GRE Prep: Data Analysis (Mean, Median, Mode, Standard Deviation, Probability, Counting, Combinatorics)

GRE – Data Analysis (Mean, Median, Mode, Standard Deviation, Probability, Counting, Combinatorics)

GRE Study Guide – Data Analysis (Mean, Median, Mode, Standard Deviation, Probability, Counting, Combinatorics)

What This Is

Data‑analysis questions on the GRE test your ability to interpret, manipulate, and draw conclusions from numerical information. You’ll be asked to compute measures of central tendency (mean, median, mode), variability (standard deviation), work with basic probability, and use counting/combinatorial techniques (permutations, combinations). For example, a typical Quantitative Comparison (QC) item might give the average score of a group of test‑takers and ask you to compare it with the median score of the same group.

Key Terms & Rules

• Mean (Arithmetic Average): Sum of all observations ÷ number of observations.
• Median: The middle value when data are ordered; if even n, median = average of the two middle numbers.
• Mode: The most frequently occurring value(s); a set can have zero, one, or multiple modes.
• Standard Deviation (σ): Measures spread; for a sample, (s = \sqrt{\frac{\sum (x_i-\bar{x})^2}{n-1}}).
• Probability (P): Favorable outcomes ÷ total equally‑likely outcomes; (0 \le P \le 1).
• Permutation (nPr): Number of ordered arrangements of r objects from n; formula (nPr = \frac{n!}{(n-r)!}).
• Combination (nCr): Number of unordered selections of r objects from n; formula (nCr = \frac{n!}{r!(n-r)!}).
• Counting Principle: If one event can occur in m ways and a second independent event in n ways, the pair can occur in m × n ways.
• Data Sufficiency (DS) Answer Choices: A – B alone sufficient; B – A alone sufficient; C – both together sufficient; D – each alone insufficient; E – insufficient even together.
• Quantitative Comparison (QC) Symbols: “>”, “<”, “=”, or “cannot be determined”.
• Common Trap – “Off‑by‑One” in Counting: Forgetting that permutations of identical items reduce the total count (divide by duplicate factorials).

Step‑by‑Step / Process Flow

1. Read the stem carefully – note whether the question asks for a specific value (mean, median, etc.) or a comparison.
2. Identify the data type – is the information given as a list, a frequency table, or a description? Write the numbers in order if needed.
3. Choose the quickest formula – for mean use Σx / n; for median sort and pick the middle; for standard deviation decide if a full calculation is required or if the “range rule of thumb” (σ ≈ range/4) suffices.
4. Apply counting rules – when the problem involves selections, decide between permutations (order matters) and combinations (order irrelevant). Use factorial shortcuts (e.g., 5! = 120).
5. Check the answer choice – eliminate any options that violate basic constraints (e.g., probabilities > 1, negative standard deviations). Confirm with a quick back‑of‑the‑envelope estimate to avoid arithmetic slip‑ups.

Common Mistakes

• Mistake: Treating a median problem as a mean problem and adding all numbers together.
  Correction: Sort the data first; the median depends only on position, not on the sum.

• Mistake: Using (n!) for combinations instead of the (nCr) formula, inflating the count.
  Correction: Remember that order does not matter for combinations; divide by (r!(n-r)!).

• Mistake: Assuming independence in probability when events are actually mutually exclusive.
  Correction: Verify whether events can occur together; if not, add probabilities only for mutually exclusive events.

• Mistake: Forgetting the “‑1” denominator in the sample standard deviation formula, leading to a slightly larger σ.
  Correction: Use (n-1) for sample SD; the GRE rarely requires the exact decimal, but the distinction can affect answer‑choice elimination.

• Mistake: In Data‑Sufficiency, treating “at least one” statements as “exactly one.”
  Correction: Pay close attention to wording; “at least one” leaves open the possibility of more, which may affect sufficiency.

Exam Insights

1. Most‑tested concept: Comparing mean vs. median; the GRE loves to ask which is larger when the distribution is skewed.
2. Typical distractor: Providing the mode as an answer choice when the question asks for the median—remember mode is about frequency, not position.
3. Probability pitfall: Offering a probability > 1 or a negative probability as a “wrong” answer; these are easy eliminations.
4. Counting nuance: Questions that involve “arrangements of letters with repeats” often require dividing by duplicate factorials (e.g., arranging the word LEVEL → 5! / 2! = 60).

Quick Check Questions

1. QC: The mean of five numbers is 12. If one of the numbers is 18, what is the sum of the remaining four numbers?

2. Answer: 42. (Explanation: Total sum = 5 × 12 = 60; subtract 18 → 60 − 18 = 42.)

3. DS: A bag contains red, blue, and green marbles. Statement 1: There are twice as many red marbles as blue marbles. Statement 2: The total number of marbles is 12. Is the number of green marbles uniquely determined?

4. Answer: B – Statement 2 alone is sufficient. (Explanation: Knowing total = 12 and red = 2 × blue lets you solve for green = 12 − (blue + 2 × blue) = 12 − 3 × blue; blue must be an integer ≤ 3, but only one integer makes the total integer for green.)

5. Multiple‑Choice (Counting): How many ways can a committee of 3 be chosen from 7 people if Alice must be on the committee?

6. Answer: 15. (Explanation: Choose Alice + any 2 of the remaining 6 → 6C2 = 15.)

Last‑Minute Cram Sheet (10 One‑Liners)

1. Mean = Σx ÷ n; Median = middle value (average the two middles if n is even).
2. σ (sample) = √[ Σ(x − (\bar{x}))² ÷ (n − 1) ]; use range ÷ 4 as a quick estimate.
3. nPr = n! / (n − r)! (order matters); nCr = n! / [r!(n − r)!] (order irrelevant).
4. Probability = favorable outcomes ÷ total outcomes; never exceed 1.
5. ⚠️ Counting trap: Forgetting to divide by duplicate factorials when objects repeat.
6. In Data‑Sufficiency, treat each statement independently before combining; answer choice “D” means each alone is insufficient.
7. QC shortcut: If both sides simplify to the same expression, answer is “=”.
8. For “at least one” vs. “exactly one,” remember “at least” includes the possibility of more.
9. When a problem gives a frequency table, compute the mean as Σ(value × frequency) ÷ total frequency.
10. ⚠️ Probability trap: Adding probabilities of overlapping events without subtracting the intersection (use inclusion‑exclusion).

Good luck—master these tools, and the data‑analysis portion will become a quick‑fire strength on test day!