**GMAT Focus Edition: Statistics – Averages, Median, Range, Counting Ideas**

GMAT Focus Edition: Statistics – Averages, Median, Range, Counting Ideas

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What This Is

Statistics questions test your ability to analyze data sets quickly and accurately—core skills for business school. The GMAT Focus Edition emphasizes efficiency and logical reasoning over complex calculations. You’ll face questions on averages (mean), median, range, and counting principles in both Problem Solving (PS) and Data Sufficiency (DS) formats.

Real-GMAT Example:
A set of 5 distinct integers has a median of 12 and a range of 15. If the smallest integer is 5, what is the largest possible integer in the set? (Answer: 20)

Mastering these concepts will save you 3–5 minutes per Quant section and help you avoid careless errors that cost 50+ points.

Key Concepts & Techniques

1. Mean (Average) = Total / Number of Terms
2. When to use: Any question involving "average" or "arithmetic mean."

3. Pro tip: If the mean is given, always calculate the total sum first (Total = Mean × Number of Terms).

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

5. When to use: Questions about the "middle" of a data set, especially when outliers are present.

6. Key insight: The median is resistant to outliers (unlike the mean).

7. Range = Largest Value – Smallest Value

8. When to use: Questions about spread or variability in a data set.

9. Trap: Range does not tell you about the distribution of values between min and max.

10. Counting Distinct Values (for Median/Mean Questions)

11. When to use: When the question asks for the number of possible values (e.g., "How many integers satisfy X?").

12. Shortcut: For consecutive integers, use Max – Min + 1.

13. Weighted Averages

14. When to use: When different groups contribute differently to the average (e.g., test scores, mixture problems).

15. Formula: Weighted Avg = (Sum of (Value × Weight)) / (Sum of Weights).

16. Data Sufficiency (DS) Logic for Statistics

17. When to use: DS questions asking if a statement is sufficient to determine mean/median/range.

18. Rule: Never assume symmetry unless explicitly stated. A median of 10 does not mean the mean is 10.

19. Maximizing/Minimizing Values in a Set

20. When to use: "What is the largest/smallest possible value?" questions.
21. Strategy: To maximize one value, minimize all others (and vice versa).

Step-by-Step Strategy

For Problem Solving (PS) Questions:

1. Read the question carefully. Identify what’s given (mean, median, range, count) and what’s asked.
2. List known values. Write down the smallest/largest values, median, and any constraints.
3. Calculate the total sum (if mean is involved). Total = Mean × Number of Terms.
4. Determine missing values. Use the median/range to find possible values.
5. Check for traps. Are the numbers distinct? Integers only? Positive/negative?
6. Solve and verify. Plug in extreme values to test your answer.

For Data Sufficiency (DS) Questions:

1. Understand the question. What exactly are you being asked to find (mean, median, range)?
2. Evaluate Statement 1 alone. Can you determine the answer with only this info?
3. Evaluate Statement 2 alone. Same as above.
4. Combine statements if needed. Do they together provide enough info?
5. Watch for sufficiency traps. A statement may give a range of possible values but not a single answer.

Fully Worked Example (PS)

Question: A set of 7 distinct integers has a median of 10 and a range of 18. If the smallest integer is 2, what is the largest possible integer in the set?

Step-by-Step Solution:
1. Given:
- 7 distinct integers → odd number of terms → median = 4th term = 10.
- Range = 18 → Largest – Smallest = 18 → Largest = 2 + 18 = 20.
- Smallest = 2.

1. List the set in order:

2. _ , _ , _ , 10 , _ , _ , _

3. Maximize the largest value:

4. To make the largest value as big as possible, minimize the other values (but they must be distinct and greater than 2).
5. The 3 values before the median must be as small as possible but greater than 2 and distinct.
   • Possible values: 3, 4, 9 (since 10 is the median, the 3rd term must be ≤ 9).

6. The 2 values after the median must be as small as possible but greater than 10 and distinct.

   
   
   • Possible values: 11, 12.

7. Final set:

8. 2, 3, 4, 10, 11, 12, 20
9. Range = 20 – 2 = 18 (matches given).
10. Median = 10 (matches given).

Answer: 20

Common Mistakes

1. Mistake: Assuming the mean = median.
2. Why it happens: Students confuse symmetric distributions with arbitrary data sets.

3. Correct approach: The mean and median are only equal in symmetric distributions (e.g., {5, 10, 15}). In skewed sets (e.g., {5, 10, 100}), they differ.

4. Mistake: Forgetting "distinct" integers.

5. Why it happens: Questions often specify "distinct" but students overlook it.

6. Correct approach: If the set must have unique values, you cannot repeat numbers (e.g., {2, 2, 10} is invalid if distinctness is required).

7. Mistake: Misapplying the median formula for even/odd counts.

8. Why it happens: Students forget whether to average the two middle numbers (even) or pick the middle one (odd).

9. Correct approach:

   
   
   • Odd # of terms: Median = middle term.
   • Even # of terms: Median = average of two middle terms.

10. Mistake: Ignoring constraints on values (e.g., positive integers, even/odd).

11. Why it happens: Students solve for general cases but miss specific restrictions.

12. Correct approach: Always check if the question specifies integers, positive numbers, etc.

13. Mistake: Overcomplicating weighted averages.

14. Why it happens: Students try to memorize complex formulas instead of using the total sum approach.
15. Correct approach: For weighted averages, multiply each value by its weight, sum them, then divide by the total weight.

GMAT Traps & Timing

Traps:

1. Hidden "Distinct" Constraint
2. Trap: The question says "a set of integers" but doesn’t specify distinctness.

3. Avoid: Assume distinct unless stated otherwise (GMAT often tests this).

4. Median vs. Mean Confusion

5. Trap: A question asks for the mean, but you calculate the median (or vice versa).

6. Avoid: Circle the word (mean/median) in the question to stay focused.

7. Range Misinterpretation

8. Trap: Students think range tells them about the spread of most values (it only tells you about the extremes).

9. Avoid: Range = Max – Min, nothing more.

10. Data Sufficiency Over-Solving

11. Trap: In DS, students calculate the exact answer when they only need to check sufficiency.
12. Avoid: Ask: "Do I have enough info to find the answer?" Not: "What is the answer?"

Timing:

• Problem Solving (PS): 1.5–2 minutes per question.
• Data Sufficiency (DS): 1–1.5 minutes per question.
• If stuck: Guess and move on—never spend >2.5 minutes on a single question.

Quick Practice

1. Question:
   A set of 5 distinct positive integers has a median of 8 and a range of 10. What is the smallest possible value in the set?
   Answer: 1
   Explanation: To minimize the smallest value, set it to 1. Then the largest value = 1 + 10 = 11. The set could be {1, 2, 8, 9, 11}.

2. Question:
   If the average of 5 numbers is 12, and the average of 3 of them is 10, what is the average of the remaining 2 numbers?
   Answer: 15
   Explanation: Total sum of 5 numbers = 5 × 12 = 60. Sum of 3 numbers = 3 × 10 = 30. Sum of remaining 2 = 60 – 30 = 30. Average = 30 / 2 = 15.

Last-Minute Cram Sheet

1. Mean = Total / Number of Terms → Always calculate the total sum first.
2. Median = Middle value (odd) or average of two middle values (even).
3. Range = Max – Min → Only tells you about extremes, not distribution.
4. For "largest possible value" questions, minimize all other values.
5. For "smallest possible value" questions, maximize all other values.
6. Weighted Average = (Sum of (Value × Weight)) / (Sum of Weights).
7. In DS, a statement is sufficient if it gives a single value (not a range).
8. Distinct integers = no repeats (unless stated otherwise).
9. Median is resistant to outliers; mean is not.
10. ⚠️ Trap: "Average of 10 and 20" is not the same as "median of 10 and 20" (mean = 15, median = 15, but in larger sets, they differ).

Final Tip: On test day, write down the given values and constraints first—this prevents 90% of careless errors. Good luck! ?