GMAT Statistics, Counting and Probability

GMAT Quant Reasoning blends statistics, set theory, counting and probability into multi-step word problems instead of testing each topic in isolation. The section measures arithmetic and elementary algebra foundational knowledge, and the GMAT Focus Edition dropped geometry from Quant. Every remaining Problem Solving question leans on the numeric-reasoning skills in this guide. A single question often stacks two of them, such as a set-overlap count feeding a probability fraction.

Official GMAC guidance directly names mean, median, weighted average and standard deviation as tested statistics concepts, alongside Venn diagrams for visualizing overlapping sets. Combinatorics and set problems sit inside a published Counting, Sets and Series skill group. This guide walks through mean, median, mode, range and standard deviation, weighted averages, set overlap, permutations and combinations, and probability rules. Every worked example below is original, written to mirror the step count of real Problem Solving items.

How GMAT Quant Tests These Topics

Quant Reasoning holds 21 Problem Solving questions in 45 minutes, inside a 2-hour-15-minute exam of 64 total questions across Quant Reasoning, Verbal Reasoning and Data Insights. No calculator, personal or on-screen, is allowed in Quant Reasoning or Verbal Reasoning. An on-screen calculator appears only in Data Insights, and personal calculators are barred exam-wide unless GMAC Accommodations approves one. Statistics and probability questions therefore reward clean fractions and mental shortcuts over long division.

For hand calculations in Quant, test-center candidates get a provided noteboard and marker, with no personal paper or pens allowed. Online candidates use an approved erasable whiteboard for the same work. Set up your noteboard before the first question, because a weighted average or a three-set Venn problem needs running totals. The table below summarizes the section format.

Quant Reasoning is scored from 60 to 90 in 1-point steps and contributes equally with Verbal Reasoning and Data Insights toward your 205 to 805 total score. Total scores move in 10-point steps and every value ends in 5. A handful of extra statistics or counting questions therefore moves both your section score and your total. Track your practice accuracy against this scale with our GMAT score calculator.

Mean, Median, Mode and Range

These four measures describe a data set with one number each. GMAT word problems give you a data set directly or ask how one measure shifts after a value is added, removed or changed. Mean reacts to every value in the set, while median reacts only to position. Knowing which measure moves under which change answers many questions without full computation.

Mean

Mean is the average of a data set. Add every value, then divide by the count of values. A set of five test scores reads 78, 82, 85, 90 and 95. Add the five scores for 430, then divide by 5 for a mean of 86. Adding one more score above 86 pulls the mean up, and adding one below 86 pulls it down.

Median, Mode and Range

Median is the middle value once a data set is ordered from low to high. With an even number of values, average the two middle numbers. Mode is the value appearing most often in a set, and a set holds no mode, one mode or several modes. Range is the spread between the highest and lowest values in a set. The table below sums up how to find each measure.

Take the set 3, 5, 5, 8, 9. The median is 5, the mode is 5, and the range is 6, since 9 minus 3 equals 6. The mean of the same set is 6, above the median, because the high value of 9 pulls the average up. Compare mean against median whenever a question hints at one extreme value.

Standard Deviation

Standard deviation measures how spread out a data set sits around its mean. A small standard deviation means values cluster close to the mean, and a large standard deviation means values scatter widely. Adding a value equal to the mean lowers the standard deviation, and adding an extreme value raises it. Most GMAT questions test direction and comparison rather than an exact figure. Follow these steps to build a standard deviation by hand.

1. Find the mean of the set.
2. Subtract the mean from each value to get each deviation.
3. Square each deviation.
4. Average the squared deviations for the variance.
5. Take the square root of the variance for the standard deviation.

Take the set 2, 4, 6, 8, 10. The mean is 6, so the deviations are negative 4, negative 2, 0, 2 and 4. Squared, these read 16, 4, 0, 4 and 16, for an average of 8. The standard deviation equals the square root of 8, about 2.83. The set 5, 6, 6, 6, 7 shares the same mean of 6 but sits far tighter, so its standard deviation is much smaller.

Weighted Averages

A weighted average accounts for groups of different sizes contributing unevenly to an overall average. A regular mean formula undercounts or overcounts a group when group sizes differ. GMAT questions build these around class averages, price mixes and speeds across two legs of a trip. The giveaway is two averages paired with two group sizes.

Weighted average equals the sum of each value multiplied by its weight, divided by the sum of the weights. A class of 20 students averages 80 on a test, and a class of 30 students averages 90 on the same test. Multiply 20 by 80 for 1600 and 30 by 90 for 2700, add the two totals for 4300, then divide by the 50 total students for a weighted average of 86. A simple average of 80 and 90 gives 85 instead, an incorrect result, since it ignores the size difference between the two classes. The weighted result always sits closer to the average of the larger group.

Venn Diagrams and Set Overlap Problems

GMAT set problems describe two or more overlapping groups and ask for a total, an overlap or a group left out entirely. A Venn diagram is one of the official data-visualization types on the GMAT, and it maps directly onto a set-overlap formula. Draw the circles before writing any equation, because the diagram shows which region each number describes. Watch the words both, only and neither, since each one points at a different region.

Two-Group Overlap

For two overlapping groups, the total equals each group size added together, minus the group counted twice in the overlap, plus any items outside both groups. Out of 100 students, 60 take Spanish, 45 take French, and 20 take both languages. Add 60 and 45 for 105, subtract the 20 counted twice for 85 students taking at least one language, leaving 15 students who take neither. The 20 students in both groups appear inside the 60 and inside the 45, so removing them once fixes the double count. Only 40 students take Spanish alone, since 60 minus 20 equals 40.

Three-Group Overlap

Three overlapping groups add each single-group count, subtract each pairwise overlap, then add back the group counted in all three sets. Take 50 students where 25 play soccer, 20 play basketball and 18 play tennis. Among them, 8 play soccer and basketball, 6 play soccer and tennis, 5 play basketball and tennis, and 3 play all three sports. Add the three single-sport counts for 63, subtract the three pairwise overlaps of 8, 6 and 5 for 44, then add back the 3 students counted in all three groups for 47 students playing at least one sport, leaving 3 students who play no sport. The add-back step exists because the three pairwise subtractions strip out the triple-overlap group three times.

Permutations vs Combinations

Counting problems ask how many ways a set of choices combines. GMAC groups this skill, along with sequences and series, under a published Counting, Sets and Series skill group, covering combinatorics such as permutations, combinations and counting paths in a grid. The first decision in every counting question is whether order changes the answer. Settle the order question first and the rest is arithmetic.

The Counting Principle

The counting principle finds the total number of outcomes across two or more independent choices. Multiply the number of options at each stage together. The principle sits underneath both permutations and combinations, so it is the safest starting point on an unfamiliar setup. Follow these steps to apply the counting principle.

1. Break the scenario into separate stages or choices.
2. Count the number of options available at each stage.
3. Multiply the counts together for the total number of outcomes.
4. Adjust for restrictions, such as an item barred from repeating.

A traveler picks one of 4 flights and one of 3 hotels. Multiply 4 by 3 for 12 possible flight-and-hotel combinations. Adding a third stage of 2 rental-car options multiplies again for 24 outcomes. Each new independent choice scales the total instead of adding to it.

Permutations

A permutation counts arrangements where order matters. Choosing a president and a treasurer from 6 candidates is a permutation, since the two roles differ. Multiply 6 by 5 for 30 ways, because the first pick leaves 5 people for the second role. Swapping the two names produces a different outcome, so both orders count separately.

Combinations

A combination counts groups where order does not matter. Choosing 2 members for a committee from the same 6 candidates is a combination, since the two spots are identical. Start from the 30 ordered pairs, then divide by 2 because each pair was counted in both orders, for 15 committees. Dividing out the repeated orders is the only difference between the two formulas. The table below compares the two.

Probability Rules

GMAC folds probability questions into the Counting, Sets and Series skill group and other problem content, without naming probability as its own stand-alone skill label. Probability problems still build directly on the counting and set-overlap skills above. A favorable-outcome count is a counting problem, and the total-outcome count is often one too. Treat every probability question as a counting question with a division at the end.

Basic Probability

Probability equals the number of favorable outcomes divided by the total number of possible outcomes. The result always falls between 0 and 1. A bag holds 5 red marbles and 7 blue marbles. The probability of drawing a red marble equals 5 divided by 12. The probability of drawing a blue marble equals 7 divided by 12, and the two fractions add up to 1.

Combined Probability: And/Or Rules

Compound events combine two or more single events. The rule to apply depends on whether the events happen together or as alternatives. Read the question for the words and, or, at least, and none, because each one maps to a specific rule. The table below lists the three combined-probability rules.

Rolling a standard die twice and getting a 6 on both rolls uses the independent-events rule: 1/6 times 1/6 equals 1/36. Drawing without replacement breaks independence, so the second fraction shifts once the first item leaves the pool. For at-least-one questions, find the probability of none and subtract it from 1. The shortcut turns a long or-chain into a single multiplication.

Sample Problem Walkthroughs

Each walkthrough below combines two or more of the skills above, matching the multi-step style of GMAT Quant word problems. Every example here is original, written for practice. Work each one on your own paper first, then compare your steps against the answer. Each answer names the rule it applies and the trap it avoids.

Weighted Average Walkthrough

A store sells 40 shirts at an average price of 18 dollars and 60 shirts at an average price of 28 dollars. Find the average price across all 100 shirts. Two averages arrive with two different group sizes, so a plain average of 18 and 28 misses.

The weighted average rule applies here, since each price carries a different number of shirts. Multiply 40 by 18 for 720, multiply 60 by 28 for 1680, add the two totals for 2400, then divide by the 100 total shirts for an average price of 24 dollars. The answer sits closer to 28 than to 18 because the 28-dollar group is larger. A simple average would give 23 dollars, the trap answer.

Set Overlap Walkthrough

Among 80 employees, 50 speak Spanish, 35 speak German, and 15 speak both languages. Find how many employees speak neither language. The both-languages count signals an overlap problem.

The two-group overlap rule applies, so the double-counted group comes out once. Add 50 and 35 for 85, subtract the 15 counted twice for 70 employees speaking at least one language, leaving 10 employees who speak neither. Subtracting 70 from the 80-employee total gives the neither group. Skipping the subtraction would give 85 speakers among 80 employees, an impossible result.

Combination Walkthrough

A committee of 3 people is chosen from a group of 8 candidates. Find the number of possible committees. No roles are named, so order does not matter.

The combination rule applies, since the same 3 people form one committee no matter the pick order. Multiply 8 by 7 by 6 for 336 ordered selections. Divide by 3 times 2 times 1, or 6, the number of orders for any single group of 3, for 56 possible committees. Treating the setup as a permutation would give 336, six times too many.

Probability Walkthrough

A jar holds 4 green, 3 yellow and 5 orange candies, 12 candies total. Find the probability of drawing a yellow candy first and a green candy second, without replacement. The phrase without replacement changes the second denominator.

The and rule applies, with the second probability adjusted because the first candy is never returned. The probability of yellow first equals 3 divided by 12, or 1/4. After removing one candy, 11 remain with 4 green ones, so the probability of green second equals 4 divided by 11. Multiply the two probabilities, 1/4 times 4/11, for 1/11. Holding the denominator at 12 for the second draw is the common error here.

Put these formulas to work with our free GMAT drills. Try the Descriptive Statistics drill for mean, median, mode, range and standard deviation reps, or the Sets, Counting and Probability drill for Venn diagrams, permutations, combinations and probability practice. Both run free, with no signup required. For sequences and series, the other half of GMAC's counting skill group, see our GMAT Sequences guide, or see the full section breakdown in our GMAT Quant Reasoning guide. All GMAT drills, module practice sets and full-length mocks sit on our GMAT practice page.

Sources

• GMAC: GMAT Exam Structure
• GMAC: GMAT Exam Content
• GMAC: Why Take the GMAT Focus Edition? To Prove Your Quant Skills
• GMAC: GMAT Quantitative Skill Groups
• GMAC: GMAT Calculator and Scratch Paper/Note-Taking Policies
• GMAC: Understanding Your Score