Statistics, Probability, and Distributions

What GRE Data Analysis Actually Tests

Data Analysis blends descriptive statistics, probability, counting, and reading data displays. The GRE keeps the statistics conceptual — you interpret the standard deviation, you do not compute it by hand — and off-syllabus tools (hypothesis testing, confidence intervals, regression) never appear.

Descriptive Statistics

Measure	Definition	GRE nuance
Mean	sum ÷ count	Moves toward outliers
Median	middle value when ordered	Resistant to outliers
Mode	most frequent value	A set can have none or several
Range	max − min	Crude measure of spread
Standard deviation	typical distance from the mean	Bigger spread ⇒ bigger SD

Mean questions are often algebra in disguise. If five numbers have a mean of 8, their sum is $5 \times 8 = 40$; a missing value is then $40$ minus the known values.

Standard deviation without arithmetic. The GRE rarely wants a numeric SD. Instead it asks which set has the larger SD — the answer is whichever data is more spread out around its mean. $\{10, 20, 30, 40, 50\}$ has a much larger SD than $\{28, 29, 30, 31, 32\}$, even though both have mean 30.

Probability

$P(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}, \qquad 0 \le P \le 1.$

Situation	Rule
Independent events (both happen)	multiply: $P(A \text{ and } B) = P(A)\,P(B)$
Mutually exclusive (either happens)	add: $P(A \text{ or } B) = P(A) + P(B)$
Complement	$P(\text{not } A) = 1 - P(A)$
Without replacement	the second probability uses the reduced pool

Without-replacement trap: drawing 2 red from a bag of 3 red and 5 blue is $\frac{3}{8} \times \frac{2}{7} = \frac{6}{56} = \frac{3}{28}$ — the second fraction has both a smaller numerator (2 red left) and a smaller denominator (7 total left).

The Normal Distribution and the Empirical Rule

For a normal (bell-shaped) distribution with mean $\mu$ and standard deviation $\sigma$:

• about 68% of data lies within $1\sigma$ of the mean ($\mu \pm \sigma$),
• about 95% within $2\sigma$,
• about 99.7% within $3\sigma$.

So scores that are normally distributed with mean 500 and $\sigma = 100$ have roughly 68% of values between 400 and 600.

Counting: Permutations vs. Combinations

• Order matters ⇒ permutation: $P(n, k) = \dfrac{n!}{(n - k)!}$.
• Order doesn't matter ⇒ combination: $C(n, k) = \dfrac{n!}{k!\,(n - k)!}$.

A committee (a set of people) is a combination; a ranked finish (1st, 2nd, 3rd) is a permutation. Choosing 3 people from 8 for a committee is $C(8, 3) = 56$, while awarding gold/silver/bronze to 3 of 8 is $P(8, 3) = 336$.

Data Interpretation Sets

Two or three consecutive questions share one table or graph. Read the title, axis labels, and units first — the GRE hides traps in units (thousands vs. millions) and in "percent of" versus "percent change." Compute only what a given question asks.