When should I use the median instead of the mean on the GRE?

Use the median when the data is skewed or contains an outlier, because the median is resistant to extreme values while the mean is pulled toward them. For roughly symmetric data, the mean and median are close and either works.

How do I compute a weighted mean?

Multiply each value by its weight (or group size), add those products, and divide by the total of the weights. A weighted mean is required whenever the groups differ in size or importance, such as combining two class averages of different enrollments.

Does the GRE make me calculate standard deviation by hand?

Almost never. The GRE tests standard deviation conceptually — comparing which set is more spread out, or predicting how shifting or scaling the data changes the spread. You reason about typical distance from the mean rather than plugging into the full formula.

What happens to spread when I add a constant to every value?

Adding the same constant to every value shifts the mean by that constant but leaves the standard deviation, range, and IQR unchanged. Spread only responds to scaling (multiplying every value), which multiplies the spread by the same positive factor.

CramClub • Descriptive Statistics — GRE® General Test Guide & Practice

$\bar{x} = \dfrac{\text{sum of values}}{\text{number of values}}$

Middle value of ordered data (average the two middles if $n$ is even)

$\bar{x}_w = \dfrac{\sum w_i x_i}{\sum w_i}$

$\text{Range} = \max - \min, \quad \text{IQR} = Q_3 - Q_1$

Typical distance of values from the mean

Summarizing a Data Set

Descriptive statistics answer two questions about a list of numbers: where is the center, and how spread out are the values. The GRE Data Analysis domain leans on these ideas constantly — in stand-alone problem-solving items, in Quantitative Comparison, and inside data-interpretation sets built on tables and graphs. The good news is that a handful of definitions cover almost everything the test asks.

Measures of Center

The three classic measures of center each describe "typical" differently.

Measure	Definition	Best when
Mean	Sum of the values divided by the count	Data is roughly symmetric
Median	The middle value once the data is ordered	Data is skewed or has outliers
Mode	The value that occurs most often	You care about the most common outcome

The mean is $\bar{x} = \dfrac{\text{sum of values}}{\text{number of values}}$ . To find the median, first order the data. With an odd count the median is the single middle value; with an even count it is the average of the two middle values. A set can have one mode, several modes, or none at all.

Here is the trap the GRE loves. Consider five salaries: $40{,}000, $42{,}000, $44{,}000, $46{,}000, and $300{,}000. The median is the middle value, $44{,}000, but the mean is $\dfrac{472{,}000}{5} = 94{,}400$ , i.e. $94{,}400. One large outlier drags the mean far above the median. This is why the median is called resistant: it barely moves when an extreme value is added, while the mean can swing wildly.

Weighted Mean

When groups have different sizes or different importance, a plain average is wrong — you need a weighted mean:

$\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$

Suppose a course grade is 30% homework, 20% quizzes, and 50% a final, and a student earns 90, 80, and 70 on those parts. The overall grade is $0.30(90) + 0.20(80) + 0.50(70) = 27 + 16 + 35 = 78$ — not the unweighted average of $80$ . Whenever a problem says "combined," "overall," or gives two groups of different sizes, reach for the weighted mean.

Adding or Removing a Value

The GRE frequently asks how the mean changes when the data set changes. The key identity is that the sum equals $\bar{x} \times n$ . If ten numbers average $50$ , their sum is $500$ ; add an eleventh value of $61$ and the new mean is $\dfrac{561}{11} = 51$ . Adding a value above the current mean pulls the mean up; adding one below pulls it down; adding a value equal to the mean leaves it unchanged.

Measures of Spread

Two data sets can share the same mean yet look completely different, so the GRE also tests spread.

Range $= \text{maximum} - \text{minimum}$ . Simple, but sensitive to a single outlier.
Quartiles split ordered data into four equal parts. $Q_1$ (first quartile) is the median of the lower half; $Q_2$ is the overall median; $Q_3$ is the median of the upper half.
Interquartile range $\text{IQR} = Q_3 - Q_1$ captures the spread of the middle 50% of the data and ignores outliers.

Standard Deviation — Spread Around the Mean

Standard deviation measures the typical distance of the values from the mean. You will almost never compute it by hand on the GRE, but you must reason about it:

A set whose values are tightly clustered near the mean has a small standard deviation.
A set whose values are spread far from the mean has a large standard deviation.
Adding the same constant to every value shifts the mean but leaves the standard deviation unchanged — spread does not move when you slide the whole set.
Multiplying every value by a positive constant multiplies the standard deviation by that constant.

For example, $\{2, 4, 6, 8, 10\}$ and $\{102, 104, 106, 108, 110\}$ have identical standard deviations, because the second set is just the first shifted up by 100. By contrast, $\{6, 6, 6, 6, 6\}$ has a standard deviation of exactly $0$ — no spread at all. Knowing the range alone is not enough to compare standard deviations; two sets can share a range and still differ in spread.

What the GRE Actually Tests

Compute a center or spread from a small list.
Predict how the mean shifts when a value is added or removed.
Compare the standard deviations of two sets conceptually, without a formula.
Read quartiles from a boxplot or a table.

The GRE never asks you to run a hypothesis test, build a confidence interval, or fit a regression — those are outside the tested syllabus. Stay with center, spread, and how they respond to changes in the data.

Find the mean, median, and mode of the data set $\{4, 9, 4, 12, 6\}$ .

A small firm reports five salaries: $38{,}000, $41{,}000, $44{,}000, $47{,}000, and $230{,}000. Which better describes a typical salary, the mean or the median, and what is it?

Find the interquartile range of $\{5, 8, 10, 15, 18, 20, 25\}$ .

A class of 30 students averaged 72 on a test; a second class of 20 students averaged 82. What is the mean score of all 50 students combined?

The mean of six numbers is 25. When one number is removed, the mean of the remaining five is 27. What number was removed?

Set $P = \{20, 20, 20, 20, 20\}$ , set $Q = \{18, 19, 20, 21, 22\}$ , and set $R = \{10, 15, 20, 25, 30\}$ . Rank the three sets by standard deviation, smallest to largest.

Reading the median off the unordered list. For $\{7, 2, 9\}$ , calling $2$ the median because it is written in the middle.

Always sort first. Ordered, $\{2, 7, 9\}$ has median $7$ . The median is the middle of the ordered data, never the original order.

Averaging two group means directly when the groups have different sizes, e.g. calling the combined average of a 30-student and a 20-student class the plain mean of their averages.

Use a weighted mean: multiply each group mean by its count, add, then divide by the total count. The bigger group counts more.

Assuming two sets with the same range have the same standard deviation, or that a bigger range always means a bigger standard deviation.

Range uses only the two extremes; standard deviation uses every value. Two sets can share a range yet differ in how the interior values cluster, so range does not determine standard deviation.

Thinking that adding a constant to every value changes the spread. Students often recompute the standard deviation after a uniform shift.

Adding the same amount to every value slides the whole set — the mean moves but the spread (standard deviation, IQR, and range) is unchanged. Only scaling multiplies the spread.

What is the median of the data set $\{12, 7, 23, 7, 15, 20\}$ ?

A course grade weights homework 30%, quizzes 20%, and the final exam 50%. A student scores 90 on homework, 80 on quizzes, and 70 on the final. What is the course grade?

List $L$ consists of the five numbers $1, 50, 51, 52, 53$ .

Quantity A: The median of list $L$

Quantity B: The mean of list $L$

What is the interquartile range of the data set $\{3, 7, 8, 12, 14, 18, 21\}$ ?

List $M$ contains 5 numbers with a range of 10. List $N$ contains 5 numbers with a range of 10.

Quantity A: The standard deviation of list $M$

Quantity B: The standard deviation of list $N$

A set of five numbers has a mean of 20. A sixth number, 8, is added to the set. What is the mean of the six numbers?