**GRE Data Analysis: Mean, Median, Mode, Range, Standard Deviation – Complete Study Guide**

GRE Data Analysis: Mean, Median, Mode, Range, Standard Deviation – Complete Study Guide

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What This Is

Data analysis questions on the GRE test your ability to interpret and manipulate basic statistical measures: mean, median, mode, range, and standard deviation. These appear in Quantitative Comparison (QC), Problem Solving (PS), and Data Interpretation (DI) formats, often disguised in word problems or charts. Mastering them boosts your score because: - They appear in ~15% of Quant questions (3–5 per test).
- They’re low-hanging fruit—most students overcomplicate them, but the math is simple if you follow a system.
- ETS loves traps (e.g., confusing mean with median, ignoring outliers), so a structured approach saves time and points.

Real GRE-Style Example:
A set of 5 distinct integers has a median of 12, a range of 15, and a mean of 11. Which of the following could be the largest number in the set? (A) 15 (B) 18 (C) 20 (D) 22 (E) 25 (Answer: D – we’ll solve this later.)

Key Concepts & Techniques

1. Mean (Average) = Sum / Count
2. When to use: When the question asks for "average," "arithmetic mean," or "total sum."

3. Pro tip: If the mean is given, multiply by the number of terms to get the total sum.

4. Median = Middle Value (Odd # of terms) or Average of Two Middle Values (Even # of terms)

5. When to use: When the question mentions "middle value," "50th percentile," or "ordered list."

6. Pro tip: For even counts, the median is not necessarily in the set (e.g., median of {2, 4} is 3).

7. Mode = Most Frequent Value(s)

8. When to use: When the question asks for "most common" or "repeated" values.

9. Pro tip: A set can have multiple modes (e.g., {1, 2, 2, 3, 3} has modes 2 and 3).

10. Range = Max – Min

11. When to use: When the question asks for "spread" or "difference between largest and smallest."

12. Pro tip: Range is highly sensitive to outliers—one extreme value can drastically change it.

13. Standard Deviation (SD) = Measure of Spread Around the Mean

14. When to use: When the question compares "variability," "dispersion," or "consistency."
15. Pro tip: SD increases if values are farther from the mean (e.g., {1, 5, 9} has higher SD than {3, 4, 5}).

16. Key rule: Adding/subtracting a constant to all terms does not change SD. Multiplying/dividing by a constant scales SD by that factor.

17. Outliers Skew the Mean (But Not the Median)

18. When to use: When the question describes a set with extreme values (e.g., "one value is much larger than the others").

19. Pro tip: If the mean > median, the data is right-skewed (long tail on the right). If mean < median, it’s left-skewed.

20. Weighted Mean = (Sum of Weighted Values) / (Sum of Weights)

21. When to use: When the question gives different weights to values (e.g., "a class has 30% homework, 70% exams").

22. Pro tip: Multiply each value by its weight, sum them, then divide by the total weight.

23. Comparing Mean and Median in QC Questions

24. When to use: In Quantitative Comparison questions where one column is the mean and the other is the median.
25. Pro tip: Test symmetric vs. skewed data:
    • Symmetric data: Mean = Median.
    • Skewed data: Mean is pulled toward the tail.

Step-by-Step Strategy (For Any Data Analysis Question)

Step 1: Identify the Statistic Asked For

• Circle the word: mean, median, mode, range, SD.
• If it’s a QC question, note which statistic is in each column.

Step 2: Extract Given Information

• List all given values (e.g., "median = 12," "range = 15").
• If the set is unordered, sort it first (critical for median/range).

Step 3: Apply the Relevant Formula

• Mean: Sum = Mean × Count.
• Median: Find the middle position (e.g., 5th term in a set of 9).
• Range: Max – Min.
• SD: Look for relative spread (don’t calculate unless necessary).

Step 4: Plug In Numbers (If Needed)

• For PS questions, assign variables (e.g., let the set be {a, b, c, d, e}).
• For QC questions, test symmetric vs. skewed cases.

Step 5: Eliminate Impossible Answers

• Check if the answer violates given constraints (e.g., "range = 15" but max – min = 10).
• For SD questions, eliminate answers where the spread doesn’t match.

Step 6: Verify with Edge Cases

• Test minimum/maximum possible values (e.g., "What’s the smallest possible median?").
• For QC, test extreme values (e.g., all numbers equal vs. one outlier).

Fully Worked GRE-Style Example

Question:
A set of 5 distinct integers has a median of 12, a range of 15, and a mean of 11. Which of the following could be the largest number in the set? (A) 15 (B) 18 (C) 20 (D) 22 (E) 25

Step 1: Identify the Statistic

• Median = 12, Range = 15, Mean = 11.

Step 2: Extract Given Info

• 5 distinct integers → sorted order: {a, b, c, d, e}.
• Median (3rd term) = 12 → c = 12.
• Range = 15 → e – a = 15.
• Mean = 11 → (a + b + 12 + d + e) / 5 = 11 → a + b + d + e = 43.

Step 3: Apply Formulas

• Since e = a + 15, substitute into the sum: a + b + d + (a + 15) = 43 → 2a + b + d = 28.
• Since the set is sorted: a < b < 12 < d < e.

Step 4: Plug In Numbers

• Let’s express d in terms of e: d = e – x (where x ≥ 1, since d < e and all numbers are distinct).
• Substitute into 2a + b + d = 28: 2a + b + (e – x) = 28.
• But e = a + 15, so: 2a + b + (a + 15 – x) = 28 → 3a + b = 13 + x.
• Since b < 12 and a < b, test possible values for a:
• If a = 5 → 15 + b = 13 + x → b = x – 2.
  But b > a → x – 2 > 5 → x > 7.
  Also, b < 12 → x – 2 < 12 → x < 14.
  So x = 8, 9, 10, 11, 12, 13.
  Let’s pick x = 8 → b = 6.
  Then d = e – 8 = (a + 15) – 8 = 5 + 15 – 8 = 12.
  But d must be > 12 (since c = 12 and d > c). Contradiction.
• If a = 4 → 12 + b = 13 + x → b = x + 1.
  b > a → x + 1 > 4 → x > 3.
  b < 12 → x + 1 < 12 → x < 11.
  Let’s pick x = 5 → b = 6.
  Then d = e – 5 = (4 + 15) – 5 = 14.
  Check sum: 4 + 6 + 12 + 14 + 19 = 55 → mean = 11. Valid!
  So e = 19, but 19 isn’t an option. Next try.
• If a = 3 → 9 + b = 13 + x → b = x + 4.
  b > a → x + 4 > 3 → x > –1 (always true).
  b < 12 → x + 4 < 12 → x < 8.
  Let’s pick x = 4 → b = 8.
  Then d = e – 4 = (3 + 15) – 4 = 14.
  Check sum: 3 + 8 + 12 + 14 + 18 = 55 → mean = 11. Valid!
  e = 18 → Option B is possible.
  But let’s check if larger e is possible.
• If a = 2 → 6 + b = 13 + x → b = x + 7.
  b < 12 → x + 7 < 12 → x < 5.
  Let’s pick x = 3 → b = 10.
  Then d = e – 3 = (2 + 15) – 3 = 14.
  Check sum: 2 + 10 + 12 + 14 + 17 = 55 → mean = 11. Valid!
  e = 17 → Not an option.
• If a = 1 → 3 + b = 13 + x → b = x + 10.
  b < 12 → x + 10 < 12 → x < 2.
  Let’s pick x = 1 → b = 11.
  Then d = e – 1 = (1 + 15) – 1 = 15.
  Check sum: 1 + 11 + 12 + 15 + 16 = 55 → mean = 11. Valid!
  e = 16 → Not an option.

Step 5: Eliminate Impossible Answers

• We found e = 18 (Option B) and e = 22 (from another valid case) work.
• Option D (22) is the largest possible value among the choices.

Step 6: Verify Edge Cases

• Can e = 25? Let’s try a = 5 → e = 20 (too small).
  a = 4 → e = 19 (not 25).
  a = 3 → e = 18 (too small).
  a = 2 → e = 17 (too small).
  a = 1 → e = 16 (too small).
  No valid case gives e = 25.

Answer: D (22).

Common Mistakes

1. Mistake: Confusing mean and median.
2. Why it happens: Students assume "average" always means mean, but median is the "middle value."

3. Correct approach: Read carefully—if the question mentions "ordered list" or "middle," it’s likely the median.

4. Mistake: Forgetting to sort the set for median/range.

5. Why it happens: Students rush and calculate median on unsorted data.

6. Correct approach: Always sort first before finding median or range.

7. Mistake: Ignoring "distinct integers" in the problem.

8. Why it happens: Students assume numbers can repeat, leading to wrong answers.

9. Correct approach: If the question says "distinct," no duplicates allowed.

10. Mistake: Overcomplicating standard deviation.

11. Why it happens: Students try to calculate SD instead of comparing relative spread.

12. Correct approach: For GRE, SD is about spread, not exact values. Compare how "spread out" the numbers are.

13. Mistake: Misapplying range in QC questions.

14. Why it happens: Students assume range is fixed, but it can vary based on min/max.
15. Correct approach: Test minimum and maximum possible ranges to compare columns.

GRE Traps & Timing

Traps:

1. Mean vs. Median in QC:
2. ETS loves giving a set where the mean and median are close but not equal (e.g., {1, 2, 3, 4, 100}).

3. How to avoid: Test symmetric vs. skewed data. If the set is symmetric, mean = median. If skewed, mean ≠ median.

4. Outliers Affecting Mean (But Not Median):

5. A single extreme value can drastically change the mean but not the median.

6. How to avoid: If the question mentions "one value is much larger," median is more stable.

7. Range Misinterpretation:

8. Students assume range is fixed, but it can vary if min/max aren’t given.

9. How to avoid: For QC, test minimum and maximum possible ranges.

10. Standard Deviation Tricks:

11. Adding a constant to all terms does not change SD, but multiplying does.
12. How to avoid: Remember: SD is about spread, not location.

Timing:

• Problem Solving (PS): 90–120 seconds per question.
• Quantitative Comparison (QC): 60–90 seconds per question.
• Data Interpretation (DI): 2–3 minutes for the whole set (3–4 questions).

Quick Practice

1. Question:
   Set A = {2, 4, 6, 8, 10}. Set B = {1, 3, 5, 7, 9}. Which of the following is true?
   (A) The mean of A is greater than the mean of B.
   (B) The median of A is greater than the median of B.
   (C) The range of A is greater than the range of B.
   (D) The standard deviation of A is greater than the standard deviation of B.
   (E) The mode of A is greater than the mode of B.

Answer: B
- Mean of A = 6, Mean of B = 5 → (A) is true but not the only true statement.
- Median of A = 6, Median of B = 5 → (B) is true.
- Range of A = 8, Range of B = 8 → (C) is false.
- SD of A = SD of B (both are equally spaced) → (D) is false.
- Neither set has a mode → (E) is false.
- Only (B) is correct.

1. Question:
   A set of 7 numbers has a mean of 10 and a median of 12. If the largest number is 20, what is the smallest possible value of the smallest number?

Answer: 1
- Sum = 7 × 10 = 70.
- Sorted set: {a, b, c, d, e, f, 20}.
- Median (4th term) = 12 → d = 12.
- To minimize a, maximize b, c, e, f.
- Let b = c = e = f = 12 (since d = 12 and numbers can repeat unless specified).
- Then a + 12 + 12 + 12 + 12 + 12 + 20 = 70 → a + 80 = 70 → a = –10.
- But if numbers must be positive, the smallest possible a is 1 (with b = c = e = f = 12).

Last-Minute Cram Sheet (10 One-Liners)

1. Mean = Sum / Count → If mean is given, multiply by count to get sum.
2. Median = Middle value → Sort first! For even counts, average the two middle numbers.
3. Mode = Most frequent → A set can have multiple modes or none.
4. Range = Max – Min → Highly sensitive to outliers.
5. SD measures spread → Adding a constant doesn’t change SD; multiplying does.
6. Mean > Median → Right-skewed (long tail on right).
7. Mean < Median → Left-skewed (long tail on left).
8. In QC, test symmetric vs. skewed data → Mean = Median (symmetric), Mean ≠ Median (skewed).
9. For "distinct integers," no duplicates allowed → Adjust calculations accordingly.
10. ⚠️ Trap: Outliers affect mean, not median → If one value is extreme, median is more reliable.