SAT Math – Problem Solving & Data Analysis
Data Representations
Interpreting tables, graphs, charts, and visual data displays
Data representations transform numbers into visual stories that reveal patterns, trends, and relationships. On the SAT, you'll extract information from tables, interpret graphs, analyze scatterplots, and draw conclusions from multiple data formats—skills that mirror how professionals in every field make data-driven decisions.
Success requires careful reading of axis labels, legends, and scales; understanding what each visual format best communicates; and performing calculations using data from charts and tables. These aren't just test questions—they're the literacy skills needed to navigate a world where data visualization appears in news articles, research papers, business reports, and social media daily.
Common Data Representation Types
Tables
Tables organize data in rows and columns for easy lookup and comparison.
Key skills: Reading row/column headers, finding intersections, calculating totals/averages
Bar Graphs & Histograms
Bar graphs compare categories using rectangular bars. Histograms show frequency distributions of continuous data.
Key skills: Reading bar heights, comparing values, understanding intervals (histograms)
Line Graphs
Line graphs show changes over time or continuous relationships by connecting data points.
Key skills: Identifying increasing/decreasing patterns, finding rates of change, reading points
Scatterplots
Scatterplots display relationships between two variables by plotting points on x-y coordinates.
Key skills: Recognizing positive/negative/no correlation, interpreting trend lines
Circle Graphs (Pie Charts)
Circle graphs show parts of a whole as percentages or proportions using sectors of a circle.
Key skills: Converting between degrees/percentages, finding sector values
Essential Reading Skills
The TAILS Method for Graphs
Title: What does the graph show?
Axes: What variables are on x and y axes?
Intervals: What does each unit represent? Check the scale!
Legend: What do symbols, colors, or patterns mean?
Source: Is there additional context or information?
Circle Graph Formulas
A complete circle = 360°
\(\text{Sector degrees} = \frac{\text{Part}}{\text{Whole}} \times 360°\)
\(\text{Sector value} = \frac{\text{Sector degrees}}{360°} \times \text{Total}\)
Two-Way Tables
Show relationships between two categorical variables
Row totals: Sum across columns
Column totals: Sum down rows
Grand total: Sum of all entries (bottom right)
Common Pitfalls & Expert Tips
❌ Not checking the scale
If the y-axis counts by 5s, don't read a bar at the 3rd mark as 3—it's 15! Always check intervals before reading values.
❌ Mixing up x and y axes
Always verify which variable is on which axis. Questions often ask about the relationship in a specific direction.
❌ Confusing correlation with causation
A scatterplot showing correlation doesn't prove one variable causes the other. Correlation ≠ causation!
❌ Misreading legends
When multiple data series appear, check the legend before interpreting. Different lines, colors, or patterns represent different groups.
✓ Expert Tip: Read ALL labels first
Before looking at the data, read the title, axis labels, legend, and scale. Understanding what you're looking at prevents misinterpretation.
✓ Expert Tip: Estimate before calculating
For bar or line graphs, estimate values visually first. This helps catch reading errors and saves time on some questions.
✓ Expert Tip: Watch for broken axes
Sometimes axes don't start at zero (shown by a break symbol). This exaggerates differences—read carefully!
Fully Worked SAT-Style Examples
The table below shows students by grade level and whether they participate in sports.
| Sports | No Sports | Total | |
|---|---|---|---|
| 9th Grade | 45 | 35 | 80 |
| 10th Grade | 60 | 20 | 80 |
| Total | 105 | 55 | 160 |
What percent of 10th graders participate in sports?
Solution:
Step 1: Find relevant values
10th graders in sports: 60
Total 10th graders: 80
Step 2: Calculate percentage
\(\frac{60}{80} \times 100\% = 0.75 \times 100\% = 75\%\)
Answer: 75% of 10th graders participate in sports
A bar graph shows book sales for four months. The y-axis represents "Books Sold (in hundreds)" and shows: January = 3, February = 4, March = 6, April = 5.
What was the total number of books sold in these four months?
Solution:
Step 1: Note the scale carefully
Y-axis is "in hundreds"
So each unit represents 100 books
Step 2: Convert values to actual books
January: 3 × 100 = 300 books
February: 4 × 100 = 400 books
March: 6 × 100 = 600 books
April: 5 × 100 = 500 books
Step 3: Calculate total
\(300 + 400 + 600 + 500 = 1{,}800\) books
Common Error:
Don't just add 3 + 4 + 6 + 5 = 18!
You must account for the scale ("in hundreds")
Answer: 1,800 books
A circle graph shows how 200 students spend their free time. The "Reading" sector measures 72°. How many students chose reading?
Solution:
Step 1: Set up proportion
Complete circle = 360°
\(\frac{\text{Reading sector}}{360°} = \frac{\text{Reading students}}{200}\)
Step 2: Substitute and solve
\(\frac{72°}{360°} = \frac{x}{200}\)
\(0.2 = \frac{x}{200}\)
\(x = 40\) students
Alternative Method:
72° is \(\frac{72}{360} = \frac{1}{5}\) of the circle
So: \(\frac{1}{5} \times 200 = 40\) students
Answer: 40 students
A line graph shows temperature over time. At 2 PM, the temperature was 68°F. At 6 PM, the temperature was 80°F. What was the average rate of change in temperature per hour?
Solution:
Step 1: Find change in temperature
\(\Delta T = 80°\text{F} - 68°\text{F} = 12°\text{F}\)
Step 2: Find elapsed time
\(\Delta t = 6 \text{ PM} - 2 \text{ PM} = 4 \text{ hours}\)
Step 3: Calculate rate of change
\(\text{Rate} = \frac{\Delta T}{\Delta t} = \frac{12°\text{F}}{4 \text{ hours}} = 3°\text{F per hour}\)
Answer: 3°F per hour
A scatterplot shows hours studied (x-axis) versus test scores (y-axis). As hours studied increases, test scores generally increase. Which statement best describes the relationship?
A) No correlation
B) Negative correlation
C) Positive correlation
D) Causation is proven
Solution:
Understanding correlation types:
Positive correlation: As x increases, y increases
Negative correlation: As x increases, y decreases
No correlation: No clear pattern
Analyzing this problem:
As hours studied (x) increases → test scores (y) increase
This describes positive correlation
Important Note:
Choice D is wrong! Correlation does NOT prove causation
Scatterplots show relationships, not cause-and-effect
Answer: C) Positive correlation
A histogram shows test score frequency: 60-69 (5 students), 70-79 (12 students), 80-89 (18 students), 90-99 (10 students). How many students scored 80 or above?
Solution:
Step 1: Identify relevant intervals
"80 or above" includes:
• 80-89 interval
• 90-99 interval
Step 2: Add frequencies
80-89: 18 students
90-99: 10 students
Total: \(18 + 10 = 28\) students
Answer: 28 students
A table shows Company A's profits: Year 1 = $50,000, Year 2 = $65,000. A bar graph shows Company B's profits: Year 1 = $45,000, Year 2 = $70,000. Which company had greater percent increase in profit from Year 1 to Year 2?
Solution:
Calculate Company A's percent increase:
Change: \(65{,}000 - 50{,}000 = 15{,}000\)
\(\frac{15{,}000}{50{,}000} \times 100\% = 30\%\)
Calculate Company B's percent increase:
Change: \(70{,}000 - 45{,}000 = 25{,}000\)
\(\frac{25{,}000}{45{,}000} \times 100\% \approx 55.6\%\)
Compare:
Company A: 30% increase
Company B: 55.6% increase
Company B had the greater percent increase
Answer: Company B
A line graph shows population over decades. The line passes through (1980, 120) and (2000, 180), where population is in thousands. During which 10-year period was the population increase the greatest?
Additional points shown: 1990 at 140, 2010 at 210
Solution:
Calculate each 10-year increase:
1980 to 1990: 140 - 120 = 20 thousand
1990 to 2000: 180 - 140 = 40 thousand
2000 to 2010: 210 - 180 = 30 thousand
Identify greatest increase:
Comparing: 20, 40, 30
Greatest increase = 40 thousand from 1990 to 2000
Visual Tip:
On line graphs, steeper slopes = faster rates of change
The 1990-2000 segment would appear steepest
Answer: 1990 to 2000
Data Representation Quick Guide
SAT Data Reading Checklist
Before Reading Data
- Read title and labels first
- Check axis scales and units
- Note any legends or keys
- Look for scale multipliers
While Reading Data
- Trace carefully from axis to value
- Watch for broken axes (≠ zero)
- Count intervals, don't estimate
- Use straight edge if needed
For Calculations
- Apply scale multipliers correctly
- Show work for complex problems
- Double-check units match question
- Verify answer makes sense
Common Mistakes
- Ignoring scale (in hundreds, etc.)
- Mixing up x and y axes
- Not reading legend
- Assuming correlation = causation
Data Representations: Visual Literacy for the Information Age
The ability to read, interpret, and analyze data visualizations is no longer optional—it's fundamental literacy in the 21st century. Every day you encounter bar graphs in news articles, line charts in business reports, scatterplots in research papers, and infographics on social media. The SAT tests these skills because they represent genuine competency: the ability to extract meaning from visual data, identify trends and patterns, compare categories, and draw evidence-based conclusions. Whether you pursue medicine, engineering, business, social science, or journalism, you'll regularly need to interpret data representations and communicate findings visually. Master the habit of checking labels, verifying scales, understanding what each format best communicates, and always questioning whether the data actually supports the claimed conclusion. These aren't just test skills—they're the critical thinking abilities that separate informed citizens from those who are misled by misleading charts and selective data presentations.
