🎯⭐ INTERACTIVE LESSON

Scatterplots and Line of Best Fit

Learn step-by-step with interactive practice!

← Back to Standard Lesson

Scatterplots and Line of Best Fit - Complete Interactive Lesson

Part 1: Data Analysis

Problem Solving: Ratios, Rates & Proportions

Part 1 of 7 — Setting Up and Solving Proportions

This is one of the most heavily tested topics on the SAT Math section. About 25-30% of Math questions fall under Problem Solving & Data Analysis.

Ratios

A ratio compares two quantities: If a recipe uses 3 cups flour to 2 cups sugar, the ratio is 3:2 or 3/2.

Setting Up Proportions

Cross-multiply to solve:

$\frac{3}{5} = \frac{x}{20} implies 3 \times 20 = 5x implies x = 12$

Unit Rates

A unit rate has a denominator of 1:

240 miles in 4 hours → 60 mph
$45 for 3 shirts → $15 per shirt

SAT Trap: Mixing Up Parts and Wholes

If the ratio of boys to girls is 3:5, there are 8 total parts (not 5).

Boys = 3/8 of total
Girls = 5/8 of total

Dimensional Analysis

Convert units by multiplying fractions: $60 \frac{\text{miles}}{\text{hour}} \times \frac{1 \text{ hour}}{60 \text{ min}} = 1 \frac{\text{mile}}{\text{min}}$

Deep Dive: Complex Ratio & Proportion Problems

Worked Example 1: Multi-Step Ratio

Step	Work
Problem	"In a mixture, the ratio of water to concentrate is 5:2. If there are 21 total cups, how much water is needed?"
Total parts	$5 + 2 = 7$ parts
Each part	$21 ÷ 7 = 3$ cups per part

Advanced Ratio & Proportion Problems 🎯

Ratio & Proportion Setup — Select the correct approach.

Part 1 Summary: Ratios, Rates & Proportions

Concept	Formula	SAT Trap
Ratio $a:b$	Part $= \frac{a}{a+b} \times$ total

Part 2: Scatterplots

Percentages: Increase, Decrease & Applications

Part 2 of 7 — Mastering Percent Problems

Percent Formula

$\text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100$

Percent Increase/Decrease

$% Change$

Part 3: Probability

Two-Way Tables & Data Interpretation

Part 3 of 7 — Reading Tables and Finding Probabilities

Two-Way Tables

These organize data by two categories. Example:

	Freshman	Sophomore	Total
Male	120	100	220
Female	130	150	280
Total	250	250	500

Conditional Probability from Tables

"What fraction of sophomores are female?"

Look at the Sophomore column: 150 female out of 250 total = 150/250 = 3/5

"Given that" = Restrict to a Subgroup

"Given that a student is male, what is the probability they are a freshman?"

Restrict to Male row: 120 freshman out of 220 male = 120/220 = 6/11

Marginal vs. Conditional

Marginal: P(Female) = 280/500 — uses the grand total
Conditional: P(Female | Sophomore) = 150/250 — uses a column/row total

Association vs. Independence

Part 4: Two-Way Tables

Statistics: Center, Spread & Shape

Part 4 of 7 — Mean, Median, Standard Deviation

Measures of Center

Mean = sum of all values / count. Sensitive to outliers.
Median = middle value when sorted. Resistant to outliers.

When to Use Mean vs. Median

Symmetric data → mean ≈ median, use either
Skewed data or outliers → median is more representative

Standard Deviation

Measures how spread out data is from the mean.

Low SD → data points close to mean (consistent)
High SD → data points far from mean (variable)

You won't calculate SD on the SAT, but you must compare SDs:

{10, 10, 10, 10, 10} → SD = 0 (no spread)
{8, 9, 10, 11, 12} → small SD
{1, 3, 10, 17, 19} → large SD

Effect of Adding/Removing Values

Adding a value equal to the mean → mean unchanged, SD decreases
Adding an outlier → mean shifts toward outlier, SD increases
Removing an outlier → mean moves away from outlier, SD decreases

Shape of Distributions

Right-skewed (tail to right): mean > median
Left-skewed (tail to left): mean < median
Symmetric: mean ≈ median

Deep Dive: Statistics in Action

Part 5: Statistical Modeling

Scatterplots & Line of Best Fit

Part 5 of 7 — Interpreting Trends and Making Predictions

Reading Scatterplots

Positive association: as x increases, y increases (upward trend)
Negative association: as x increases, y decreases (downward trend)
No association: no visible pattern

Line/Curve of Best Fit

The line that minimizes the total distance from all points. Key interpretations:

Slope = rate of change (For each 1-unit increase in x, y changes by [slope])
y-intercept = predicted y-value when x = 0

Making Predictions

Use the equation to predict values:

If y = 2.3x + 15 models study hours vs. test score:
10 hours → predicted score: 2.3(10) + 15 = 38

Interpolation vs. Extrapolation

Interpolation (within data range): reliable predictions
Extrapolation (beyond data range): unreliable — the trend may not continue

Residuals

Residual = actual – predicted

Positive residual: actual is above the line
Negative residual: actual is below the line
Random residuals → good model
Patterned residuals (curved) → wrong model type

Deep Dive: Scatterplot Analysis

Worked Example 1: Interpreting Slope in Context

Step

Part 6: Problem-Solving Workshop

Probability & Counting

Part 6 of 7 — SAT Probability Essentials

Basic Probability

$P(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$

Complement Rule

$P$

Part 7: Review & Applications

Problem Solving & Data Review

Part 7 of 7 — Mixed Practice & Strategy

Topic Checklist

✓ Ratios, rates, proportions, and unit conversion ✓ Percent increase/decrease and successive changes ✓ Two-way tables and conditional probability ✓ Mean, median, standard deviation, and outliers ✓ Scatterplots, line of best fit, and residuals ✓ Probability, expected value, and counting

SAT Strategy for This Section

Read the question last — scan the table/graph first to understand the data
Identify what the denominators should be — marginal vs. conditional probability
Watch for traps: part-to-part vs. part-to-whole ratios
Use estimation — if a scatterplot has a clear trend, estimate before calculating

Common Mistakes

Confusing "percent increase" with "percentage points"
Using the wrong total for conditional probability
Forgetting that percent change compounds (not additive)
Extrapolating beyond the data range when the question asks for interpolation

Deep Dive: Mixed SAT Data Problems

Worked Example 1: Multi-Concept Problem

Step	Work
Problem	"A dataset's mean is 50 and SD is 8. Every value is doubled then 10 is added. Find the new mean and SD."
Double	Mean $=$ , SD

Worked Example 2: Unit Conversion Chain

Step	Work
Problem	"A printer prints 12 pages per minute. How many pages in 2.5 hours?"
Convert hours → minutes	$2.5 \times 60 = 150$ minutes
Calculate	$12 \times 150 = 1{,}800$ pages

Ratio vs. Fraction — Key Difference

Statement	Ratio	Fraction of Total
"Boys to girls is 3:5"	$3:5$	Boys $= \frac{3}{8}$ , Girls $= \frac{5}{8}$
"Boys to total is 3:8"	$3:8$	Boys $= \frac{3}{8}$
"3 out of every 5 are boys"	$3:2$ (boys:girls)	Boys $= \frac{3}{5}$

Dimensional Analysis — Multi-Step

Convert 45 mph to feet per second:

$45 \frac{\text{mi}}{\text{hr}} \times \frac{5280 \text{ ft}}{1 \text{ mi}} \times \frac{1 \text{ hr}}{3600 \text{ sec}} = 66 \frac{\text{ft}}{\text{sec}}$

Next: Percentages — increase, decrease, and successive changes →

% Change = \frac{New - Original}{Original} \times 100

Shortcut multipliers:

20% increase → multiply by 1.20
15% decrease → multiply by 0.85
8% tax → multiply by 1.08

Successive Percent Changes

A 10% increase followed by a 10% decrease is NOT back to the original: $100 \xrightarrow{+10\%} 110 \xrightarrow{-10\%} 99$

SAT Classic: "What percent of X is Y?"

Translate directly: "What percent of 80 is 24?" $\frac{24}{80} \times 100 = 30\%$

Percent vs. Percentage Points

"Increased from 40% to 52%" = increase of 12 percentage points but a 30% increase (12/40 × 100).

Deep Dive: Percent Problem Strategies

Worked Example 1: Finding Original Price

Step	Work
Problem	"After a 30% discount, a jacket costs $56. What was the original price?"
Setup	You pay 70% of original: $0.70x = 56$
Solve	$x = 56 / 0.70 = 80$
Common mistake	Adding 30% of 56: $56 + 16.80 = 72.80$ ← WRONG

Worked Example 2: Successive Changes

Step	Work
Problem	"A stock rises 25% one year, then drops 20% the next. Net change?"
Year 1	$100 \times 1.25 = 125$
Year 2	$125 \times 0.80 = 100$

Percent Multiplier Quick Reference

Phrase	Multiplier	Example
15% increase	$\times 1.15$	$200 \times 1.15 = 230$
15% decrease	$\times 0.85$

"Percent OF" vs. "Percent MORE THAN"

"A is 25% of B" → $A = 0.25B$
"A is 25% more than B" → $A = 1.25B$
"A is 25% less than B" → $A = 0.75B$

Advanced Percent Problems 🎯

Percent Multiplier Check — Select the correct multiplier.

Part 2 Summary: Percentages

Concept	Formula
Percent of	$\frac{\text{Part}}{\text{Whole}} \times 100$
Percent change	$\frac{\text{New} - \text{Old}}{\text{Old}} \times 100$
x% increase	Multiply by $(1 + x/100)$
x% decrease	Multiply by $(1 - x/100)$
Successive changes	Multiply the multipliers
Finding original	Divide by the multiplier

SAT Traps

Successive equal percent changes DON'T cancel out
"A is 60% more than B" ≠ "B is 60% less than A"
Always divide by the original for percent change

Next: Two-way tables and data interpretation →

Two variables are independent if knowing one doesn't change the probability of the other.

If P(Female) = P(Female | Sophomore), gender and class year are independent
If those probabilities differ, there's an association

Deep Dive: Navigating Two-Way Tables

Worked Example 1: Filling In a Table

Step	Work
Problem	"200 employees: 120 full-time, 80 part-time. 90 have benefits; of those, 75 are full-time. Complete the table."
Full-time + benefits	$75$
Full-time, no benefits	$120 - 75 = 45$
Part-time + benefits	$90 - 75 = 15$
Part-time, no benefits	$80 - 15 = 65$

	Benefits	No Benefits	Total
Full-time	75	45	120
Part-time	15	65	80
Total	90	110	200

Worked Example 2: Testing for Independence

Step	Work
Question	"Is having benefits independent of employment type?"
P(Benefits)	$90/200 = 0.45$
**P(Benefits	Full-time)**
Compare	$0.45 \neq 0.625$ → NOT independent

Denominator Guide

Question Phrasing	Denominator
"What fraction of ALL students...?"	Grand total
"What fraction of males...?"	Row total (Males)
"What fraction of freshmen...?"	Column total (Freshman)
"Among those who passed..."	Subtotal of those who passed

SAT Trap: Joint vs. Conditional

Joint: P(male AND freshman) $= 120/500$ (out of everyone)
Conditional: P(freshman | male) $= 120/220$ (out of males only)

Advanced Two-Way Table Problems 🎯

Pick the Right Denominator — What goes in the denominator for each question?

Part 3 Summary: Two-Way Tables

Concept	Key Fact
Marginal probability	Uses the grand total as denominator
Conditional probability	Restricts to a row or column total
Joint probability	One specific cell ÷ grand total
Independence test	P(A) $=$ P(A
Filling in tables	Rows and columns must sum to their totals

SAT Strategy

Read the question word-for-word to find the correct denominator.
"Given that" or "among" = conditional → use a subtotal.
"Of all" = marginal → use the grand total.

Next: Statistics — mean, median, and standard deviation →

Worked Example 1: Finding a Missing Value

Step	Work
Problem	"Five test scores have mean 82. The first four are 78, 85, 92, 71. What is the fifth score?"
Total needed	$82 \times 5 = 410$
Sum of four	$78 + 85 + 92 + 71 = 326$
Fifth score	$410 - 326 = 84$

Worked Example 2: Effect of Removing a Value

Step	Work
Problem	"Data: {10, 12, 14, 15, 100}. How do mean and median change if 100 is removed?"
With 100	Mean $= 151/5 = 30.2$ , Median $= 14$
Without 100	Mean $= 51/4 = 12.75$ , Median $= 13$
Effect	Mean drops significantly ( $30.2 → 12.75$ ), median barely changes ( $14 → 13$ )

Adding a Constant vs. Multiplying

Operation	Effect on Mean	Effect on Median	Effect on SD
Add $k$ to all values	Mean $+ k$	Median $+ k$	SD unchanged
Multiply all by $k$	Mean $\times k$	Median $\times k$	SD $\times

SAT favorite: "If every student's score increases by 5 points, what happens to the standard deviation?" → Nothing — adding a constant shifts all values equally.

Skewness Quick Reference

Shape	Tail Direction	Relationship	Example
Right-skewed	Long tail right	Mean $>$ median	Income distribution
Left-skewed	Long tail left	Mean $<$ median	Easy test scores
Symmetric	Equal tails	Mean $\approx$ median	Heights in a population

Advanced Statistics Problems 🎯

Statistics Quick Check — Select the correct answer.

Part 4 Summary: Statistics

Measure	What It Tells You	Sensitive to Outliers?
Mean	Average value	YES
Median	Middle value	NO
SD	Spread from mean	YES
Range	Max − Min	YES

Key Rules

Add constant $k$ : mean & median shift by $k$ , SD unchanged
Multiply by $k$ : mean, median, & SD all multiply by $|k|$
Right-skewed → mean $>$ median
Outlier → use median as the better center

Next: Scatterplots and line of best fit →

Worked Example 2: Choosing the Best Model

Data Pattern	Best Model	How to Tell
Straight upward trend	Linear ( $y = mx + b$ )	Residuals are random
Curve (increasing rate)	Exponential ( $y = ab^x$ )	Residuals show U-pattern for linear
Curve (decreasing rate)	Logarithmic or square root	Curve levels off
Ups and downs	Quadratic ( $y = ax^2 + bx + c$ )	Parabolic residual pattern

Correlation Coefficient ( $r$ )

$r$ Value	Strength	Direction
$r = 1.0$	Perfect	Positive
$0.7 < r < 1.0$	Strong	Positive
$0.3 < r < 0.7$	Moderate	Positive
$0 < r < 0.3$	Weak	Positive
$r = 0$	None	—
$r < 0$	Same scale, opposite direction	Negative

SAT key fact: $r^2$ = proportion of variation in $y$ explained by $x$ . If $r = 0.8$ , then $r^2 = 0.64$ , meaning 64% of the variation is explained.

Advanced Scatterplot Problems 🎯

Scatterplot Interpretation — Select the correct answer.

Part 5 Summary: Scatterplots & Best Fit

Concept	Key Fact
Slope	Rate of change in context
y-intercept	Predicted value when $x = 0$
Residual	Actual − predicted
$r$	Strength and direction of linear relationship
$r^2$	Proportion of variation explained
Random residuals	Good model fit
Patterned residuals	Try different model type
Interpolation	Reliable (within data range)
Extrapolation	Unreliable (beyond data range)

Next: Probability and expected value →

P (not A) = 1 - P (A)

Often easier: "What's the probability of getting AT LEAST one?" = 1 − P(none).

P(A or B) = P(A) + P(B) − P(A and B)
P(A and B) for independent events = P(A) × P(B)

SAT Probability Questions — Common Types

From a table: "A randomly selected student from the table above..."
Cards/marbles: "If 3 red and 5 blue marbles..."
Surveys: "Based on the survey results, what proportion..."

If the SAT asks what value you'd "expect": $\text{Expected} = \text{Total} \times P(\text{event})$

Example: 200 people surveyed, 35% prefer A → Expected = 200 × 0.35 = 70

Just another word for proportion: $\text{Relative frequency of A} = \frac{\text{count of A}}{\text{total count}}$

Deep Dive: Probability Strategies

Worked Example 1: "At Least One" with Complement

Step	Work
Problem	"Roll a die twice. What is P(at least one 6)?"
Complement	P(no sixes) $= \frac{5}{6} \times \frac{5}{6} = \frac{25}{36}$
Answer	P(at least one 6) $= 1 - \frac{25}{36} = \frac{11}{36}$

Worked Example 2: Expected Value from a Survey

Step	Work
Problem	"In a sample, 42% prefer Brand A. If 500 people are surveyed from the same population, how many would you expect to prefer Brand A?"
Expected	$500 \times 0.42 = 210$ people

Probability Rules Summary

Rule	Formula	When to Use
Complement	$P(\text{not } A) = 1 - P(A)$	"At least one", "none"
Or (general)	$P (A \cup B) = P (A) + P (B) - P$

SAT Probability from Tables

The SAT usually gives you a two-way table and asks:

Simple probability: one cell ÷ grand total
Conditional: one cell ÷ row/column total
"At least": use complement (1 − P(none))

Advanced Probability Problems 🎯

Probability Rule Identification — Match the scenario to the correct approach.

Part 6 Summary: Probability

Rule	Formula	Key Word
Basic	Favorable ÷ Total	"probability of"
Complement	$1 - P(A)$	"at least one", "not"
Or (exclusive)	$P(A) + P(B)$	Can't happen together
Or (overlap)	$P(A) + P(B) - P(A \cap B)$	Can happen together
And (independent)	$P(A) \times P(B)$	With replacement, separate trials
And (dependent)	$P(A) \times P(B	A)$
Expected value	Total $\times$ P(event)	"How many would you expect"

Next: Comprehensive review and mixed practice →

= 50 \times 2 = 100

Worked Example 2: Comprehensive Table + Probability

Step	Work
Problem	"150 students surveyed: 60 prefer A, 50 prefer B, 40 prefer C. Of the A-preferrers, 40 are juniors. What is P(Junior
Restrict	Given "prefers A" → denominator $= 60$
Answer	$\frac{40}{60} = \frac{2}{3}$

SAT Problem Solving Cheat Sheet

Topic	Key Formula	Common Trap
Ratios	Part $= \frac{a}{a+b} \times$ total	Part:part vs. part:whole
Percents	Multiplier method	Successive changes compound
Two-way tables	Conditional → use subtotal	Wrong denominator
Mean	$\frac{\text{sum}}{n}$	Outliers distort
SD	Spread from mean	Add constant → SD unchanged
Scatterplots	Slope = rate of change	Extrapolation ≠ interpolation
Probability	Complement for "at least one"	With vs. without replacement

Time Management for This Section

Difficulty	Time Budget	Strategy
Easy (direct read from table)	30 sec	Read carefully, answer
Medium (one calculation)	60 sec	Set up, solve, check
Hard (multi-step)	90 sec	Plan approach first
Very hard (trap question)	90+ sec	Skip, flag, return

SAT Problem Solving Challenge 🎯

Problem Solving Quick Check — Select the correct answer.

Full Topic Summary: Problem Solving & Data

Part	Topic	Must-Know
1	Ratios & Proportions	Part:whole, cross-multiply, unit rates
2	Percentages	Multiplier method, successive changes compound
3	Two-Way Tables	Marginal vs. conditional vs. joint probability
4	Statistics	Mean/median/SD, outlier effects, skewness
5	Scatterplots	Slope in context, residuals, $r$ and $r^2$
6	Probability	Complement, OR/AND rules, expected value
7	Review	Decision framework, time management, traps

Top Strategies

Read the question carefully — identify what the denominator should be
Use multipliers for percent problems
Complement for "at least one" probability
Median when data has outliers
Check your answer — does it make sense in context?

🎉 Problem Solving & Data complete!

200 \times 0.85 = 170

P (A \cup B) = P (A) + P (B) - P (A \cap B)

Model	$y = 3.5x + 120$ where $x$ = years of experience, $y$ = weekly earnings ($)
Slope meaning	For each additional year of experience, weekly earnings increase by $3.50.
y-intercept	A worker with 0 years of experience earns $120/week.
SAT phrasing	"The estimated increase in weekly earnings for each additional year of experience"