Lesson 5: GRE Quantitative Reasoning - Data Analysis, Statistics & Probability

Master data interpretation, statistical measures, probability concepts, and combinatorics for GRE success

Lesson 5: GRE Quantitative Reasoning - Data Analysis, Statistics & Probability 📊

Introduction

Welcome to Lesson 5! You've conquered arithmetic, algebra, geometry, and percentages—now it's time to tackle one of the most practical and frequently tested areas on the GRE: data analysis, statistics, and probability. These topics appear not only in standalone questions but also in data interpretation sets where you'll analyze graphs, charts, and tables.

🎯 What You'll Master:

Statistical measures (mean, median, mode, range, standard deviation)
Data interpretation from graphs and tables
Probability fundamentals and conditional probability
Combinatorics (permutations and combinations)
Set theory and Venn diagrams

💡 Why This Matters: The GRE tests your ability to make informed decisions from data—a critical skill in graduate school and professional life. These questions assess logical reasoning as much as mathematical computation.

Core Concept 1: Measures of Central Tendency and Spread 📏

Mean, Median, and Mode

These three measures describe the "center" of a data set, but they tell different stories:

Mean (Average): Sum of all values divided by the count

Formula: μ = (Σx) / n
💡 Tip: The mean is sensitive to outliers. One extremely high or low value can skew it significantly.

Median: The middle value when data is arranged in order

For odd n: the middle number
For even n: average of the two middle numbers
💡 Tip: The median is resistant to outliers, making it better for skewed distributions.

Mode: The most frequently occurring value

A dataset can have no mode, one mode (unimodal), or multiple modes (bimodal, multimodal)

🤔 Did you know? Income data is typically reported using median rather than mean because a few billionaires would make the average misleadingly high!

Range and Standard Deviation

Range: The difference between the maximum and minimum values

Range = Max - Min
Simple but doesn't tell you about distribution

Standard Deviation (σ): Measures how spread out data is from the mean

Low standard deviation: data clustered near the mean
High standard deviation: data widely scattered
Formula: σ = √[Σ(x - μ)² / n]

💡 GRE Shortcut: You won't need to calculate standard deviation from scratch, but you must understand what it represents!

Visualization of Standard Deviation:

Low SD:              High SD:
    *                 *
   ***               * *
  *****             *   *
 *******           *     *
=========         =========
  mean              mean

Quartiles and Interquartile Range (IQR)

Quartiles divide ordered data into four equal parts:

Q1 (25th percentile): 25% of data falls below this
Q2 (50th percentile): The median
Q3 (75th percentile): 75% of data falls below this

Interquartile Range (IQR): Q3 - Q1

Represents the middle 50% of the data
Used to identify outliers: values below Q1 - 1.5(IQR) or above Q3 + 1.5(IQR)

 Box Plot Representation:

     |----[====|====]----|
   Min    Q1   Q2  Q3   Max
           (median)

Core Concept 2: Probability Fundamentals 🎲

Basic Probability Rules

Probability measures the likelihood of an event occurring:

P(Event) = (Number of favorable outcomes) / (Total possible outcomes)
Range: 0 ≤ P(E) ≤ 1 (or 0% to 100%)
P(certain event) = 1
P(impossible event) = 0

Complementary Events

The complement of event A (written as A' or Ā) is "A does not occur"

P(A) + P(A') = 1
Therefore: P(A') = 1 - P(A)

💡 GRE Strategy: Sometimes it's easier to calculate the probability that something DOESN'T happen!

Independent vs. Dependent Events

Independent Events: One event doesn't affect the other

P(A and B) = P(A) × P(B)
Example: Flipping a coin twice—the first flip doesn't influence the second

Dependent Events: One event affects the probability of the other

P(A and B) = P(A) × P(B|A)
P(B|A) means "probability of B given that A occurred"

Mutually Exclusive vs. Non-Mutually Exclusive Events

Mutually Exclusive: Events cannot occur simultaneously

P(A or B) = P(A) + P(B)
Example: Rolling a 3 OR a 5 on a single die

Non-Mutually Exclusive: Events can occur together

P(A or B) = P(A) + P(B) - P(A and B)
Example: Drawing a King OR a Heart from a deck (King of Hearts is both!)

 Venn Diagram - Non-Mutually Exclusive:

     +-------+-------+
     |   A   | Both  |   B
     |  only | A & B | only
     +-------+-------+

 P(A or B) = P(A only) + P(Both) + P(B only)
           = P(A) + P(B) - P(A and B)

🧠 Mnemonic: "OR means ADD, AND means MULTIPLY" (for independent events)

Core Concept 3: Combinatorics - Counting Principles 🔢

The Fundamental Counting Principle

If there are m ways to do one thing and n ways to do another, there are m × n ways to do both.

Example: A restaurant offers 4 appetizers, 5 main courses, and 3 desserts. How many different three-course meals are possible?

Answer: 4 × 5 × 3 = 60 different meals

Permutations vs. Combinations

This is where students often get confused! The key difference:

Permutations: Order MATTERS

Formula: P(n,r) = n! / (n-r)!
"n" objects, choosing "r", arrangement matters
Example: First, second, and third place in a race of 8 runners

Combinations: Order DOESN'T matter

Formula: C(n,r) = n! / [r!(n-r)!]
"n" objects, choosing "r", arrangement doesn't matter
Example: Choosing 3 toppings from 8 options for a pizza

💡 How to Remember: "Combination" sounds like "combination lock," but actually, a combination lock is a PERMUTATION lock because order matters! (This irony helps you remember.)

 Decision Tree:

 Does ORDER matter?
        |
   +----+----+
  YES       NO
   |         |
PERMUTATION COMBINATION
  P(n,r)    C(n,r)

Factorial Notation (!)

n! = n × (n-1) × (n-2) × ... × 2 × 1

5! = 5 × 4 × 3 × 2 × 1 = 120
0! = 1 (by definition)

💡 GRE Shortcut: Simplify before calculating!

Example: 10! / 8! = (10 × 9 × 8!) / 8! = 10 × 9 = 90

Arrangements with Restrictions

Arranging n distinct objects: n! ways Arranging n objects where some are identical:

Formula: n! / (n₁! × n₂! × ... × nₖ!)
Where n₁, n₂, etc. are the counts of identical items

Example: How many ways can you arrange the letters in MISSISSIPPI?

11 letters total: 1M, 4I, 4S, 2P
Answer: 11! / (1! × 4! × 4! × 2!) = 34,650

Core Concept 4: Data Interpretation 📈

The GRE presents data in various formats. Let's examine common types:

Bar Charts and Histograms

Bar Chart: Compares discrete categories Histogram: Shows frequency distribution of continuous data

 Sales by Region (Bar Chart):

 50 |     ████
 40 | ████ ████
 30 | ████ ████ ████
 20 | ████ ████ ████ ████
 10 | ████ ████ ████ ████
  0 +----+----+----+----+
     East West North South

Line Graphs

Show trends over time or continuous variables

Look for: increasing/decreasing trends, peaks, valleys, intersections

Pie Charts

Show parts of a whole (percentages or proportions)

All sectors must sum to 100%
💡 Tip: Convert percentages to actual values using the total given

Two-Way Tables (Contingency Tables)

Show relationships between two categorical variables:

+----------+--------+-------+-------+
|          | Male   | Female| Total |
+----------+--------+-------+-------+
| Freshman |   45   |   38  |   83  |
| Sophomore|   52   |   47  |   99  |
| Junior   |   40   |   45  |   85  |
| Senior   |   38   |   42  |   80  |
+----------+--------+-------+-------+
| Total    |  175   |  172  |  347  |
+----------+--------+-------+-------+

💡 GRE Strategy: Always check if the question asks for:

Actual numbers vs. percentages
Part vs. whole
Change vs. percent change

Worked Examples 🔍

Example 1: Statistical Analysis

Problem: A class of 15 students scored the following on a test: 85, 88, 90, 92, 78, 85, 95, 88, 85, 90, 82, 85, 88, 90, 75

Find: (a) mean, (b) median, (c) mode, (d) range

Solution:

(a) Mean: Sum = 85+88+90+92+78+85+95+88+85+90+82+85+88+90+75 = 1,296 Mean = 1,296 / 15 = 86.4

(b) Median: First, arrange in order: 75, 78, 82, 85, 85, 85, 85, 88, 88, 88, 90, 90, 90, 92, 95 Middle value (8th position) = 88

(d) Range: Range = 95 - 75 = 20

💡 Notice: The mean (86.4) is slightly lower than the median (88) because of the lower outliers (75, 78), indicating a slight left skew.

Example 2: Probability with Independence

Problem: A coin is flipped three times. What is the probability of getting exactly two heads?

Solution:

Method 1: List all outcomes

Possible outcomes (8 total):
HHH, HHT, HTH, HTT, THH, THT, TTH, TTT

Exactly 2 heads: HHT, HTH, THH (3 outcomes)

P(exactly 2 heads) = 3/8 = 0.375 or 37.5%

Method 2: Combinations and probability

Number of ways to choose which 2 flips are heads: C(3,2) = 3
Probability of any specific sequence (e.g., HHT): (1/2)³ = 1/8
P(exactly 2 heads) = 3 × (1/8) = 3/8

🧠 Key Insight: When dealing with multiple independent trials, use combinations to count favorable arrangements, then multiply by the probability of each arrangement.

Example 3: Permutations vs. Combinations

Problem: A committee of 4 people is to be formed from a group of 10 people. (a) How many different committees can be formed? (b) If the committee needs a president, vice-president, secretary, and treasurer (distinct roles), how many ways can it be formed?

Solution:

(a) Order doesn't matter (just selecting 4 people) → Combination C(10,4) = 10! / (4! × 6!) = (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1) = 5,040 / 24 = 210 committees

(b) Order matters (specific roles) → Permutation P(10,4) = 10! / 6! = 10 × 9 × 8 × 7 = 5,040 arrangements

💡 Notice: There are many more permutations than combinations because each group of 4 people can be arranged in 4! = 24 different ways for the roles.

Example 4: Data Interpretation

Problem: The table below shows smartphone sales (in thousands) for three brands over three quarters:

+--------+--------+--------+--------+
|        |   Q1   |   Q2   |   Q3   |
+--------+--------+--------+--------+
| Brand A|   120  |   135  |   150  |
| Brand B|   140  |   130  |   125  |
| Brand C|    95  |   110  |   115  |
+--------+--------+--------+--------+

(a) What was the percent increase in Brand A's sales from Q1 to Q3? (b) Which brand had the highest total sales across all three quarters? (c) In Q2, Brand A's sales represented what percent of total sales (all brands)?

Solution:

(a) Percent increase for Brand A: Increase = 150 - 120 = 30 Percent increase = (30/120) × 100% = 25%

(b) Total sales by brand:

Brand A: 120 + 135 + 150 = 405
Brand B: 140 + 130 + 125 = 395
Brand C: 95 + 110 + 115 = 320 Brand A had the highest total: 405 thousand

⚠️ Common Mistake: Don't confuse "percent increase" with "increase." The increase is 30, but the percent increase is 25%.

Common Mistakes to Avoid ⚠️

1. Confusing Mean and Median

❌ Wrong: Assuming the mean is always the "middle" value ✅ Right: Mean is the average; median is the middle value when ordered

Use median for skewed data or when outliers are present

2. Permutation/Combination Confusion

❌ Wrong: Using C(n,r) when order matters ✅ Right: Ask yourself: "If I swap two items, is it a different outcome?"

YES → Permutation | NO → Combination

3. Adding Probabilities Incorrectly

❌ Wrong: P(A or B) = P(A) + P(B) for all events ✅ Right: Only for mutually exclusive events! Otherwise: P(A or B) = P(A) + P(B) - P(A and B)

4. Forgetting the Complement Rule

❌ Wrong: Calculating complex "at least one" scenarios directly ✅ Right: P(at least one) = 1 - P(none)

This is especially useful for "at least one" probability questions!

5. Misreading Data Interpretation Graphs

⚠️ Watch out for:

Scale breaks or non-zero starting points (can exaggerate differences)
Percentages vs. actual numbers
Per capita vs. total figures
Whether axes are labeled correctly

6. Standard Deviation Misconceptions

❌ Wrong: Thinking standard deviation measures the range ✅ Right: It measures spread around the mean

A small standard deviation means data is clustered near the mean
Two datasets can have the same mean but very different standard deviations

7. Factorial Calculation Inefficiency

❌ Wrong: Computing 100! then dividing by 98! ✅ Right: Cancel first: 100!/98! = 100 × 99 = 9,900

Key Takeaways 🎯

✅ Mean is affected by outliers; median is resistant to them ✅ Standard deviation measures spread from the mean ✅ Probability ranges from 0 to 1; complement rule: P(A') = 1 - P(A) ✅ For independent events: P(A and B) = P(A) × P(B) ✅ For mutually exclusive events: P(A or B) = P(A) + P(B) ✅ Permutations (order matters): P(n,r) = n!/(n-r)! ✅ Combinations (order doesn't matter): C(n,r) = n!/[r!(n-r)!] ✅ Always read data interpretation questions carefully—check units, scales, and what's being asked ✅ Use the complement rule for "at least one" probability questions

Quick Reference Card 📋

+================================+
| STATISTICS FORMULAS            |
+================================+
| Mean: μ = Σx / n              |
| Range: Max - Min              |
| IQR: Q3 - Q1                  |
+================================+
| PROBABILITY RULES              |
+================================+
| Complement: P(A') = 1 - P(A)  |
| Independent AND:              |
|   P(A and B) = P(A) × P(B)    |
| Mutually Exclusive OR:        |
|   P(A or B) = P(A) + P(B)     |
| General OR:                   |
|   P(A or B) = P(A) + P(B)     |
|            - P(A and B)       |
+================================+
| COUNTING PRINCIPLES            |
+================================+
| Permutation: n!/(n-r)!        |
| Combination: n!/[r!(n-r)!]    |
| Fundamental Principle:        |
|   Multiply number of choices  |
+================================+
| QUICK CHECKS                   |
+================================+
| Order matters? → Permutation  |
| Order doesn't? → Combination  |
| At least one? → Use complement|
| Can both happen? → Not mut.ex.|
+================================+

📚 Further Study

Khan Academy - Statistics and Probability: https://www.khanacademy.org/math/statistics-probability
- Comprehensive video lessons and practice problems
GRE Official Guide - Data Analysis: https://www.ets.org/gre/test-takers/general-test/prepare.html
- Official ETS practice questions and strategies
Seeing Theory - Visual Statistics: https://seeing-theory.brown.edu/
- Interactive visualizations of statistical concepts

🎓 You're now equipped to tackle GRE data analysis questions! In the next lessons, we'll build on these skills with more advanced quantitative reasoning, word problems, and integrated concepts. Remember: the GRE tests your reasoning ability, not just calculation—always think about what the question is really asking!

💪 Practice Tip: When reviewing data interpretation questions, spend time understanding WHY each answer choice is right or wrong, not just whether you got it correct. This builds pattern recognition for test day!

📝

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