Statistics and Data Analysis

What This Module Covers

This module covers the core statistical concepts the GRE tests: mean, median, mode, and range; standard deviation and variance; weighted averages; quartiles and the interquartile range; and the normal distribution. You will not need to run complex calculations — the GRE favors reasoning and trap-avoidance over arithmetic.

Why It Matters on the GRE

Statistics questions show up in both Problem Solving and Quantitative Comparison formats, and the GRE loves to disguise them as simple "find the average" prompts that hide a trap. The most common errors come not from hard math but from mixing up mean and median, forgetting that standard deviation ignores shifts, and averaging two averages when group sizes differ. Knowing the conceptual rules cold is worth more here than computation speed.

Core Concepts

Mean, Median, Mode, and Range

The mean is the arithmetic average: add all values, then divide by the count. The GRE's favorite mean question is: "If one value changes, how does the mean change?" The shortcut is to think about the total sum — if a value increases by 10 and there are 5 values, the mean increases by 2.

The median is the middle value when the list is sorted in order. For an odd number of values, it is the exact middle. For an even number of values, it is the average of the two middle values. The critical insight is that the median is resistant to outliers — you can drag the highest value in a list to infinity and the median won't budge.

The mode is the value that appears most often. A data set can have no mode (all values appear once), one mode, or multiple modes. The GRE rarely tests mode in isolation, but it appears in combination questions.

The range is simply max − min. It is sensitive to outliers because it only looks at the two extreme values.

Must Know: In a symmetric (bell-shaped) distribution, mean = median. In a right-skewed distribution (long tail to the right), the mean is pulled higher than the median. In a left-skewed distribution, the mean is pulled lower. When you see "skewed," ask yourself which direction the tail points — the mean follows the tail.

Standard Deviation (Conceptual)

Standard deviation measures how spread out values are around the mean. A small SD means values cluster tightly; a large SD means they are spread wide. The GRE will never ask you to calculate SD from scratch — it tests whether you understand how SD behaves when the data changes.

Must Know: Adding or subtracting a constant to every value in a data set shifts the mean by that constant but does not change the standard deviation. The spread between values stays identical. Multiplying every value by a constant scales both the mean and the SD by that constant.

Example — Effect of a constant on SD:

1. Data Set A: {2, 4, 6, 8, 10}. The mean is 6.
2. Add 5 to every value → Data Set B: {7, 9, 11, 13, 15}. The mean is now 11.
3. Ask: did the SD change? No. The gap between each value and the new mean is identical to the gap from the old mean. Spread is unchanged.
4. Now multiply every value in Set A by 3 → {6, 12, 18, 24, 30}. The SD is also multiplied by 3.

Must Know: Variance = (standard deviation)². If two data sets contain the same values in any order, they have the same SD. A set where all values are equal has SD = 0.

Weighted Averages

A plain average treats every data point equally. A weighted average accounts for the fact that some groups are larger than others. The formula is:

Weighted Mean = (n₁ × avg₁ + n₂ × avg₂) / (n₁ + n₂)

The most dangerous trap is simply averaging the two group averages when the groups have different sizes. If Class A has 10 students averaging 80 and Class B has 30 students averaging 70, the combined average is NOT (80 + 70) / 2 = 75. It is (10 × 80 + 30 × 70) / 40 = (800 + 2100) / 40 = 72.5 — pulled toward the larger group.

Must Know: The weighted average always falls between the two individual averages. It is always closer to the average of the larger group. Use this as a quick sanity check — if your answer is outside the range of the two averages, you made an error.

Example — Weighted average:

1. Group 1: 20 employees, average salary $50,000.
2. Group 2: 80 employees, average salary $30,000.
3. Weighted mean = (20 × 50,000 + 80 × 30,000) / (20 + 80).
4. = (1,000,000 + 2,400,000) / 100 = $34,000.
5. Check: $34,000 is between $30,000 and $50,000, and much closer to $30,000 because that group is four times larger. Correct.

Quartiles, Percentiles, and IQR

Percentiles tell you what percentage of values fall at or below a given point. The quartiles divide a sorted data set into four equal parts:

• Q1 (1st quartile) = 25th percentile — 25% of values fall below this.
• Q2 (2nd quartile) = median = 50th percentile.
• Q3 (3rd quartile) = 75th percentile — 75% of values fall below this.

The Interquartile Range (IQR) = Q3 − Q1. It captures the spread of the middle 50% of the data and is resistant to outliers because it ignores the top and bottom 25%.

Must Know: A value is typically considered an outlier if it falls more than 1.5 × IQR below Q1 or more than 1.5 × IQR above Q3. The GRE tests IQR primarily in data interpretation questions involving box plots.

Normal Distributions (Conceptual)

A normal distribution is the classic bell curve — symmetric around the mean, with most values clustered near the center and fewer values in the tails. The GRE tests three key benchmarks:

• ~68% of data falls within 1 SD of the mean (between mean − 1SD and mean + 1SD).
• ~95% of data falls within 2 SD of the mean.
• ~99.7% of data falls within 3 SD of the mean.

Must Know: If the mean shifts, the entire bell curve slides left or right but keeps the same shape and spread. If the SD increases, the curve flattens and widens. A larger SD = more spread = a flatter bell. A smaller SD = tighter spread = a taller, narrower bell.

Common Traps

• Mean vs. Median Confusion: For symmetric data, they are equal. For skewed data, they diverge. When a question says "average," confirm whether mean or median is intended.
• Adding a Constant Changes SD: It does not. Only multiplication scales the SD. This is one of the GRE's favorite QC traps — Quantity A shifts the data by adding 10, Quantity B keeps the original set, and the answer is "equal" because SD is unchanged.
• Averaging Two Averages with Unequal Group Sizes: Never add two averages and divide by 2 unless both groups are the same size. Always weight by group size.
• Forgetting That Median Ignores Outliers: If a question tells you a single extreme value changes, the median may be completely unaffected while the mean shifts significantly.
• Misidentifying the Direction of Skew: The skew direction is the direction of the tail, not the hump. A right-skewed distribution has its hump on the left and its tail on the right.

GRE Strategy

• When a QC question changes every value by adding or subtracting a constant, the SD of both quantities is equal — this is almost always the answer.
• For weighted average questions, check group sizes first. If sizes are equal, simple average works. If unequal, always weight.
• In data interpretation questions with box plots, IQR = Q3 − Q1. The box itself visually represents the IQR.
• For normal distribution questions, memorize 68/95/99.7 — the GRE uses these benchmarks directly.
• When mean and median are both given, the gap between them tells you about skew. Mean > Median = right skew. Mean < Median = left skew.

Worked Example

Question:

Data Set A: {2, 4, 6, 8, 10} Data Set B: {4, 6, 8, 10, 12}

Quantity A: Standard deviation of Data Set A Quantity B: Standard deviation of Data Set B

(A) Quantity A is greater. (B) Quantity B is greater. (C) The two quantities are equal. (D) The relationship cannot be determined from the information given.

Solution:

Step 1 — Identify what changed between the two sets. Data Set B is Data Set A with 2 added to every value: {2+2, 4+2, 6+2, 8+2, 10+2} = {4, 6, 8, 10, 12}. Every value shifted up by the same constant.

Step 2 — Apply the SD constant rule. Adding a constant to every value in a data set does not change the standard deviation. The distances between each value and the mean stay identical. The spread is completely preserved.

Step 3 — Verify with the mean. Mean of Set A = (2+4+6+8+10)/5 = 30/5 = 6. Mean of Set B = (4+6+8+10+12)/5 = 40/5 = 8. The mean shifted by exactly 2 — confirming a constant was added. The deviations from the mean in both sets are {−4, −2, 0, +2, +4}, which are identical.

Step 4 — State the answer. Since the deviations from the mean are identical in both sets, the standard deviations are equal. The answer is (C).

Why the wrong answers are tempting:

• (A) or (B): It is natural to assume that larger numbers produce larger spread. They do not — what matters is how far apart the values are from each other, not their absolute size.
• (D): There is no ambiguity here. The relationship is fully determined by the constant-shift rule.

Takeaway: Any time a GRE QC question shifts all values in a set by adding or subtracting a constant, flag it immediately. The SD is equal. The only way SD changes is if the values are multiplied or divided by a constant, or if the actual spread between values changes.