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### Course: Digital SAT Math>Unit 7

Lesson 6: Scatterplots: medium

# Scatterplots | Lesson

A guide to scatterplots on the digital SAT

## What are scatterplots?

A scatterplot displays data about two variables as a set of points in the $xy$-plane. Each axis of the plane usually represents a variable in a real-world scenario.
In this lesson, we'll learn to:
1. Use the line of best fit to describe scatterplots
2. Make predictions using the line of best fit
3. Fit functions to scatterplots
This lesson builds upon the following skills:
• Data representations
• Graphs of linear equations and functions
• Exponential graphs
You can learn anything. Let's do this!

## How do we talk about scatterplots?

### Bivariate relationship linearity, strength and direction

Bivariate relationship linearity, strength and directionSee video transcript

## What is the line of best fit?

### Interpreting a trend line

Interpreting a trend lineSee video transcript

### The line of best fit

While each point in a scatterplot represents a specific observation, the line of best fit describes the general trend based on all of the points.
For a given data point, we expect to see a difference between its $y$-value and the $y$-value predicted by the line of best fit. These differences are used for more advanced statistical analysis; for the SAT, we only need to calculate the difference.
We can also interpret the slope and $y$-intercept of the line of best fit the same way we interpret line graphs:
• The slope represents a constant rate of change.
• The $y$-intercept represents an initial value.

### Try it!

Try: find the difference between predicted and actual values
A scatterplot and its line of best fit are shown in the $xy$-plane above.
The line of best fit passes through the point $\left(15,$
$\right)$.
Point $m$ has the coordinates $\left(15,$
$\right)$.
The positive difference in $y$-value between the data point and the line of best fit is
.

TRY: Interpret the meaning of the line of best fit
The scatterplot above shows the relative housing cost and the population density for several large US cities in the year $2005$. The equation of the line of best fit is $y=0.0125x+61$.
The constant $61$ means that when the population density is $0$ people per square mile of land area, the relative housing cost is
.
The coefficient $0.0125$ means that as the population density increases by $1,000$ people per square mile land area, the relative housing cost increases by
of the national average cost.

## How do I use the line of best fit to make predictions?

### Line of best fit: smoking in 1945

Line of best fit: smoking in 1945See video transcript

### Predicting what we can and cannot see

When making predictions based on scatterplots, always use the line of best fit instead of individual data points.
If the prediction lies within the part of the $xy$-plane shown, it must lie on the line of best fit.
If the prediction lies beyond the part of the $xy$-plane shown, we can either extend the line of best fit or use its equation to find the prediction.

### Try it!

Try: predict using the line of best fit
The scatterplot above shows the relative housing cost and the population density for several large US cities in the year $2005$. The equation of the line of best fit is $y=0.0125x+61$.
According to the graph, the predicted relative housing cost for a population density of $15,000$ people per square mile land area is approximately
of the national average cost.
According to the equation of the line of best fit, the predicted relative housing cost for a population density of $5,000$ people per square mile land area is
of the national average cost.

## How do I fit functions to scatterplots?

### Use direction and intercepts to determine the best fit

On the SAT, questions that ask you to fit a function to a scatterplot are always multiple choice, and all four choices are usually functions of the same type, e.g., four linear functions or four quadratic functions.
For linear functions in the form $f\left(x\right)=mx+b$:
• Sketch a line that fits the data and approximate its slope.
• The value of $m$ should match the slope. Make sure to pay attention to the signs!
• Approximate the $y$-intercept of the function that best fits the data. Make sure the constant term $b$ matches the $y$-intercept.
For quadratic functions in the form $f\left(x\right)=a{x}^{2}+bx+c$:
• Sketch a parabola and approximately fits the data.
• If the parabola opens upward, $a$ should be positive. If the parabola opens downward, $a$ should be negative.
• Approximate the $y$-intercept of the function that best fits the data. Make sure the constant term $c$ matches the $y$-intercept.

### Try it!

Try: describe a modeling function for a scatterplot
The scatterplot above shows the foot lengths and shoulder heights of the elephants in Kruger National Park in South Africa.
According to the scatterplot, as foot length increases, shoulder height generally
. Therefore, the slope of the line of best fit for this scatterplot is
.
If we sketch a line of best fit for the scatterplots, the $y$-intercept of the line would be close to $0$ and slightly

Practice: find the difference between data and prediction
The scatterplot above shows the dimensions of $12$ picture frames on Lee's wall along with the line of best fit. Which of the following statements about the widest picture frame is true?

Practice: interpret the line of best fit
A panel is rating different kinds of potato chips. The scatterplot above shows the relationship between their average ratings and the price of the chips. The line of best fit for the data is also shown. According to the line of best fit, which of the following is closest to the predicted increase in average rating for every $\mathrm{}0.10$ increase in price?

Practice: predict using the line of best fit
The scatterplot above shows data from a random sample of people who reported the age and mileage of their cars. A line of best fit for the data is also shown. Based on the line of best fit, which of the following is closest to the predicted mileage, in thousands of miles, of a car that is $13$ years old?

Practice: fit a quadratic function to a scatterplot
The scatterplot above shows $y$, the number of employees remaining in an office building, $x$ hours after the building's air conditioning stopped working. Of the following equations, which best models the data in the scatterplot?

## Want to join the conversation?

• In the last example we want to know the sign of the quadratic function. So let's suppose that this is the function : ax²+8x+c
First, we are going to find a :
You can see through the scatterplots above that as x increases, y decreases. That means that x is negative, right? (Ask me if you don't understand)
So we got a=-7 (not 7 because it is negative)
Now, let's find the c :
We know that c = y-intercepts
At x=0 , y= 300 (not -300)
So we have c=300
So the equation is y = -7x²+8x+300
I hope that help :)
• where's 8x from please, and how did you find the slope??
• I just want to draw straight lines like Sal does.
• can someone help me with the potato chip question pls
• Ok, so first we want to find the slope of the line of best fit. (Let me know if you need help with that). We find that the slope is 5/1. That means that the average rating goes up by 5 for every $1.00 increase. Ok, great, but we need to find how much it goes up for every$0.10 increase.
Well, to get from $1.00 to$0.10 we divided by 10, right?
So now we need to divide 5 by 10. And we get 0.5
That means that for every \$0.10 increase we get an 0.5 increase in average rating.
Does this help?
• Guessing the line between the dots=frustrating
• it's so subjective too it would never be on the SAT
• Tell me if I’m wrong, but it can be broken down to two things on the SAT: a frown face is -ax^2 and a smile is ax^2 (I’m referring to the shape of the curve - frown 🙁negative and smile 🙂positive) and the y intercept is -c if below x axis and c if above.
• Yeah, you understood it right!
• how are we going to draw the line of best fit with a pc, please?
• You won't have to draw a line of best fit on the dsat. You might have to choose between several graphs and pick which one is the best line of best fit, but you'll never have to actually draw one out.
• guys is it enough? for studying all lesson SAT in 2 months? I am geting test on November? T_T
• its a good thing to finish the sat courses on khan academy but it would be more beneficial if you practice and slove more
also you should do the full lengh tests on blue book
• The beard question in the quiz section almost killed me of laughter hahahahha
• Hahahaha sameee
• How can we formulate quadratic equations out of graphs? I don't think there has been a lesson regarding this till now
• We won't have to. We will have to choose between the options given to us, to find the quadratic equation most suitable to the scatterplot
• In the last example we want to know the sign of the quadratic function. So let's suppose that this is the function : ax²+8x+c
First, we are going to find a :
You can see through the scatterplots above that as x increases, y decreases. That means that x is negative, right? (Ask me if you don't understand)
So we got a=-7 (not 7 because it is negative)
Now, let's find the c :
So we have c=300