If you haven’t taken a statistics course, you might be terrified at the prospect of having to face SAT questions on the subject. Before you let fear take over, though, realize that there are many statistical concepts that you have already learned! And, as for the ones you haven’t, well, they can be mastered with just a bit of extra study time. In this article, we’ll give you a solid review of the basics and a preview of the more challenging statistics questions that you might encounter on the SAT.

**Here are the topics we’ll cover:**

- An Overview of SAT Math Topics
- The Average (Arithmetic Mean)
- The Weighted Average
- The Median
- The Mode
- The Range
- The Standard Deviation
- Summary: SAT Statistics Problems
- What’s Next?

Before we look at some practice questions, let’s take a look at the subtopics that we’ll encounter in SAT statistics.

## An Overview of SAT Math Topics

On the digital SAT, there have been significant changes from the pencil-and-paper test. First, the number of questions in the Math section has been reduced from 58 to 44. Second, students may use a calculator in both modules of the Math section. Third, the test-maker has implemented what is called “adaptive scoring.” This means that the difficulty of Math module 2 is based on your performance in Math module 1. Module 2 could be harder, easier, or about the same as module 1.

There are four major categories of SAT math questions. Let’s look at each category and list the math topics that are covered in each:

**Algebra:**linear equations, linear functions, systems of linear equations, and linear inequalities**Advanced Math:**nonlinear equations, systems of nonlinear equations, nonlinear functions**Problem Solving and Data Analysis****(PSDA):**ratios, proportions, percentages, data distributions, measures of center and spread, data interpretation, scatterplots, probability, statistical significance, margin of error, evaluating statistical claims**Geometry and Trigonometry:**area, volume, lines, angles, triangles, right triangle trigonometry, circles

Questions from the first two categories (Algebra and Advanced Math) each constitute about 35% of SAT math questions. The third and fourth categories (Problem Solving and Data Analysis, and Geometry and Trigonometry) each constitute about 15% of SAT math questions.

We see that statistics questions are in the Problem Solving and Data Analysis (PSDA) category, and the bulk of questions in that category are statistics questions. Thus, we can expect to see as many as 6 statistics questions on the SAT. This is sufficient reason to continue reading this article and learning the facts and concepts presented!

KEY FACT:

You can expect to see as many as 6 statistics questions in the SAT Math section.

Let’s start with the average.

## The Average (Arithmetic Mean)

On the SAT, you will see the terms “average” and “arithmetic mean” used interchangeably. So, to calculate the average or the arithmetic mean, you’ll use the basic formula:

average = sum of values / number of values

Here is a basic arithmetic mean question:

What is the average of Erica’s test scores of 80, 62, 74, and 76?

We use the formula for the average:

average = sum of values / number of values

average = (80 + 62 + 74 + 76) / 4

average = 292 / 4 = 73

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

Five basketball players scored points in last night’s game. The average number of points scored was 16. Jeremiah, Winston, Lonnie, and Morris scored 7, 19, 30, and 12 points, respectively. How many points did Alfred, the fifth player, score?

- 12
- 13
- 16
- 17

#### Solution:

We know that the average of the 5 players is 16. To find the sum of the points scored by all five players, we can use the formula for the average:

average = sum of values / number of values

16 = sum / 5

80 = sum

We see that the sum of the 5 players’ scores is equal to 80. If we let x represent Alfred’s unknown score, we can express the sum as:

80 = 7 + 19 + 30 + 12 + x

We can then solve for x:

80 = 68 + x

12 = x

We see that Alfred scored 12 points.

**Answer: A**

In this next example, the values contain variables. We can still use the formula for the average.

### Example 2: Average — Variable Values

The arithmetic mean of (2x + 5), (x + 2), 3, and x is 8. What is the value of x?

- 2
- 4.5
- 5.5
- 6

#### Solution:

We have 4 terms with an arithmetic mean (average) of 8. Let’s use the formula for the average:

average = sum of values / number of values

8 = [(2x + 5) + (x + 2) + 3 + x] / 4

32 = 4x + 10

22 = 4x

5.5 = x

**Answer: C**

Let’s now consider the weighted average.

## The Weighted Average

The SAT doesn’t always test 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 see how this works.

Consider the following 6 values: 1, 1, 7, 7, 7, 7.

If we were asked to calculate the average, we could use the average formula:

average = sum of values / number of values

average = (1 + 1 + 7 + 7 + 7 + 7) / 6 = 30 / 6 = 5

Another way of computing the average would be to note that there are two 1s and four 7s. We could then use multiplication to find the sum:

average = sum of values / number of values

average = [(2 x 1) + (4 x 7)] / 6 = (2 + 28) / 6 = 5

### The Weighted Average Formula

When we have repeated values and we use multiplication to make the arithmetic easier, we are calculating the weighted average, or weighted mean. We can use the following formula:

weighted average = [(f1)(x1) + (f2)(x2) + … + (fn)(xn)] / sum of frequencies

The formula tells us to multiply each value by its frequency, then add the products, and finally divide by the number of values (the sum of the frequencies).

Consider the following question:

During a fund drive, six donors gave $50 each, three donors gave $100 each, and four donors gave $175 each. What was the average donation?

**Solution:**

We can use the weighted average formula:

weighted average = [(f1)(x1) + (f2)(x2) + … + (fn)(xn)] / sum of frequencies

weighted average = [(6)(50) + (3)(100) + (4)(175)] / (6 + 3 + 4)

weighted average = (300 + 300 + 700) / 13

weighted average = 1300 / 13 = 100

The average donation is $100.

KEY FACT:

Weighted average = [(f1)(x1) + (f2)(x2) + … + (fn)(xn)] / sum of frequencies

Let’s consider a more challenging example.

### Example 3: Weighted Average

A team of ten wildlife biologists counted the number of sea turtle nests at various locations in the Southeast, and the average number of nests reported was 420. Five biologists reported 350 nests each and four biologists reported 500 nests each. How many nests were reported by the tenth biologist?

- 400
- 420
- 435
- 450

#### Solution:

Let’s let x = the number of nests reported by the tenth biologist. We can then use the weighted average formula:

weighted average = [(f1)(x1) + (f2)(x2) + … + (fn)(xn)] / sum of frequencies

420 = [(5)(350) + (4)(500) + (1)(x)] / (10)

4200 = (1750 + 2000 + x)

4200 = 3750 + x

x = 450

**Answer: D**

## The Median

Another measure of the center of a set is the median. The median is found by ordering the values in a set and finding the middle value. The following mini-examples illustrate how to manually determine the value of the median.

### Median Mini-Example 1

What is the median of the set {6, 10, 13, 1, 4, 3, 18}?

**Solution: **

We first list the values in ascending order.

1 3 4 **6** 10 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 a set has an even number of values, there is no single middle value. As a result, we must calculate the average of the two middle values.

### Median: Mini-Example 2

What is the median of the set {20, 1, 6, 14, 13, 9}?

**Solution:**

First, we put the values in ascending order:

1 6 **9 13** 14 20

Because we have an even number of values, there is no single middle value. So, we must calculate the average of the two middle values to determine the median:

Median = (9 + 13) / 2 = 22 / 2 = 11

The median is 11.

The technique we just performed is fast and easy if we have a small data set. But if our set contains a large number of values, this technique is time-consuming. We need an easier method for determining which value is the middle one.

Below is a shortcut formula for finding which value is the median:

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; instead, it indicates 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

Members of the school band sold tickets to an upcoming concert. Eight students sold 12 tickets each, fourteen students sold 20 tickets each, ten students sold 25 tickets each, and five students sold 50 tickets each. What is the median number of tickets sold by the band members?

- 9
- 19
- 20
- 29

#### Solution:

First, we note that the number of students is 8 + 14 + 10 + 5 = 37. Thus, n = 37. We can use the formula for the position of the median to determine the number of tickets that is the median value.

Position of median = (n + 1) / 2 = (37 + 1) / 2 = 38 / 2 = 19

The median is the 19th value in the list when the values are ordered.

We see that 8 students (the 1st through the 8th student) sold 12 tickets each. The next 14 students, who sold 20 tickets each, include the 9th through the 22nd student. Since the median is the 19th value, or the number of tickets sold by the 19th student, we see that this student sold 20 tickets.

**Answer: C**

KEY FACT:

For large data sets, it may be easier to determine the position of the median by using the formula: position of median = (n + 1) / 2.

## The Mode

In a set of values, the mode is the one that occurs most often. The mode is sometimes confusing because a set can have no mode, one mode, or multiple modes. Let’s look at how this works.

What is the mode of the set {7, 5, 2, 1, 4, 7, 4, 4}?

Here, the mode is 4. It occurs three times.

What is the mode of the set {1, 12, 9, 2, 3, 7, 15}?

Here, there is no mode: no value occurs more often than any other.

What is the mode of the set {4, 4, 2, 11, 6, 7, 11, 7}?

Here, there are three modes: 4, 7, and 11. Each occurs twice.

KEY FACT:

A set can have no mode, one mode, or multiple modes.

On the SAT, you will probably not be asked such basic 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, using a scenario that we encountered earlier in this article.

### Example 5: Mode

Members of the school band sold tickets to an upcoming concert. Eight students sold 12 tickets each, fourteen students sold 20 tickets each, ten students sold 25 tickets each, and five students sold 50 tickets each. What is the mode of this distribution?

- 12
- 14
- 20
- 25

#### Solution:

If we listed the numbers of tickets sold by the 37 students, we would have:

12, 12, 12, 12, 12, 12, 12, 12, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 25, 25, 25, 25, 25, 25, 25, 25, 25, 25, 50, 50, 50, 50.

We see that 20 occurs more often than any other value, so 20 is the mode.

Rather than listing the numbers individually, it is more efficient to note that the value with the greatest frequency is the mode. From the original problem, we are told that fourteen students sold 20 tickets each. Thus, since 14 is the greatest frequency, we know that 20 is the mode.

**Answer: C**

The mean, median, and mode are the three main measures of the center of a set because they focus on the middle of a distribution. Let’s now consider what are called “measures of dispersion,” which describe how the data are spread around the center of a set. The two main measures of spread are the range and the standard deviation.

## The Range

The range is the difference between the greatest value and the least value in a data set. A small range generally indicates that the values are relatively close to the center, whereas a large range generally indicates that the values are more widely spread around the center.

KEY FACT:

Range = highest value – lowest value

Let’s do a practice problem.

### Example 6: Range

The quarterback of a high school football team completed 16, 12, 22, and 10 passes, respectively, in the first four games of the season. After the fifth game, the team statistician stated that the range of the number of completed passes for the season had doubled, compared to the range of just the first four games. How many passes did the quarterback complete during the fifth game?

- 14
- 24
- 34
- 44

#### Solution:

First, let’s calculate the range for the first four games of the season:

Range = highest value – lowest value

Range = 22 – 10 = 12

We know that after the fifth game, the range of the number of passes completed doubled. Thus, 24 is the range after the fifth game.

We know that the new range is 24 and the lowest value is 10. If we let x = the number of completed passes during the fifth game, we have:

Range = highest value – lowest value

24 = x – 10

34 = x

**Answer: C**

The last statistical measure is also a measure of the spread of a set of data: the standard deviation.

## The Standard Deviation

The standard deviation is another way to measure the spread of values in a set. More specifically, the standard deviation measures how far individual data points are from the mean of the set. It is more widely used by statisticians than the range.

The SAT tests you on limited aspects of the standard deviation. For example, you will not be required to calculate the actual value of the standard deviation, but you must have a basic knowledge of its properties.

KEY FACT:

The standard deviation is a measure of how far individual data points are from the average of the set.

### Standard Deviation Facts

Several facts about the standard deviation can be used to help us answer SAT questions:

- If all the values in a set are the same, the standard deviation is 0.
- The standard deviation can never be negative.
- The farther away the values are from the mean, the greater the standard deviation.

TTP PRO TIP:

Know the basic standard deviation facts.

### Example 7: Standard Deviation

Consider the following three sets:

Set A: {8, 9, 10, 13}

Set B: {10, 10, 10, 10}

Set C: {0, 0, 20, 20}

Which of the following statements must be true?

I. The averages of the three sets are equal

II. The standard deviations of all three sets are positive

III. Set C has the greatest standard deviation.

- I and II only
- I and III only
- II and III only
- I, II, and III

#### Solution:

Let’s consider each statement individually.

**Statement I**

The mean of Set A is (8 + 9 + 10 + 13) / 4 = 40 / 4 = 10.

The mean of Set B is (10 + 10 + 10 + 10) = 40 / 4 = 10.

The mean of Set C is (0 + 0 + 20 + 20) / 4 = 40 / 4 = 10.

Statement I is true.

**Statement II**

Recall that if all the values in a set are equal, the standard deviation is 0. Thus, the standard deviation of Set B, in which all values are equal to 10, 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 its mean of 10 and the values in Set C to its mean of 10, we see that the individual values in Set C are much farther from the mean of 10 than the values in Set A are. Thus, Set C has the greatest standard deviation.

Statement III is true

**Answer: B**

## Summary: SAT Statistics Problems

In this article, we covered six statistics SAT concepts that you need to know, and we have provided practice problems for each.

The measures of central tendency include the average (or arithmetic mean), the weighted average, the median, and the mode. Measures of spread include the range and the standard deviation. Being familiar with these basic statistical measures will aid you significantly in answering statistics questions on the SAT quickly and accurately.

There are additional SAT statistics review topics, including probability, box plots, linear regression, sampling methods, statistical significance, and margin of error, that were not covered here. But you can access them, and all SAT topics, in our SAT self-study course.

## What’s Next?

Statistics is just one of the 20+ topics in the SAT Math section. You want a great score on the exam, so it’s important to spread your time, energy, and focus over all the topics. And you can read our article about improving your math score on the SAT to get some useful score-enhancing tips for getting a great score.

Good luck!