Desmos for the SAT · Technique 22 of 23

Find the Residual for a Data Point

The question gives you the model. Evaluate it at the x the question names, then subtract observed minus predicted.

SolvesWhat is the residual for this data point?Observed minus predictedFind the observed value from a residualCompare two models at the same point

Watch the lesson

The whole move worked on one real SAT question, narrated.

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The line of best fit is already given, so the work is one evaluation and one subtraction, with the sign attached.

A residual is how far one real data point misses the model, and which direction it missed. The definition is one line: residual = observed minus predicted. Every SAT question on this move is that one line with different numbers in it.

One thing to know before you start, because it changes what you do first. These questions almost never ask you to fit anything. The line of best fit arrives already written, as an equation in the question, along with one data point that really happened. So there is no table to build and no regression to run. You evaluate a model that someone else already produced.

In Desmos the essentials are two typed lines, name the model and then subtract, plus a plotted point if you want to see which side of the line the data fell on. The reason to type it rather than work it on paper is not the arithmetic, it is the sign. Subtract the two numbers the other way round and you get the same size with the opposite sign, and that reversed value is printed among the answer choices every time.

Nova

These questions look like data questions and they are not. The line of best fit is already printed, so you evaluate the model at one x, subtract, and the answer is the gap. Be careful which number goes first, because the other order is always one of the four choices.

Now you try it

The rest of the page takes that move slowly. First how it compares with doing the algebra, then the steps in order, then a real SAT question with a live Desmos calculator to try it in.

The algebra way vs the Desmos way

This technique sidesteps substituting by hand and keeping the subtraction in the right order.

The question

The line of best fit is $y=4.5x+62$ , and one observed point is $(8,95)$

What is the residual for that point?

The algebra way

Multiply: 4.5(8) = 36.
Add the intercept: 36 + 62 = 98, which is the predicted score.
Subtract, in the right order: 95 - 98 = -3.
Reverse those two numbers by accident and the value you land on is also printed in the choices.

The Desmos way

Type f(x) = 4.5x + 62.
Type 95 - f(8) and read the -3.

The payoff: Both routes do the same three operations. The difference is that Desmos holds the model and the arithmetic, so the only thing left for you to get right is which number you type first.

Quick reference

What the question hands you, and the line that finishes it

You are given	What you type	What comes back
A model and an observed point	`f(x) = ...`, then `observed - f(x)`	The residual, signed
A model and a residual	`f(x) = ...`, then `f(x) + residual`	The observed value
Two models and one observed point	both models named, then one subtraction each	Two different residuals for the same point
A point that lands on the model	nothing to type	A residual of zero
A residual plot and a question about its shape	nothing to type	Scattered means the model fits; a curve or a U means it does not

Step by step

The whole move, all 7 steps

Type the model the question printed, as a named function: f(x) = .... Copy it exactly, decimals and all. Desmos stores the name and draws the line at the same time.
Find the two numbers the question is actually handing you: the x it names, and the observed y at that x. A point, a table row, or a sentence about one measurement all carry the same pair.
Plot the observed point on its own line as an ordered pair, so you can see which side of the model it falls on.
Type the prediction as a point, (x, f(x)), using the x the question named. Desmos plots it on the model and prints its coordinates, which is the predicted value.
Do the subtraction on one line: the observed value, a minus sign, then f( the named x ). Desmos returns the residual with its sign already attached.
Read the sign against the picture before you answer. Observed point above the model is positive, below is negative, and on it is zero.
Given the residual and asked for the observed value instead? Add: observed = predicted + residual, which is f( the named x ) plus the residual on one line.

The question

A researcher collected data on the relationship between study time, $x$ hours, and test scores, $y$ points. The line of best fit is $y=4.5x+62$ . When $x=8$ , the actual test score was 95.

What is the residual for this data point?

A $-3$
B $3$
C $-5$
D $5$

Step 1 of 6

Type the line of best fit as a named function: f(x) = 4.5x + 62. Copy it exactly as printed, decimal and all.

In the calculator: f(x)=4.5x+62

Naming it f is what makes every line after this one short.

Syntax: Name the model f, then call it: f(8) is the predicted value at x = 8, (8, f(8)) is the predicted point, and 95 - f(8) is the residual. The letter is yours to pick, so use one the question is not already using.

Now try it on a real SAT question

Head or scratch paper. The prediction here is 5(6) + 60, or 90, which is faster in your head than at the keyboard. Knowing which questions to read instead of type is worth as many points as the typing is, and this is one of them. The order of the subtraction is still the thing to slow down for.

A data set shows the relationship between hours of study, $x$ , and test score, $y$ , for 10 students. The line of best fit is $y=5x+60$ . One student studied for 6 hours and scored 85 on the test.

What is the residual for this student's data point?

Pick an answer to get instant coaching from Nova.

Stuck?

Your turnstarts empty

Set this question up yourself, then read the answer straight off the graph.

Worked it out on scratch paper? Type it into Desmos anyway and check. It takes a couple of seconds and it catches the slip you cannot spot by re-reading your own work.

Straight from Desmos

Desmos Help Center: Functions and Function NotationNaming a model and calling it at a value, which is the whole mechanic on this page.

The Desmos trap: when to close the calculator

The question nearly always prints the line of best fit for you. If you find yourself typing data into a table, re-read it: building a fit from raw data is a different move, and it lives in Fit a Line or Curve to a Table of Values.
Type the observed value first. Observed minus predicted is the definition, the reversed order gives the same size with the opposite sign, and that reversed value is printed among the choices.
When the question shows a residual plot and asks what its shape means, there is nothing to type. Scattered above and below zero means the model fits; a clear curve or U-shape means another model fits better.
Check what is being asked for. Some questions hand you the residual and want the observed value back, which is an addition rather than a subtraction.
A residual belongs to one model. Swap the model and the same observed point has a different residual, which is exactly what a two-model question is testing.
Evaluate the model at the same x as the observed point. A prediction taken at the wrong x is a number with nothing to compare it to.

Nova

If the numbers are small and whole, do the prediction in your head and move on. The calculator is worth the trip when there are decimals, an exponent, or a second model to keep straight.

All Desmos techniques

Put this move into real practice

"Find the Residual for a Data Point" shows up all over the real SAT. SAT Prep Quest drills the skill at your level, so reaching for it becomes automatic by test day.

Practice free

SAT is a registered trademark of the College Board, which was not involved in producing, and does not endorse, this product. Score estimates are Learner Labs projections, not official College Board scores.