Scatter Plots

📈 Scatter Plots

Part 1 of 5 — Plotting & Reading Two Variables

Topics in This Part

🔑 Key Concept: A scatter plot shows the relationship between two different measurements taken on the same things — like a person's height and their shoe size. Each dot is one "thing" with two numbers attached.

What Is a Scatter Plot?

A scatter plot is a graph of dots on a coordinate grid. We use it to look for a pattern between two quantities.

The data behind it is called bivariate data — bi means "two", so it's two measurements per item:

Each row becomes one point: Ava is $(1, 60)$, Ben is $(3, 74)$, Cara is $(5, 88)$.

• The first number ($x$) goes across — the horizontal axis.
• The second number ($y$) goes up — the vertical axis.

🔑 Key Idea: One dot = one item, carrying two values written as an ordered pair $(x, y)$. We do not connect the dots with lines.

Concept Check 🎯

Plotting and Reading Points

To plot $(3, 74)$: start at the origin, move right $3$ (the $x$-value), then up $74$ (the $y$-value), and place a dot.

To read a point you already see, do the reverse:

• Drop straight down to the $x$-axis to find the first number.
• Slide straight across to the $y$-axis to find the second number.

Example

A dot sits directly above $4$ on the $x$-axis and lines up with $80$ on the $y$-axis. That dot is the point $(4, 80)$ — a student who studied $4$ hours and scored $80$.

💡 The $x$-value tells you the cause/input (hours), and the $y$-value tells you the result/output (score). Keep the order straight: it's always $(x, y)$.

Read the Points 🧮

Use the data table from earlier (Hours $x$, Score $y$).

1) Ben studied $3$ hours and scored $74$. His $x$-coordinate is $\,?$ 2) Ben's $y$-coordinate is $\,?$ 3) A new student is plotted at $(2, 68)$. What was that student's score?

Match the Word to the Axis 🔽

A scatter plot compares temperature and ice-cream sales.

📈 Scatter Plots

Part 2 of 5 — Patterns: Association & Correlation

🔑 The Big Question: When you look at the cloud of dots, do they trend in a direction? The pattern we see is called the association (or correlation) between the two variables.

Direction of the Association

A positive association slopes uphill from left to right. A negative association slopes downhill from left to right.

💡 Think of reading left-to-right like reading a sentence. Uphill = positive, downhill = negative, flat/messy = none.

Concept Check 🎯

Form, Strength, and Outliers

Beyond direction, describe three more things:

• Form — Are the dots roughly along a straight line (linear) or do they curve (nonlinear)?
• Strength — Are the dots tightly packed near a line (strong) or widely spread out (weak)?
• Outliers — Is there a dot that sits far away from the rest of the pattern?

⚠️ Watch out: "Strong" is about how tight the dots are, not how steep they are. A gentle line with dots glued to it is strong; a steep but scattered cloud is weak.

Describe the Scatter 🔽

Choose the best word for each described scatter plot.

Spot the Outlier 🎯

📈 Scatter Plots

Part 3 of 5 — The Trend Line (Line of Best Fit)

🔑 The Idea: When a scatter plot looks roughly linear, we draw a single straight line that follows the middle of the cloud. It's called the trend line or line of best fit, and it summarizes the whole pattern.

How to Draw a Good Trend Line

A line of best fit does not have to touch any of the dots. Instead it should:

1. Follow the overall direction of the data (uphill for positive, downhill for negative).
2. Pass through the middle of the cloud, with roughly as many dots above the line as below it.
3. Stay as close as possible to all the dots overall.

⚠️ Common mistake: A good trend line is not drawn just to connect the first and last dots, and it does not have to hit the most points. Balance the dots above and below.

Why bother?

Once we have a line, we can predict: estimate a $y$-value for an $x$ we never measured. That's the real power of a scatter plot.

Concept Check 🎯

Judge the Line 🔽

For each scatter plot, decide whether a straight line of best fit is appropriate and what direction it should slope.

Balance the Points 🧮

A scatter plot has $10$ data points. You want your trend line to split them as evenly as possible.

1) For a well-balanced line, about how many points should sit above the line? $\,?$ 2) About how many should sit below it? $\,?$ 3) A different plot has $7$ points and one lands exactly on the line. If $3$ are above, how many are below? $\,?$

📈 Scatter Plots

Part 4 of 5 — The Equation: Slope, Intercept & Predicting

🔑 Big Payoff: Once the trend line has an equation $y = mx + b$, the slope $m$ and intercept $b$ each tell a story about the real situation, and the equation lets us predict.

Reading the Equation $y = mx + b$

The trend line is just a line, so it has the slope-intercept form you already know:

$y = mx + b$

• $m$ = slope = the change in $y$ for each increase of $1$ in $x$ (the rate).
• $b$ = $y$-intercept = the predicted $y$ when $x = 0$ (the starting value).

Example: Studying vs Score

Suppose the line of best fit is $y = 7x + 50$, where $x$ is hours studied and $y$ is the test score.

💡 The slope is always a rate: "so much $y$ per $1$ unit of $x$." Saying it in real words ("$7$ points per hour") is exactly what Grade 8 problems ask for.

Interpret the Line 🎯

Predicting With the Line

To predict, substitute the $x$-value into the equation and compute $y$.

Example: $y = 7x + 50$

Predict the score for a student who studies $4$ hours:

$y = 7(4) + 50 = 28 + 50 = 78$

Predict the score for $6$ hours:

$y = 7(6) + 50 = 42 + 50 = 92$

⚠️ Interpolation vs Extrapolation: Predicting inside the range of your data (interpolation) is fairly safe. Predicting far outside it (extrapolation) is risky — the pattern may not continue. Studying $40$ hours won't give a score of $7(40)+50 = 330$; scores cap at $100$!

Predict With the Equation 🧮

Use the trend line $y = 3x + 12$, where $x$ is weeks of practice and $y$ is words typed per minute.

1) Predict the speed after $5$ weeks: $y = \,?$ 2) Predict the speed after $10$ weeks: $y = \,?$ 3) What is the predicted starting speed (at $0$ weeks)? $\,?$

Find the Slope From Two Points 🧮

A trend line passes through the points $(2, 8)$ and $(6, 20)$. Use $m = \dfrac{y_2 - y_1}{x_2 - x_1}$.

1) Change in $y$ (the numerator): $20 - 8 = \,?$ 2) Change in $x$ (the denominator): $6 - 2 = \,?$ 3) The slope $m = \,?$

📈 Scatter Plots

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) plot and read bivariate points, (2) describe direction, form, and strength, (3) draw a line of best fit, and (4) read its slope/intercept and predict. Let's put it all together.

Quick Reference

⚠️ Remember: correlation describes a pattern, not proof that one thing causes the other — and don't extrapolate far past your data.

Vocabulary Round-Up 🔽

Lock in the key terms one more time.

Correlation vs Causation 🎯

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.