMAT doesn’t test advanced statistics—it tests decision-making under pressure. Specifically: 1. Can you identify the correct statistical measure? (Mean vs. median vs. mode vs. range vs. standard deviation) 2. Can you avoid misreading the question? (E.g., confusing "average" with "median" or "sum" with "count") 3. Can you handle conditional data? (E.g., "If the highest value is removed, what happens to the mean?")

ANATOMY OF THE QUESTION

Structure Breakdown

1. Stem: Describes a dataset (numbers, variables, or a table).
2. Conditions: Often adds constraints (e.g., "If one value is removed," "If the mean is 10").
3. Answer Choices: 4-5 options, usually testing:
4. A numerical value (e.g., "What is the new mean?")
5. A relationship (e.g., "Which is greater, the mean or the median?")
6. A change (e.g., "How does the standard deviation change if X is added?")

What to Ignore

• Overcomplicating calculations (GRE/GMAT rarely requires exact SD or variance).
• Assuming symmetry (unless explicitly stated, data isn’t always normally distributed).
• Ignoring units (e.g., "dollars" vs. "percentages").

Representative Example Question

Set S consists of 5 distinct integers. The mean of S is 20, and the median is 25. If the range of S is 30, which of the following could be the largest element of S? (A) 30 (B) 35 (C) 40 (D) 45 (E) 50

THE DECISION FRAMEWORK (Step-by-Step)

Run this every time—no exceptions.

1. Identify the measure being tested.
2. Underline: mean, median, mode, range, standard deviation.

3. If multiple are mentioned, prioritize the one the question asks about.

4. Extract the given data.

5. List all numbers, variables, and constraints.

6. Example: Mean = 20, Median = 25, Range = 30, 5 distinct integers.

7. Determine if the question is about:

8. A value (e.g., "What is the mean?") → Calculate.
9. A relationship (e.g., "Is the mean > median?") → Compare.

10. A change (e.g., "If X is removed, what happens to the SD?") → Analyze impact.

11. Apply the correct formula/rule.

12. Mean: Sum / Count
13. Median: Middle value (odd count) or average of two middle values (even count).
14. Range: Max – Min

15. Standard Deviation: Only need to know if it increases/decreases (not exact value).

16. Check for traps.

17. Are the numbers distinct? Ordered? Symmetrical?

18. Does the question imply a change (e.g., "removed," "added")?

19. Eliminate wrong answers.

20. Use logic (e.g., "The mean can’t be larger than the max value").

21. Test edge cases (e.g., "What if the smallest number is 0?").

22. Confirm the answer.

23. Does it satisfy all conditions?
24. If stuck, plug in numbers (backsolve).

Worked Examples

Example 1 – Straightforward (Mean & Median)

Set T has 6 numbers: 10, 15, 20, 25, 30, x. If the mean of T is 20, what is the median? (A) 17.5 (B) 20 (C) 22.5 (D) 25 (E) 27.5

Step-by-Step: 1. Measure tested: Mean → Median. 2. Given data:
- Numbers: 10, 15, 20, 25, 30, x
- Mean = 20 → Sum = 6 20 = 120 3. Calculate x:
- Sum of known numbers = 10 + 15 + 20 + 25 + 30 = 100
- x = 120 – 100 = 20 4. New set: 10, 15, 20, 20, 25, 30 5. Median: Average of 3rd and 4th terms = (20 + 20)/2 = 20 6. Answer: (B) 20

Example 2 – Common Trap (Ignoring "Distinct" Numbers)

Set Q has 5 distinct positive integers. The mean is 10, and the median is 12. What is the smallest possible value of the largest integer in Q? (A) 12 (B) 13 (C) 14 (D) 15 (E) 16

Step-by-Step: 1. Measure tested: Mean → Median → Range (implied by "largest integer"). 2. Given data:
- 5 distinct integers, mean = 10 → Sum = 50
- Median = 12 (3rd term in ordered set) 3. Structure: [a, b, 12, d, e], where a < b < 12 < d < e 4. Minimize e (largest integer):
- Maximize a and b to reduce e.
- Smallest possible a = 1 (positive integer)
- Next, b = 11 (must be < 12 and distinct)
- Sum so far: 1 + 11 + 12 = 24
- Remaining sum: 50 – 24 = 26
- d and e must be >12, distinct, and d < e.
- Smallest possible d = 13 → e = 26 – 13 = 13 → But d and e must be distinct.
- Next, d = 13 → e = 14 (sum = 27, but we need 26) → Doesn’t work.
- d = 13 → e = 13 → Invalid (not distinct).
- d = 14 → e = 12 → Invalid (e must be > d).
- Correct approach: d = 13 → e = 14 (sum = 27) → Too high.
- Adjust a and b: a = 1, b = 10 → Sum = 1 + 10 + 12 = 23 → Remaining = 27
- d = 13 → e = 14 (sum = 27) → Still too high.
- a = 1, b = 9 → Sum = 1 + 9 + 12 = 22 → Remaining = 28
- d = 13 → e = 15 (sum = 28) → Works! 5. Smallest possible e = 15 6. Answer: (D) 15

Trap: Students assume a = 1, b = 2, etc., without maximizing a and b to minimize e.

Example 3 – Hard Variant (Standard Deviation Change)

Set A has 4 numbers: 2, 4, 6, 8. If a new number x is added to Set A, which of the following will result in the standard deviation decreasing? (A) x = 0 (B) x = 5 (C) x = 6 (D) x = 10 (E) x = 12

Step-by-Step: 1. Measure tested: Standard deviation (SD) change. 2. Key rule: SD decreases if the new number is closer to the mean than most existing numbers. 3. Calculate mean of Set A:
- Mean = (2 + 4 + 6 + 8)/4 = 5 4. Evaluate each option:
- x = 0: Distance from mean = |0 – 5| = 5 → Increases SD (far from mean).
- x = 5: Distance = 0 → Decreases SD (same as mean).
- x = 6: Distance = 1 → Closer than most numbers (2,4,8 are 3,1,3 away) → Decreases SD.
- x = 10: Distance = 5 → Increases SD.
- x = 12: Distance = 7 → Increases SD. 5. Options that decrease SD: (B) and (C). 6. Which decreases SD more? x = 5 (distance = 0) decreases it more than x = 6 (distance = 1).
- But the question asks for any decrease, so both (B) and (C) work.
- However, only (C) is listed as a single correct answer (likely a typo in options).
- Re-evaluate: If only one answer is correct, (C) is the safer choice (x = 5 is the mean, but the question may imply "strictly decreasing"). 7. Answer: (C) 6

Trap: Students overcomplicate SD calculations—focus on distance from the mean.

WRONG ANSWER PATTERNS

1. Mean/Median Confusion
2. Looks right: "The mean is 10, so the median must be 10."

3. Why wrong: Mean and median are equal only in symmetric distributions.

4. Ignoring "Distinct" Numbers

5. Looks right: "The median is 12, so the numbers are 10, 11, 12, 13, 14."

6. Why wrong: Numbers must be distinct (e.g., 10, 11, 12, 13, 14 is valid, but 10, 12, 12, 13, 14 is not).

7. Range Misapplication

8. Looks right: "The range is 30, so the largest number is 30."

9. Why wrong: Range = Max – Min. If Min = 5, Max = 35.

10. SD Overestimation

11. Looks right: "Adding a number far from the mean always increases SD."
12. Why wrong: If the new number is closer to the mean than most existing numbers, SD decreases.

Common Mistakes

1. Mistake: Assuming data is ordered.
2. Why it happens: Students forget to sort numbers before finding median/range.

3. Correct approach: Always order the set first.

4. Mistake: Calculating exact SD.

5. Why it happens: Overcomplicating—GRE/GMAT only tests if SD increases/decreases.

6. Correct approach: Compare distances from the mean.

7. Mistake: Forgetting "distinct" numbers.

8. Why it happens: Overlooking the word "distinct" in the question.

9. Correct approach: Highlight "distinct" and ensure no duplicates.

10. Mistake: Misapplying mean/median changes.

11. Why it happens: Not recalculating after adding/removing numbers.

12. Correct approach: Recompute mean/median after any change.

13. Mistake: Ignoring units or context.

14. Why it happens: Skimming the question (e.g., "dollars" vs. "percentages").
15. Correct approach: Circle units and ensure consistency.

TIME STRATEGY

• Target time: 1.5–2 minutes per question.
• When to skip: If stuck after 2 minutes, flag and return.
• Minimum work:
• For mean/median: Write down the formula and plug in numbers.
• For SD: Compare distances from the mean.
• For range: Identify min/max.

BACKSOLVING AND SHORTCUTS

1. Plug in numbers: If the question gives variables, pick numbers that fit the conditions.

2. Example: Set S has mean 10 and median 12. Pick numbers like 8, 10, 12, 12, 18 (sum = 60, mean = 12—wrong). Adjust to 8, 10, 12, 14, 16 (sum = 60, mean = 12—correct).

3. Eliminate extremes: For "which could be" questions, test the smallest/largest answer choices first.

4. Example: Largest element in Set S (from earlier) could be 50? Test: Min = 50 – 30 = 20. Set: 1, 2, 25, 26, 50 (sum = 104, mean = 20.8—too high). Eliminate (E).

5. Use symmetry: If the mean = median, the data is symmetric (e.g., 2, 4, 6, 8, 10).

1-Minute Recap

"Here’s the deal: Statistics questions on the GRE/GMAT are about speed and precision, not math. Every time you see one, ask yourself: What measure are they testing? Mean, median, range, or SD? Write down the formula, plug in the numbers, and eliminate wrong answers. For mean/median, always order the set. For SD, just compare distances from the mean—no calculations needed. If the question says ‘distinct,’ double-check for duplicates. And if you’re stuck, plug in numbers or test the answer choices. This isn’t about being a math genius—it’s about following the steps and avoiding traps. Now go crush it."

Final Note: Bookmark this guide. Revisit the Decision Framework before every practice session. Statistics questions are predictable—master the process, and you’ll gain 30+ points.