SAT Math: Problem Solving and Data Analysis 📊

SAT Math Overview

The Problem Solving & Data Analysis portion of the SAT Math exam accounts for about 17 out of 58 questions, roughly 30% of the math section. These questions test your ability to convert units, work with percentages, read data from charts and graphs, and determine whether relationships are linear or exponential. This guide covers the three main skill groups you'll need to master.

Main Problem Solving & Data Analysis Topic Areas

College Board organizes these skills into several categories. We've grouped them into three focus areas:

Group 1: Ratio, Proportion, Units, and Percentage

• These skills revolve around comparing two numbers and predicting unknown values.

Group 2: Interpreting Relationships Presented in Scatterplots, Graphs, Tables, and Equations

• You'll need to analyze how two variables interact and change. These relationships can be modeled as linear, quadratic, or exponential graphs and equations.
• This group includes table data, scatterplots, and linear and exponential growth.

Group 3: More Data and Statistics

• These questions ask you to interpret provided data by identifying key values or calculating probabilities.
• This group includes data inferences, statistical distributions, and data collection and conclusions.

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Ratio, Proportion, Units, and Percentage

Ratios and proportions are all about comparing two numbers. These questions can get tricky if you mix up which number goes where, so pay close attention to what's being compared.

What You Need to Know: Ratio, Proportion, Units, and Percentage

For this first group of skills, you should be able to:

• Identify the difference between ratios, proportions, and percents.
  • A ratio is a comparison of two different values, or parts (e.g., 4 apples to 3 oranges, written as 4:3).
  • A proportion is an equation stating that two ratios are equal (e.g., $\frac{1}{2} = \frac{3}{6}$). You can also think of it as a part-to-whole relationship, like 3 out of 5 stars.
  • A percent expresses a value as a part of 100 (e.g., 75% means 75 out of 100).
• Apply the properties of proportion to word problems.
• Manipulate fractions and decimals.
• Convert between units and between percentages, fractions, and decimals.
• Simplify ratios. Ratios should always be written in their simplest form.
• Calculate percent change: Percent Change = $\frac{\text{difference}}{\text{original}} \times 100$

Applying Your Knowledge: Ratio, Proportion, Units, and Percentage

Unit Conversion Practice

Some common conversions involve measures of distance, like yards to inches. You can set up a conversion table to work through the problem step by step.

Here's how to convert 4 yards into inches:

1. Place the given value on the far left. That's 4 yards.
2. Multiply by a fraction equal to 1 that converts yards to feet. Since 3 feet = 1 yard, write it as $\frac{3 \text{ feet}}{1 \text{ yard}}$.
3. Multiply: $4 \text{ yards} \times \frac{3 \text{ feet}}{1 \text{ yard}} = 12 \text{ feet}$. Notice how "yards" cancels out. Always place the unit you want to cancel in the denominator.
4. Repeat the process to convert feet to inches. Since 12 inches = 1 foot: $12 \text{ feet} \times \frac{12 \text{ inches}}{1 \text{ foot}} = 144 \text{ inches}$.

Percentage Practice

To calculate a percentage, divide the part by the total and multiply by 100.

Say there are 32 students in a class and 12 of them are on a field trip. The question asks for the percentage of students that were present.

1. Calculate the number of students present: $32 - 12 = 20$ students in class.

2. Divide the part by the total: $\frac{20}{32} = 0.625$.

3. Multiply by 100: $0.625 \times 100 = 62.5\%$ of students were in class that day.

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Interpreting Relationships Presented in Scatterplots, Graphs, Tables, and Equations

These questions require you to look at how two variables interact and change. The relationship between them can be modeled as linear, quadratic, or exponential.

What You Need to Know: Scatterplots, Graphs, Tables, and Equations

For this second group of skills, you should be able to:

• Interpret data from charts, graphs, or tables.
• Find and interpret a line of best fit for given data.
• Place data within the context of word problems.
• Determine whether a function is linear or exponential:
  • If there is a common difference between consecutive output values (you add or subtract the same amount each time), the relationship is linear. Linear relationships follow the form $y = mx + b$.
  • If there is a common ratio between consecutive output values (you multiply by the same amount each time), the relationship is exponential. Exponential relationships follow the form $y = a(b)^x$.

Applying Your Knowledge: Scatterplots, Graphs, Tables, and Equations

Linear vs. Exponential Functions

One tip for drawing a line of best fit: sketch a line that has roughly an equal number of data points above and below it. It doesn't have to be perfect, but it should capture the overall trend.

To tell whether data is linear or exponential, look at the pattern in the output values.

Table A:

Look at the changes between consecutive Y values in Table A. Each Y value increases by 3 to produce the next one. Since there's a constant difference being added each time, this is a linear relationship.

Table B:

In Table B, adding or subtracting a constant doesn't produce the next output value. Instead, each Y value is multiplied by 2 to get the next one. Since there's a common ratio (multiply by 2 each time), Table B models an exponential relationship.

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More Data and Statistics

These questions ask you to interpret data by identifying key values or calculating probabilities.

What You Need to Know: More Data and Statistics

For this third group of skills, you should be able to:

• Answer questions involving measures of central tendency:
  • The mean is the mathematical average of a data set (sum of all values divided by the number of values).
  • The median is the middle number when the data set is arranged from least to greatest.
  • The mode is the most frequently occurring number in a data set.
• Use data to calculate probability:
  • If event A and event B must both occur, multiply the individual probabilities together.
  • If event A or event B can occur, add the individual probabilities together.
• Understand sampling procedures in statistics:
  • Samples must be randomly chosen and representative of the whole population in order to generalize conclusions to that population.
  • Participants must be randomly assigned to treatment groups in an experiment if you want to determine cause and effect.

Applying Your Knowledge: More Data and Statistics

Measures of Central Tendency Practice

Take a look at this data set: 3, 6, 7, 9, 1, 22, 4, 6, 8.

• To find the mean, add all the numbers and divide by the count: $\frac{3 + 6 + 7 + 9 + 1 + 22 + 4 + 6 + 8}{9} = \frac{66}{9} \approx 7.33$.
• To find the median, first arrange the values in order: 1, 3, 4, 6, 6, 7, 8, 9, 22. With 9 values, the middle one is the 5th value: 6.
• The mode is the most frequently occurring value. Since 6 appears twice and every other number appears once, the mode is 6.

Probability Practice

Imagine a jar of 20 cookies: 12 chocolate chip, 5 peanut butter, and 3 Snickerdoodle.

• If you're asked the probability of randomly choosing a peanut butter or Snickerdoodle cookie, add the individual probabilities: $\frac{5}{20} + \frac{3}{20} = \frac{8}{20} = \frac{2}{5}$.

You may also be asked to calculate the probability of event A and event B both happening. In that case, multiply the two probabilities. For example, imagine rolling two dice and wanting both to show a number less than 3 (that's a 1 or a 2).

• Probability of getting less than 3 on die A: $\frac{2}{6}$
• Probability of getting less than 3 on die B: $\frac{2}{6}$
• Probability of both events occurring: $\frac{2}{6} \times \frac{2}{6} = \frac{4}{36} = \frac{1}{9}$

Note: this multiplication rule applies when the two events are independent (one outcome doesn't affect the other), which is the case with two separate dice.

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SAT Math Sample Questions + Explanations

SAT Math Questions 1-3: Line of Best Fit Practice

The following graph will be used for questions 1-3. These fit under the second group of skills (interpreting relationships). You can open the image in a new tab to see it larger.

Image Courtesy of College Board's SAT Student Guide

The line on this graph is a line of best fit, which represents the overall pattern. The individual dots are the actual data points representing real people and their measurements.

Question 1: Line of Best Fit

This question is considered "easy" in difficulty. You need to compare individual data points to the line of best fit.

Image Courtesy of College Board's SAT Student Guide

The correct answer is B.

Each individual's actual height (the y-value of the data point) must differ from the predicted height (the y-value on the line of best fit at the same x) by at least 3 centimeters. Four individuals meet this criterion: those with metacarpal bones of approximately 4, 4.3, 4.8, and 4.9 cm.

Question 2: Analyzing Slope

This question is considered "easy" in difficulty. You need to interpret what the slope means in context.

Image Courtesy of College Board's SAT Student Guide

The correct answer is A.

Slope represents the change in the y-value for each unit change in the x-value. Here, height is on the y-axis and metacarpal bone length is on the x-axis, so the slope represents the predicted change in height for each 1 cm increase in metacarpal bone length.

This eliminates choices B and D, which reverse the relationship. Choice C describes what happens when the input is 0, which is the y-intercept, not the slope. That leaves A as the correct answer.

Question 3: Estimating with the Line of Best Fit

This question is considered "easy." You need to use the line of best fit to estimate a value.

Image Courtesy of College Board's SAT Student Guide

The correct answer is C.

You're given the input: a metacarpal bone length of 4.45 cm. Find 4.45 on the x-axis (between the 4.4 and 4.5 gridlines). Draw a vertical line up to where it meets the line of best fit, then read across to the y-axis. The predicted height is approximately 170 cm. Don't be afraid to mark up your test paper when doing this.

SAT Math Question 4: Unit Conversion Practice

This question is considered "medium" in difficulty. It requires converting between megabits and gigabits and between hours and seconds. This fits under the first group of skills.

Image Courtesy of College Board's SAT Practice Problems

The correct answer is B.

Before starting, note the key facts: 1 hour = 3,600 seconds, 1 gigabit = 1,024 megabits, and 1 image = 11.2 gigabits.

The key to dimensional analysis is making sure each original unit cancels out as you multiply by conversion factors, leaving you with the desired units. Using this setup, you find that approximately 10 images per day (rounded down) can be sent.

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Closing

That covers the three main skill areas for SAT Math Problem Solving and Data Analysis. You should now have a solid understanding of ratios, proportions, and percentages; interpreting relationships in graphs and tables; and working with statistics and probability.

Need more SAT resources? Check out our SAT Math Section Tips and Tricks. Want to see more practice questions? Take a look at "What are the SAT Math Test Questions Like?"

Helpful SAT Math Resources

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