Descriptive Statistics

Concise answer

Keep the three jobs separate: center (mean, median, mode), location (quartiles and percentiles), and spread (range, interquartile range, standard deviation) — and on normal-distribution questions, lean on symmetry and the 68–95–99.7 benchmarks instead of computation.

Definitions

Mean: The arithmetic average: the sum of the values divided by how many there are.
Median: The middle value of the ordered data; with an even count, the average of the two middle values.
Mode: The value that occurs most frequently; a data set can have more than one mode or none.
Range: The largest value minus the smallest value.
Standard deviation: A measure of how far data values typically lie from the mean; larger spread means larger standard deviation.
Percentile: A location marker: a value at the $p$th percentile is at the point below which about $p$ percent of the ordered data fall.
Quartiles: The 25th, 50th, and 75th percentiles ($Q_1$, the median, and $Q_3$), which divide ordered data into quarters; $Q_3 - Q_1$ is the interquartile range.

Intuition

The mean is the balance point of the data, so it chases outliers; the median only cares about position in the ordered list, so a few extreme values drag the mean without moving the median.

Standard deviation answers one question — how far do values typically sit from the mean? Data bunched near the mean has a small standard deviation no matter how large the values themselves are, and shifting every value by the same amount changes the mean but not the spread.

Concept walkthrough

Measures of center summarize a data set with one number. The mean is the sum divided by the count; the median is the middle of the ordered data (average the two middle values when the count is even); the mode is the most frequent value. In symmetric data the mean and median agree, while skewed data pulls the mean toward the long tail — a favorite GRE comparison.

Measures of location and spread describe where values sit and how widely they scatter. Quartiles divide the ordered data into quarters: $Q_1$ is the 25th percentile, the median is the 50th, $Q_3$ is the 75th, and the interquartile range $Q_3 - Q_1$ spans the middle half. The range is simply maximum minus minimum. The standard deviation measures typical distance from the mean: it is never negative, it is zero only when every value is identical, and adding the same constant to every value leaves it unchanged. GRE questions almost always test this comparative understanding rather than the computing formula.

The GRE uses the normal distribution as a stock of fixed facts rather than a computation topic. A normal distribution is symmetric about its mean, so the mean and median coincide, and the Empirical Rule pins down the proportions: about 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three. A z-score converts a value into 'how many standard deviations from the mean,' which is how GRE problems ask you to compare positions in two different distributions. ETS lists these descriptive-statistics and normal-distribution topics under the data analysis content of the Quantitative Reasoning measure.

After this page, you should be able to

Compute and compare the mean, median, and mode of a small data set.
Locate quartiles and interpret percentiles as positions in ordered data, not scores.
Compare standard deviations of data sets by their spread around the mean without calculating them.
Apply normal-distribution symmetry and the 68–95–99.7 benchmarks to GRE-style questions.

Formulas and assumptions

Arithmetic mean

\bar{x} = \dfrac{x_1 + x_2 + \cdots + x_n}{n}

Variables

x_1, ..., x_n: the data values
n: how many values there are
x-bar: the mean

Assumptions

n is not zero.
Every value is included exactly once; with a frequency table, weight each value by its frequency.

Range and interquartile range

\text{range} = x_{\max} - x_{\min} \qquad \text{IQR} = Q_3 - Q_1

Variables

x_max, x_min: the largest and smallest data values
Q_1, Q_3: the first and third quartiles (25th and 75th percentiles)

Assumptions

The data are ordered before quartiles are located.
The range uses only the two extreme values, so it says nothing about how the middle of the data is spread.

Worked example

Center and spread of a five-value data set

For the data set $\{4,\ 7,\ 7,\ 9,\ 13\}$ , find the mean, median, mode, and range.

1The data are already ordered, so read off the positions directly.
2Mean: the sum is $4 + 7 + 7 + 9 + 13 = 40$ , and $40 \div 5 = 8$ .
3Median: with $5$ values the middle (third) value is $7$ .
4Mode: $7$ appears twice and every other value once, so the mode is $7$ .
5Range: $13 - 4 = 9$ .
6Consistency check: the mean ( $8$ ) sits above the median ( $7$ ) because the high value $13$ pulls the average up — as expected for data with a longer right tail.

Mean $= 8$ , median $= 7$ , mode $= 7$ , range $= 9$ .

Common traps

Finding the median of unordered data by grabbing the middle of the list as written instead of sorting first.
Assuming the mean and median are equal in skewed data — outliers move the mean much more than the median.
Treating a percentile as a score: being at the 75th percentile means outranking about 75% of the group, not scoring 75%.
Believing a data set with larger values must have a larger standard deviation — spread around the mean, not size, is what counts, and adding a constant to every value changes the mean but not the standard deviation.

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Question depth and domain coverage vary by exam. Practice answers are checked after submission.

Sources

GRE General Test Quantitative Reasoning Overview — ETS. Accessed 2026-08-02. Use as a cited source for exam facts; do not imply affiliation or reproduce protected test material.
Introductory Statistics 2e, Section 2.5: Measures of the Center of the Data — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
Introductory Statistics 2e, Section 2.3: Measures of the Location of the Data — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
Introductory Statistics 2e, Section 2.7: Measures of the Spread of the Data — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
Introductory Statistics 2e, Section 6.1: The Standard Normal Distribution — OpenStax. Accessed 2026-08-02. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.

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Last source check: 2026-08-02
Next scheduled review: 2026-11-02

Recheck the ETS Quantitative Reasoning content page and the OpenStax statistics references before each major GRE preparation cycle.

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