Scatter Plots and Trend Lines

📈 Scatter Plots and Trend Lines

Part 1 of 5 — What a Scatter Plot Shows

Topics in This Part

🔑 Key Concept: A scatter plot is a graph of paired numbers — like a student's hours studied and their test score. Each dot is one ordered pair $(x, y)$. The shape made by all the dots tells a story about how the two quantities are related.

Bivariate Data

When you measure two things about each subject, you have bivariate data ("bi" = two). For example, for each car you might record its age and its price.

Each row becomes one dot on the scatter plot, placed at $(x, y)$.

💡 The $x$-axis is the quantity you treat as the "input" (here, age). The $y$-axis is the quantity that may respond to it (here, price). Choosing axes is a judgment call, but the input usually goes on the horizontal axis.

Concept Check 🎯

Plotting a Point

To plot $(x, y)$: start at the origin, move right $x$ units, then up $y$ units.

Example: To plot $(3, 8)$ — move right $3$, then up $8$, and place a dot.

⚠️ Order matters! The first number is always $x$ (horizontal). $(3, 8)$ and $(8, 3)$ are different dots. A common mistake is reversing them.

Read the Data 🧮

A scatter plot was made from this table:

1) How many dots are on this scatter plot? $\,?$ 2) What is the $y$-coordinate (height) of the dot for Plant 2? $\,?$ 3) For the dot $(6, 19)$, how far right from the origin is it? $\,?$

Name That Axis 🔽

A researcher studies how a sunflower's days since planting affects its height. Choose the best label for each part of the scatter plot.

You Can Now Build a Scatter Plot

You can turn a table of paired numbers into a cloud of dots. That cloud is about to become very useful: its shape reveals whether the two quantities move together, move opposite, or have nothing to do with each other.

In Part 2 we learn to read that shape — the association between the variables.

📈 Scatter Plots and Trend Lines

Part 2 of 5 — Describing Association

🔑 The Big Question: When $x$ goes up, what does $y$ tend to do? The answer is the association between the two variables. We describe it in three ways: direction, form, and strength.

Direction: Positive, Negative, or None

Real examples:

• Hours studied vs. test score → positive (more study, higher score).
• Car age vs. car price → negative (older car, lower price).
• Shoe size vs. math grade → no association (unrelated).

💡 "Positive" does not mean good and "negative" does not mean bad. It only describes the direction the dots tilt.

Concept Check 🎯

Form: Linear or Nonlinear

Form describes the shape of the cloud of dots.

• Linear — the dots roughly follow a straight line.
• Nonlinear — the dots follow a curve (they bend).

This lesson focuses on data that is roughly linear, because a straight trend line fits it well. If the dots clearly curve, a straight line would be a poor description.

Strength

Strength is how tightly the dots hug the pattern.

• Strong — dots cluster very close to a line; the trend is obvious.
• Weak — dots are spread out; you can sense a trend but it's loose.

💡 A scatter plot can be a strong positive linear association (tight, rising, straight) or a weak negative one (loose, falling), and every combination in between.

Describe the Association 🔽

For each described scatter plot, pick the best label.

Clusters and Outliers

Two features can stand out in a scatter plot:

• A cluster is a group of dots bunched together, separate from the rest. It can signal two different types of subjects in one plot.
• An outlier is a dot that lies far away from the overall pattern. It often comes from an unusual case or a recording error.

⚠️ Outliers can be misleading. A single far-off dot can pull a trend line toward it, so always look for one before drawing your line.

Concept Check 🎯

📈 Scatter Plots and Trend Lines

Part 3 of 5 — Drawing the Trend Line

🔑 The Idea: A trend line (also called a line of best fit) is a single straight line drawn through the middle of a linear scatter plot. It summarizes the whole cloud of dots with one simple rule.

How to Draw a Good Trend Line

1. Make sure the association is roughly linear (a straight line makes sense).
2. Draw a line that runs through the center of the dots.
3. Aim for about half the dots above the line and about half below it.
4. Get the line as close as possible to as many dots as you can.

⚠️ A trend line is not "connect the dots," and it does not have to pass through any actual data point. It passes through the middle of them.

💡 Because people draw it by eye, two students may get slightly different trend lines — and both can be reasonable. What matters is that the line follows the overall pattern.

Is It a Good Trend Line? 🔽

A linear scatter plot rises from lower-left to upper-right. Judge each proposed line.

Finding the Slope from the Line

Once your trend line is drawn, pick two points that lie ON the line (corner gridpoints are easiest — they do not need to be real data dots). Then use the slope formula:

$m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$

Worked Example

Suppose a trend line passes through $(0, 2)$ and $(4, 10)$.

$m = \frac{10 - 2}{4 - 0} = \frac{8}{4} = 2$

The slope is $2$: every time $x$ increases by $1$, the line predicts $y$ increases by about $2$.

💡 A positive slope confirms a positive association; a negative slope confirms a negative one.

Find the Slope 🧮

Each trend line passes through the two given points. Find the slope $m = \dfrac{y_2 - y_1}{x_2 - x_1}$.

1) Through $(1, 5)$ and $(5, 25)$: $\;m = \,?$ 2) Through $(2, 8)$ and $(6, 20)$: $\;m = \,?$ 3) Through $(0, 30)$ and $(10, 10)$: $\;m = \,?$ (it falls, so $m$ is negative)

Finding the $y$-Intercept

The $y$-intercept $b$ is the $y$-value where the trend line crosses the $y$-axis — that is, where $x = 0$.

In the worked example through $(0, 2)$, the line already passes through the $y$-axis at $y = 2$, so $b = 2$.

Now you have both pieces of the line's equation: $y = mx + b = 2x + 2$

🔑 Slope-intercept form $y = mx + b$ packs the whole trend line into one equation: $m$ is the slope, $b$ is the $y$-intercept. In Part 4 we use this equation to make predictions.

Concept Check 🎯

📈 Scatter Plots and Trend Lines

Part 4 of 5 — Predicting & Interpreting

🔑 The Payoff: Once the trend line is written as $y = mx + b$, you can predict a $y$-value for any $x$, and you can explain what the slope and intercept mean in the real situation.

Making a Prediction

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

Worked Example

A study finds that the trend line for hours studied ($x$) vs. test score ($y$) is: $y = 8x + 40$

Predict the score for a student who studies $4$ hours: $y = 8(4) + 40 = 32 + 40 = 72$

The model predicts about a $72$.

💡 Predictions are estimates, not guarantees. A real student who studies $4$ hours might score $68$ or $77$ — the trend line gives the expected value, near the center of the dots.

Concept Check 🎯

Interpreting Slope and Intercept in Context

The two numbers in $y = mx + b$ carry real-world meaning:

💡 Always attach the units when you interpret: the slope here is "$8$ points per hour," not just "$8$."

⚠️ Interpolation vs. extrapolation. Predicting inside the range of the data (interpolation) is fairly safe. Predicting far outside it (extrapolation) is risky — the pattern may not continue. Predicting a score for $40$ hours of study with this line would be extrapolation.

Interpret the Model 🔽

A trend line for ice cream sales uses $x =$ temperature (°F) and $y =$ cones sold: $y = 12x + 30$

Predict It 🧮

Use each trend line to make the prediction.

1) $y = 5x + 3$. Predict $y$ when $x = 6$: $\,?$ 2) $y = -2x + 50$. Predict $y$ when $x = 10$: $\,?$ (a negative slope means $y$ falls) 3) $y = 4x + 12$. Each $1$-unit increase in $x$ raises $y$ by how much? $\,?$

📈 Scatter Plots and Trend Lines

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) build a scatter plot, (2) describe its association, (3) draw a trend line and find its equation, and (4) predict and interpret. Let's put it all together.

Quick Reference

⚠️ Remember: a trend line passes through the middle of the dots (not every dot), and predicting far outside the data (extrapolation) is risky.

Mixed Practice 🎯

Apply It 🧮

A trend line for minutes exercised ($x$) vs. calories burned ($y$) is: $y = 9x + 15$

1) Predict calories burned for $x = 10$ minutes: $\,?$ 2) How many calories does the model add per extra minute of exercise? $\,?$ 3) What does the model predict for $x = 0$ minutes (the $y$-intercept)? $\,?$

Exit Quiz ✅

Answer all three to finish the lesson.