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If you’ve looked at the list of Quant topics tested on the GMAT, there’s no doubt that you noticed that statistics is one of roughly 20 major topics represented on the exam. For some students, this allows for a sigh of relief, but for others, it may create a sense of foreboding! No matter what your initial reaction is, you know that mastery of statistics questions is mandatory if you want to get that great GMAT score. We’ll take you through the intricacies of GMAT statistics questions in this article, from the basics of the arithmetic mean to the standard deviation. We’ll show you how to solve statistics math problems on the GMAT, step by step.
Here are the topics we’ll cover:
- GMAT Statistics Topics
- The Average (Arithmetic Mean)
- The Weighted Average
- The Median
- The Mode
- The Range
- The Standard Deviation
- Summary: GMAT Statistics Questions
- What’s Next?
Before we look at some practice questions, let’s take a look at the subtopics that we’ll encounter in GMAT statistics.
GMAT Statistics Topics
GMAT students have a mixed reaction to reviewing statistics during their GMAT preparation. They think back to their introductory statistics course during their undergraduate years. Some recall being thoroughly confused from day one until the final exam, and others say it was a course that helped them better understand the complex world of data and analysis. No matter which experience you had, you’ll be happy to know that the GMAT tests only rudimentary concepts from that statistics course. You will not need to worry about such things as confidence intervals or sampling techniques or hypothesis testing. You will only need to review such topics as mean, median, mode, range, and standard deviation.
In this article, we’ll look at each of these topics in detail, preparing you to hit each of them head-on so that when you encounter GMAT statistics questions on the exam, you’ll smile and not shudder! Let’s get started!
The Average (Arithmetic Mean)
You will see the terms “average” and “arithmetic mean” used interchangeably. So, whether you’re asked to calculate the average or the arithmetic mean, you’ll use the basic formula:
average = sum of values / number of values
An easy, basic question about the average is this:
What is the average of Javier’s test scores of 79, 88, 91, and 84?
We use the formula for the average:
average = sum of values / number of values
average = (79 + 88 + 91 + 84) / 4
average = 342 / 4 = 85.5
KEY FACT:
The formula for the average, or arithmetic mean, is average = sum of values / number of values.
Let’s look at a more challenging example.
Example 1: Average — Missing Value
The average race time of 5 runners was 172 seconds. Jeremiah, Kia, Lonnie, and Morris had times of 172, 147, 180, and 211 seconds, respectively. What was the finishing time, in seconds, of Nabik, the fifth runner?
- 142
- 150
- 165
- 172
- 187
Solution:
We know that the average time of the 5 runners is 172. Thus, we can use the formula for the average:
average = sum of values / number of values
172 = sum / 5
860 = sum
We see that the sum of the 5 times must equal 860. If we let Nabik’s (unknown) time be x, we can express the sum as:
860 = 172 + 147 + 180 + 211 + x
We can then solve for x:
860 = 710 + x
150 = x
Nabik’s finishing time is 150 seconds.
Answer: B
Let’s try another question about the average, but this time, the terms contain variables.
Example 2 : Average — Variable Values
The arithmetic mean of (x + 3), (2x + 7), 2, and x is 9. What is the value of x?
- 6
- 8
- 10.5
- 12
- 14.25
Solution:
We have 4 values, and the arithmetic mean is 9. Using the average formula, we have:
average = sum of values / number of values
9 = [(x + 3) + (2x + 7) + 2 + x] / 4
36 = 4x + 12
24 = 4x
6 = x
Answer: A
Let’s now consider the weighted average.
The Weighted Average
The GMAT seldom tests you on the easiest version of a concept. Rather than being asked to compute a simple average, you might be given a question that tests you on the weighted average. Let’s look at the basics and then answer a couple of example questions.
Consider the following 6 values: 2, 2, 5, 5, 5, 5.
If we were asked to calculate the average, we could use the formula we’ve already covered for the average:
average = sum / number
average = (2 + 2 + 5 + 5 + 5 + 5) / 6 = 24 / 6 = 4
Another way of computing the average would be to note that there are two 2s and four 5s. We could then use multiplication to find the sum:
average = sum / number
average = [(2 x 2) + (4 x 5)] / 6 = (4 + 20) / 6 = 4
The Weighted Average Formula
When we have repeated values and we use multiplication to simplify the arithmetic, we calculate the weighted average, or weighted mean. We can use the following formula:
weighted average = [(f1)(x1) + (f2)(x2) + … + (fn)(xn)] / sum of frequencies
In other words, we multiply each value by its frequency, add the products, and finally divide by the total number of values (the sum of the frequencies).
Let’s consider the following question:
On a math quiz, four students scored 70, eight students scored 80, and three students scored 90. To the nearest whole number, what was the average score on the quiz?
Solution:
We use the weighted average formula:
weighted average = [(f1)(x1) + (f2)(x2) + … + (fn)(xn)] / sum of frequencies
weighted average = [4)(70) + (8)(80) + (3)(90)] / (4 + 8 + 3)
weighted average = (280 + 640 + 270) / 15
weighted average = 1190 / 15 = 79.3
Rounded to the nearest whole number, the quiz average is 79.
KEY FACT:
Weighted average = [(f1)(x1) + (f2)(x2) + … + (fn)(xn)] / sum of frequencies
Let’s consider a more challenging example.
Example 3: Weighted Average — Missing Value
Last week, AAA Tutoring, Inc. paid its tutors an average hourly rate of $25.00. The company paid two tutors an hourly rate of $15, five tutors an hourly rate of $32, and the remaining tutors an hourly rate of $22. How many tutors were paid an hourly rate of $22?
- 3
- 4
- 5
- 6
- 7
Solution:
Let’s let x = the number of tutors who were paid an hourly rate of $22. We can then use the weighted average formula:
weighted average = [(f1)(x1) + (f2)(x2) + … + (fn)(xn)] / sum of frequencies
25 = [(2)(15) + (5)(32) + (x)(22)] / (7 + x)
25 = (30 + 160 + 22x) / (7 + x)
25(7 + x) = 190 + 22x
175 + 25x = 190 + 22x
3x = 15
x = 5
Answer: C
The Median
During your GMAT prep or in your basic statistics class, you probably learned that the median is the middle value of a set (or the average of the two middle values, if the set contains an even number of values). If the set is small, you can simply order the values and easily find the value of the median.
Let’s do a quick practice problem as a review.
Median Mini-Example 1
What is the median of the set {6, 9, 13, 2, 5, 3, 18}?
Solution:
We first list the values in ascending order.
2 3 5 6 9 13 18
Because there are 7 values (an odd number), we know that the median is the middle value, with three values below it and three values above it. Thus, the median is 6.
If we have a set with an even number of values, we see that there is no single middle number. So, we take the average of the two middle numbers.
Median Mini-Example 2
What is the median of the set {14, 2, 7, 12, 11, 9}?
Solution:
Here, we order the numbers:
2 7 9 11 12 14
We have an even number of values, so there is no single middle number. Thus, we take the average of the two middle numbers to find the value of the median:
Median = (9 + 11) / 2 = 20 / 2 = 10
The median is 10.
The above technique is quick and easy if we have a small set. But what if our set contains, say, 31 values? We need a shortcut for determining which value is the middle one.
The formula for finding which value is the median is:
Position of median = (n + 1) / 2
In this formula, n = the number of values. Be sure to note that this formula does not compute the value of the median; rather, it tells us which value is the median. Let’s look at an example question that uses this formula to identify the median.
Example 4: Median — Large Data Set
At the end of the fifth-grade candy bar sales project, students turned in their candy bar sales counts. 16 students sold 12 bars each, 10 students sold 20 bars each, 9 students sold 25 bars each, and 10 students sold 50 bars each. What is the median number of candy bars sold by the fifth-grade students?
- 12
- 20
- 22.5
- 25
- 50
Solution:
First, we note that the number of students is 16 + 10 + 9 + 10 = 45. Thus, n = 45. We can now use the formula for the position of the median:
Position of median = (n + 1) / 2 = (45 + 1) / 2 = 46 / 2 = 23
The median is the 23rd value in the list.
We see that 16 students (the 1st through the 16th student) sold 12 bars each. The next group of 10 students that sold 20 bars each includes the 17th through the 26th student, and thus the 23rd (median) value is in this category. Because all students in this group sold 20 bars each, we see that the median number of candy bars sold was 20.
Answer: B
KEY FACT:
For large data sets, it may be easier to determine the position of the median by using the formula.
The Mode
The mode is the most frequently occurring value in a set. Sometimes this can present confusion because a set could have no mode, one mode, or multiple modes. Let’s look at how this works.
What is the mode of the set {5, 5, 2, 1, 3, 7, 3, 3}?
Here, the mode is 3. It occurs three times.
What is the mode of the set {1, 6, 9, 11, 3, 5, 8}?
Here there is no mode, as no value occurs more frequently than any other.
What is the mode of the set {2, 4, 2, 6, 6, 7, 11, 7}?
Here, there are three modes: 2, 6, and 7. Each occurs two times.
KEY FACT:
A set can have no mode, one mode, or multiple modes.
Of course, the GMAT will not ask such simple questions about the mode as those posed above. Let’s look at a question about the mode that we might encounter on the actual exam.
Example 5: Mode
In the set {3, 8, 4, 6, 3, 1, 3, 9, 4, x}, x represents the only mode. Which of the following is a possible value of x?
I. 3
II. 4
III. 6
- I only
- II only
- III only
- I and II only
- II and III only
Solution:
First, let’s list the values in order:
3, 3, 3, 4, 4, 6, 1, 8, 9, x (Note that we list x at the end, just for convenience.)
Now, let’s consider each option.
If x = 3, then the only mode would be 3, with a frequency of 4. Thus, x could equal 3.
If x = 4, then the set would be {3, 3, 3, 4, 4, 4, 6, 1, 8, 9}. The two modes would be 3 and 4. We are told that the set has only one mode, so x cannot equal 4.
If x = 6, then the set is {3, 3, 3, 4, 4, 6, 6, 1, 8, 9}. The set’s mode is 3. Thus, x cannot equal 6.
Answer: A
So far, we have looked at statistical measures, referred to as “measures of center,” which focus on the center, or middle, of a set. Now, let’s consider what are called “measures of dispersion,” which describe how the data are spread around the center. The two main measures of spread are the range and the standard deviation.
The Range
Simply stated, the range is the difference between the greatest value and the least value in a set. If a set has a large range, it’s an indication that the values are generally spread out from the center.
KEY FACT:
Range = highest value – lowest value
Let’s do a practice problem.
Example 6: Range
The proprietor of a food truck recorded the number of people in line at noon each day. The results for Monday through Friday were 6, 14, 8, 10, and 9, respectively. On Saturday, there were a lot of people in line at noon because a soccer game near the food truck had just ended.
After including the Saturday data in his calculations, he calculated that the range of the number of people in line had increased by 50%. How many people were in line at noon on Saturday?
- 14
- 15
- 16
- 17
- 18
Solution:
First, let’s calculate the range for the period Monday through Friday, inclusive.
Range = highest value – lowest value
Range = 14 – 6 = 8
We are told that after the Saturday value was included, the range increased by 50%. Thus, we multiply the old range of 8 by 1.5, to get (8)(1.5) = 12 as the new range.
We know the new range is 12 and the lowest value is 6. Letting c = the number of people in line on Saturday, we have:
12 = c – 6
18 = c
Answer: E
Our final statistical measure is also a measure of the spread, or dispersion, of a set of data: the standard deviation.
The Standard Deviation
In this article, we cannot cover every type of standard deviation GMAT problem that you might see on the exam, so we will focus on some basic ones. For complete coverage of standard deviation and all other GMAT topics, you might wish to use our self-study GMAT prep course.
The standard deviation, like the range, is a measure of dispersion, or spread, of a set. More specifically, it is a measure of the distance of the data points from the mean. It is more accurate than the range, and it is used extensively as a tool in statistical analysis.
You will not be required to calculate the actual value of the standard deviation, but you must have a working knowledge of its properties and how it might be used.
KEY FACT:
The standard deviation is a measure of the distance of data points from the mean of the set.
The Mean and the Standard Deviation
We might describe a particular value of a data set as being 1, 2, or 3 standard deviations from the mean. For example, if the mean hourly wage for workers at a theme park is $15 with a standard deviation of $3, then we would know the following facts:
- A wage that is 1 standard deviation above the mean is $15 + $3 = $18
- A wage that is 2 standard deviations below the mean is $15 – (2)($3) = $15 – $6 = $9
- A wage that is within 2 standard deviations of the mean is $15 +/- (2)($3) = $15 +/- 6. Thus, a wage between $9 and $21 is within 2 standard deviations of the mean.
It is unlikely that a value is more than 2 standard deviations away from the mean, as about 95% of values are within 2 standard deviations of the mean.
Let’s see how this concept could be tested on the GMAT.
Example 7: Standard Deviation
The finishing times, in minutes, of 7 students in a study group taking a calculus exam were 46, 39, 50, 52, 36, 41, and 43. The entire class of 30 students had a mean finishing time of 46 minutes with a standard deviation of 5 minutes. How many of the 7 students in the study group finished within 1 standard deviation of the mean?
- 3
- 4
- 5
- 6
- 7
Solution:
Within 1 standard deviation of the mean means that we are looking for values within 5 minutes of the mean. Thus, we have 46 +/- 5 = 41 to 51, inclusive. Any score between 41 and 51, inclusive, is within 1 standard deviation of the mean. Of the 7 students in the study group, we see that four times (46, 50, 41, and 43) are within 1 standard deviation.
Answer: B
Standard Deviation Facts
Several facts about the standard deviation can help us answer GMAT questions:
- If all the values in a set are equal, the standard deviation is 0.
- The farther away the values are from the mean, the greater the standard deviation.
- The standard deviation can never be negative.
TTP PRO TIP:
Know the basic facts about the standard deviation.
Example 8: Standard Deviation
Consider the following three sets:
Set A: {0, 0, 10, 10}
Set B: {5, 5, 5, 5}
Set C: {4, 4, 6, 6}
Which of the following statements is/are true?
I. The means of the three sets are equal
II. The standard deviations of all three sets are positive
III. Set A has the greatest standard deviation.
- I and II only
- I and III only
- II and III only
- I, II, and III
- None are true
Solution:
Let’s consider each statement individually.
Statement I
The mean of Set A is (0 + 0 + 10 + 10) / 4 = 5.
The mean of Set B is (5 + 5 + 5 + 5) = 20 / 4 = 5.
The mean of Set C is (4 + 4 + 6 + 6) / 4 = 5.
Statement I is true.
Statement II
Recall that if all of the values in a set are equal, the standard deviation is 0. Thus, the standard deviation of Set B, in which every value is 5, is 0, which is not a positive number.
Statement II is not true.
Statement III
We have already established that the standard deviation of Set B is 0. Now, if we compare the values in Set A to the values in Set C, we see that the values in Set A are much farther from the mean of 5 than the values in Set C are. Thus, Set A has the greatest standard deviation.
Statement III is true
Answer: B
Summary: GMAT Statistics Questions
This article has focused on six main subtopics of statistics and solving statistics math problems.
Measures of center include the average (or arithmetic mean), the weighted average, the median, and the mode. Measures of dispersion, or spread, include the range and the standard deviation. Several related topics — probability, combinations, permutations, and quartiles — are covered in depth in our GMAT self-study course.
Having familiarity with the formulas, facts, and rules of these basic statistical measures will aid you significantly in answering statistics questions on the GMAT quickly and accurately.
What’s Next?
Statistics is just one of the 20+ topics on the Quant section of the GMAT. You want a great score on the exam, so it’s important to spread your time, energy, and focus over all the topics. You can get a look at the big picture by reading this breakdown of the GMAT Quant section or checking out some tips for improving your GMAT Quant score.
Good luck!
