GMAT (Graduate Management Admission Test) Quantitative Reasoning: Word Problems, Statistics & Combinatorics Syllabus

How do you compute the range of a data set?

Range = maximum value - minimum value.

What are the quartiles and how is the interquartile range (IQR) defined?

Q1 = median of the lower half, Q3 = median of the upper half. IQR = Q3 - Q1, measuring the spread of the middle 50% of the data.

What does standard deviation measure conceptually?

It measures how spread out values are around the mean; a larger standard deviation means values are more dispersed, a smaller one means they cluster near the mean.

How does adding a constant to every value, or multiplying every value by a constant, affect the standard deviation?

Adding a constant to every value leaves the standard deviation unchanged. Multiplying every value by a constant k multiplies the standard deviation by |k|.

Which data set has a larger standard deviation: one with values tightly clustered near the mean or one with values far from the mean?

The set with values farther from the mean has the larger standard deviation; clustering near the mean gives a smaller standard deviation.

What is the formula for a weighted average of values with different weights?

Weighted average = (sum of weight x value) / (sum of weights) = (w1x1 + w2x2 + ...) / (w1 + w2 + ...).

When combining two groups, why is the overall average not simply the average of the two group averages?

Because group sizes (weights) differ; the overall average is the weighted average, pulled toward the average of the larger group.

What happens to the mean when you add a data point equal to the current mean?

The mean stays the same; adding a value equal to the mean does not change the average (though it can change other measures).

How does adding a new data point above the current mean affect the mean, and one below it?

A value above the current mean raises the mean; a value below the current mean lowers it.

How can removing a data point change the median versus the mean?

The mean changes whenever the removed value differs from the mean. The median may shift to a new middle position; it is far less affected by removing an extreme (outlier) value than the mean is.

State the fundamental counting principle.

If one task can be done in m ways and a second independent task in n ways, the two together can be done in m x n ways. Multiply the number of choices at each independent stage.

What is the formula for the number of permutations of n distinct items taken r at a time?

P(n,r) = n! / (n - r)!. It counts ordered arrangements of r items chosen from n.

What is the formula for the number of combinations of n distinct items taken r at a time?

C(n,r) = n! / [r!(n - r)!]. It counts unordered selections of r items from n.

How do you decide whether a problem is a permutation or a combination?

If order matters (arrangements, rankings, sequences, assigning distinct roles), use permutations. If order does not matter (selecting a group, committee, or subset), use combinations.

How many ways can n distinct objects be arranged in a row, and how does this relate to permutations?

n! ways. This is P(n,n) = n!/0! = n!, the permutation of all n items.

What is the relationship between C(n,r) and C(n,n-r)?

C(n,r) = C(n,n-r); choosing r items to include is equivalent to choosing the n-r items to leave out.

What is the basic probability of a single event for equally likely outcomes?

P(event) = (number of favorable outcomes) / (total number of possible outcomes). Probability is always between 0 and 1.

Define independent events and give the formula for both occurring.

Two events are independent if one occurring does not change the probability of the other. P(A and B) = P(A) x P(B).

How does a dependent event differ from an independent one, and how do you compute the joint probability?

For dependent events, the first outcome changes the probability of the second. P(A and B) = P(A) x P(B given A); recompute the second probability after the first occurs (e.g., draws without replacement).

Define mutually exclusive events and the probability that either occurs.

Mutually exclusive (disjoint) events cannot happen at the same time, so P(A and B) = 0. Then P(A or B) = P(A) + P(B).

What is the general addition rule for the probability of A or B when events can overlap?

P(A or B) = P(A) + P(B) - P(A and B). Subtract the overlap so it is not counted twice.

What is complementary probability and when is it useful?

P(not A) = 1 - P(A). It is useful when the complement is easier to count, especially for 'at least one' problems: P(at least one) = 1 - P(none).

How do you compute the probability of a sequence of events happening one after another?

Multiply the probability of each step, using conditional probabilities when steps are dependent: P(A then B then C) = P(A) x P(B|A) x P(C|A and B).

What is the formula for conditional probability P(B given A)?

P(B|A) = P(A and B) / P(A), defined for P(A) > 0. It is the probability of B restricted to outcomes where A has occurred.

In a two-set Venn diagram, how do you find the number in 'exactly one' set and how does the inclusion-exclusion principle read?

Total in either = |A| + |B| - |both|. Exactly one = (|A| - both) + (|B| - both) = |A| + |B| - 2(both).

State the inclusion-exclusion principle for three sets.

|A or B or C| = |A| + |B| + |C| - |A and B| - |A and C| - |B and C| + |A and B and C|.

In group (overlapping sets) problems, how do you account for the 'neither' category?

Total = (in at least one set) + (neither). So Total = |A| + |B| - |both| + neither for two sets; solve for the unknown.

When should you use the double-matrix (two-variable) method, and how is it structured?

Use it when each item is classified by two independent yes/no attributes. Build a 2x2 grid with row/column totals; each cell plus the margins must sum consistently, letting you solve for missing cells.

In a double-matrix problem, how do the cell entries relate to the row and column totals?

Each row's two cells sum to the row total, each column's two cells sum to the column total, and the four cells sum to the grand total; the row totals and column totals each sum to the grand total.

What is the difference between the union and intersection of two sets?

Union (A or B) = all elements in A, in B, or in both. Intersection (A and B) = only elements that are in both sets simultaneously.