SAT Problem Solving and Data Analysis

The College Board has instituted some significant changes to the SAT, the most obvious being the switch from a paper-and-pencil test to an online digital format. Some other notable changes are combining the Reading section and the Writing section into one section, allowing a calculator for all math questions, and making the exam section-adaptive.  However, one area of the SAT that has seen little change is the material that is tested in the SAT Mathematics section. There are still 4 overarching categories into which the math material is organized, one of which is SAT Problem-Solving and Data Analysis (PSDA).

In this article, we will look at the concepts and topics that are included in the PSDA category, giving you some background and some example problems. You’ll have a better understanding of the material that is tested and the types of questions you can expect to encounter on this important part of the exam.

Here are the topics we’ll cover:

• The SAT Math Section — A Quick Overview
• The 4 Math Categories
• The Problem-Solving and Data Analysis (PSDA) Category
• PSDA Topic #1: Ratios
  • Example 1: Ratios
• PSDA Topic #2: Percentages
  • Example 2: Percent Change
• PSDA Topic #3: Rates
  • Example 3: Average Rate
• PSDA Topic #4: Measures of Center (Mean, Median, Mode)
  • Example 4: Measures of Center
• PSDA Topic #5: Scatterplots
  • Example 5: Scatterplots
• PSDA Topic #6: Probability
  • Example 6: Probability
• PSDA Topic #7: Statistical Inference
  • Example 7: Statistical Inference
• PSDA Topic #8: Margin of Error
  • Example 8: Margin of Error
• Summary
• What’s Next?

Let’s begin by reviewing the changes to the SAT Math section.

The SAT Math Section — A Quick Overview

While the topics tested in the Math section of the SAT have not changed significantly, there are changes in the structure and format. First, the number of questions in the Math section has been reduced from 58 to 44, and the total time to solve them has gone from 80 minutes to 70 minutes. Second, you may use a calculator for all math questions. Third, the College Board has implemented a testing methodology called “adaptive scoring.” This means that the difficulty of Math module 2 is based on your performance in Math module 1. If you did well in module 1, the questions in module 2 will be more difficult, but they will be worth more points.

KEY FACT:

Performing well in Math module 1 gives you the chance to earn more points in Math module 2.

Next, let’s discuss the types of math that the SAT Math section tests.

The 4 Math Categories

The College Board uses 4 descriptive headings to categorize SAT math questions. Let’s look at each category and the math topics that are covered in each:

• Algebra: linear equations in 1 or 2 variables; linear functions; systems of 2 linear equations in 2 variables; linear inequalities in 1 or 2 variables
• Advanced Math: equivalent expressions; nonlinear equations in 1 or 2 variables; systems of nonlinear equations in 2 variables; nonlinear functions
• Problem-Solving and Data Analysis (PSDA): ratios and proportions; percentages; rates and unit conversion; data distributions; measures of center and spread; scatterplots; probability and conditional probability; statistical analysis and inference; margin of error; evaluating statistical claims; observational studies and experiments
• Geometry and Trigonometry: area; volume; lines, angles, and triangles; right triangles and trigonometry; circles

There are 13-15 questions each from the first 2 categories (Algebra and Advanced Math). There are 5-7 questions each from the last 2 categories (Problem-Solving and Data Analysis, and Geometry and Trigonometry).

KEY FACT:

The SAT Math section tests 4 math categories, and each type covers multiple topics.

Now, let’s examine the Problem-Solving and Data Analysis category in detail.

The Problem-Solving and Data Analysis (PSDA) Category

If you look at the 4 categories above, you’ll see that the PSDA category has more individual topics tested than any other category. You’ll note, too, that this category is much more than just SAT graphs and charts. We can subdivide the category into 2 sub-topics: (1) Arithmetic and (2) Statistics. In this article, we will look at facts and detailed example questions from both sub-categories.

KEY FACT:

We can expect to be presented with 5-7 questions from the Problem-Solving and Data Analysis category in the quantitative reasoning section of the SAT.

PSDA Topic #1: Ratios 

A ratio is a numerical comparison between 2 or more quantities. For example, if we have 5 almonds and 10 walnuts, the ratio of almonds to walnuts is 5 to 10. This ratio is called a part-to-part ratio, and it can also be expressed as the fraction 5/10. This can be reduced to create the ratio 1/2.

KEY FACT:

A part-to-part ratio of two quantities a and b can be expressed as a/b.

Similarly, a part-to-whole ratio for 2 quantities compares one of the quantities to the total. For example, in the earlier example, we had 5 almonds and 10 walnuts, for a total of 15 nuts. The part-to-whole ratio of almonds would then be 5/15, or 1/3.

KEY FACT:

A part-to-whole ratio of two quantities a and b to their total can be expressed as a/(a + b) or as b/(a + b).

Let’s consider the following example.

The next topic in the list is percentages.

PSDA Topic #2: Percentages

There are quite a few ways that the SAT can ask you questions about percentages: “percent of,” “what percent,” “percent greater than,” “percent less than,” and a few others. Here, let’s focus our attention on percent change.

The most important thing when you are asked a percent change question is to make sure you use the percent change formula correctly. Here is the formula:

Percent change = (New Value – Old Value) / Old Value

For example, if last year a lake had 13 resident ducks and this year the lake has 25 resident ducks, we can calculate the percent increase as follows:

Percent Change = (25 – 13) / 15 = 12/15 = 0.80, or 80%

Thus, the percent increase in the number of ducks is 80%.

KEY FACT:

Memorize the percent change formula. Percent Change = (New Value – Old Value) / Old Value.

Let’s work on an example problem.

Next, let’s discuss rates.

PSDA Topic #3: Rates 

It’s highly likely that you have used the basic rate formula: distance = rate x time. For example, if you drive at 30 miles per hour for 2 hours, you have traveled a total distance of 30 x 2 = 60 miles.

The basic rate formula can be useful for simple SAT rate questions, but let’s consider a more challenging type of rate question, that of the average rate. Let’s look at a scenario that requires using the average rate formula: Average Rate = Total Distance / Total Time.

KEY FACT:

The average rate formula is: Average Rate = Total Distance / Total Time.

Let’s now discuss the various ways to measure the center of a data set.

PSDA Topic #4: Measures of Center (Mean, Median, Mode)

You have most likely already learned how to calculate the mean, median, and mode of a data set. Let’s review each measure of center.

• Mean = sum / number
• The median is the middle value when the data values are listed in order. If there is no unique median, then the median is the average of the 2 middle numbers.
• The mode is the data value that occurs most often.

KEY FACT:

The common measures of center are the mean, median, and mode.

Let’s work on a sample question.

Next, let’s discuss scatterplots.

PSDA Topic #5: Scatterplots

A scatterplot is a graph that displays the relationship between 2 numeric variables. For example, we might want to consider the relationship between the number of hours studied and the score on an exam. A scatterplot might show that the more hours studied, the higher the score; this would indicate a positive relationship. Alternatively, a negative relationship might exist between calories consumed each day and weight loss. The higher the caloric intake, the lower the amount of weight lost.

KEY FACT:

A scatterplot is a graph that displays the relationship between 2 numeric variables.

Let’s look at an example question.

Example 5: Scatterplots

A 401(k) is a type of savings account for retirement. A sample of 20 individuals working at a large company was taken, and each person’s age and the total amount in his or her 401(k) was recorded. The scatterplot above displays the results.

Based on the scatterplot, which of the following statements is true?

• There is no relationship between one’s age and the amount of money in one’s 401(k).
• The greater the age, the greater the amount in one’s 401(k).
• The greater the age, the less the amount in one’s 41(k).
• As age increases, the amount in a 401(k) tends to remain constant.

Solution:

Looking at the scatterplot, we see that younger individuals, those in their 20s, have $20,000 or less in their 401(k) accounts. But, as age increases, the amount in the 401(k) increases. Thus, we see that the relationship between age and amount is positive. As age increases, the amount in the 401(k) is greater.

Answer: B

Let’s now discuss probability.

PSDA Topic #6: Probability

The SAT will often present you with what is called a 2-way table, and your job will be to determine a probability based on the data in the table. Use the following example problem to discern the technique for calculating a basic probability from a data table.

KEY FACT:

The SAT may ask probability questions by presenting you with a 2-way table.

Example 6: Probability

During a busy weekend, a grocery store recorded the grocery bag preference of each customer, noting the gender of each customer and whether he or she preferred paper or plastic. Shoppers who brought their own reusable bags or those who did not express a preference were not included in this survey. If one shopper is randomly selected from those surveyed, what is the probability that the shopper is a female who prefers paper bags?

• 0.156
• 0.255
• 0.398
• 0.610

Solution:

This 2-way data table displays information about both the gender of a respondent (male or female) as well as his or her shopping bag preference (paper or plastic). We are to calculate the probability that one randomly-selected shopper is both a female and one who prefers paper bags.

From the table, we see that 200 females prefer paper bags, which is the cell where the two categories — female and paper bag — overlap. This is out of a total number of respondents of 1,286. Thus, the probability of randomly selecting a person from the group who is both female and prefers paper bags is 200 / 1,286 = 0.156.

Answer: A

Our next topic is statistical inference.

PSDA Topic #7: Statistical Inference

Statisticians frequently use data to generalize their sample results to an entire population. This is called statistical inference. In making an inference, we must be sure that the sample taken was randomly selected, that the sample was large enough (generally greater than 30), and that the results are generalized only to the population from which the sample was taken.

KEY FACT:

We perform statistical inference when we generalize from a sample to a population.

Consider the following example.

Our final PSDA topic is the margin of error.

PSDA Topic #8: Margin of Error

The margin of error tells you to what degree your sample results may differ from the true value. For example, if you determine that 45% of a sample of 90 children from a school with 500 students prefer gummy worms as their favorite snack, you cannot be sure that the true percentage is 45%. You didn’t ask all of the students, so 45% is a good estimate, but it’s probably not perfect.

To compensate for this lack of certainty, statisticians use what is called the “margin of error,” a kind of “fudge factor.” For the gummy worm example, we might say: “We are highly confident that the true proportion of students who prefer gummy worms is 45% +/- 10%.” This would result in our estimate of the true proportion of students preferring gummy worms to be between 35% and 45%.

KEY FACT:

the margin of error is a “plus or minus” value used to calculate the interval of values that a statistical measure might be expected to take on.

Let’s try an example question.

Summary:

The SAT Math section covers everything from algebra and functions to probability and statistics, and many topics in between!

The Math section of the SAT has 4 major topics:

• Algebra
• Advanced Math
• Problem-Solving and Data Analysis (PSDA)
• Geometry and Trigonometry

In this article, we learned some background about the SAT Math section and answered 8 problem solving and data analysis SAT practice questions. Here are the key points:

• A ratio is a numerical comparison between 2 or more quantities.
• The percent change formula is: Percent Change = (New Value – Old Value) / Old Value
• The average rate formula is: Average Rate = Total Distance / Total Time.
• The common measures of center are the mean, median, and mode.
• A scatterplot is a graph that displays the relationship between 2 numeric variables
• The SAT may ask probability questions by presenting you with a 2-way table.
• We perform statistical inference when we generalize from a sample to a population.
• The margin of error is a “plus or minus” value used to calculate the interval of values that a statistical measure might be expected to take on.

What’s Next?

To do well on the SAT and other college entrance exams, you’ll need to ensure your test preparation is solid and that you use proven SAT test strategies to get your best score. Read our article to get additional practice with SAT math questions.

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