Fit Lines to Scatter Plots
Help Questions
8th Grade Math › Fit Lines to Scatter Plots
A scatter plot shows the relationship between the number of laps swum ($x$) and total time in minutes ($y$). A student drew a line of best fit.
To judge how well the line fits, which distances should you mainly look at?
The horizontal distances from the points to the line (how far left or right each point is from the line).
Only whether the line touches at least two points.
The distance from each point to the origin.
The vertical distances from the points to the line (how far up or down each point is from the line).
Explanation
Tests informally fitting straight line to scatter plot showing linear association and assessing fit by judging point closeness to line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of laps swum vs time, assessing fit by measuring vertical distances from points to the line, judging smaller averages as better. The correct distances to look at are the vertical distances from the points to the line (choice A). A common error is using horizontal distances or distances to the origin, but vertical residuals matter for predicting y from x. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.
A science class measured the temperature outside ($x$ in °F) and the number of cups of lemonade sold at lunch ($y$). The scatter plot points trend upward in a roughly straight pattern.
Should a straight line be a reasonable model for this relationship?
No, because upward trends must be modeled with a curve, not a line.
Yes, because the best-fit line should always be horizontal to balance points.
Yes, because the points show a positive linear association (they follow an upward straight trend).
No, because a line is only reasonable if every point is exactly on the line.
Explanation
Tests informally fitting straight line to scatter plot showing linear association and assessing fit by judging point closeness to line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of temperature vs lemonade sales showing upward linear trend, fitting line through approximate center sloping upward, assessing fit by observing most points close to the line vertically (good fit). The correct assessment is yes, a straight line is reasonable because the points show a positive linear association following an upward straight trend (choice A). A common error is thinking a line is only reasonable if it passes through every point or that upward trends require a curve, but lines approximate linear trends even with scatter. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.
A student drew a line of best fit for a scatter plot of $x$ = practice minutes and $y$ = free-throw percentage. The student wants to check the fit.
Which method is the best way to judge whether the line fits well?
Check whether the line touches the highest and lowest points
Check whether the points are horizontally close to the line (small left-and-right distances)
Check whether the line passes through every point
Check whether the points are vertically close to the line (small up-and-down distances)
Explanation
This question tests informally fitting a straight line to a scatter plot showing linear association and assessing fit by judging point closeness to the line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of practice minutes vs free-throw percentage showing upward linear trend, fitting line through approximate center sloping upward, assessing fit by observing most points within ±5% of line vertically (good fit), or comparing two possible lines where one passes closer to majority of points (better fit). The correct method is to check whether the points are vertically close to the line (small up-and-down distances), as this measures prediction accuracy for y-values. A common error is judging by horizontal distances, but fit is assessed vertically since the line predicts y from x. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.
A scatter plot shows the relationship between number of pages read ($x$) and minutes spent reading ($y$). A student drew the line $y=2x+5$ as a model.
Data points: $(5,16), (6,17), (7,19), (8,21), (9,24), (10,25), (11,27), (12,29), (13,31), (14,33)$
How good is the fit of the line $y=2x+5$?
Poor fit, because many points are about 10–20 minutes away from the line
Good fit, because most points are within about 1–2 minutes vertically from the line
Poor fit, because the line must pass through every point to be a good fit
Good fit, because you should judge fit using how close the points are horizontally to the line
Explanation
This question tests informally fitting a straight line to a scatter plot showing linear association and assessing fit by judging point closeness to the line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of pages read vs minutes spent reading showing upward linear trend, fitting line through approximate center sloping upward, assessing fit by observing most points within ±2 minutes of line vertically (good fit), or comparing two possible lines where one passes closer to majority of points (better fit). The correct assessment is a good fit, because most points are within about 1–2 minutes vertically from the line, indicating accurate predictions. A common error is claiming poor fit because the line doesn't pass through every point, but lines approximate trends in scatter plots, allowing for some deviation. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.
A class measured how many pages they read ($x$) and how many minutes it took ($y$). The scatter plot shows a clear positive linear trend.
A student drew the line $y=2x+5$ on the scatter plot. Most points are within about 3 minutes vertically of the line.
How would you describe the fit of this line?
Good fit, because most points are close to the line (small vertical distances).
Poor fit, because the line does not pass through every point.
Poor fit, because the points are above the line more often than below it.
Good fit, because the points are spread out widely from the line.
Explanation
Tests informally fitting straight line to scatter plot showing linear association and assessing fit by judging point closeness to line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of pages read vs minutes taken showing positive linear trend, fitting line through approximate center sloping upward, assessing fit by observing most points within ±3 minutes of line vertically (good fit), or comparing to alternatives where points are farther (poorer fit). The correct assessment is good fit because most points are close to the line with small vertical distances (choice A). A common error is claiming poor fit because the line does not pass through every point, but scatter plots show variability and lines approximate the trend. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.
A student drew a line of best fit for a scatter plot of (x = number of texts sent, y = phone battery percent used). One point is far away from the rest (an outlier). Which choice best describes how the line should be drawn?
Draw a horizontal line, because outliers mean there is no relationship at all.
Draw the line to fit the main cluster of points, without letting the one outlier determine the slope.
Draw a curved line so it hits every point exactly, including the outlier.
Draw the line through the outlier and ignore the rest of the points.
Explanation
This question tests informally fitting a straight line to a scatter plot showing linear association and assessing fit by judging point closeness to the line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of texts sent vs battery used with one outlier, fitting line through the main cluster's center, assessing fit by closeness of majority points, ignoring the outlier's pull. The best approach is to draw the line to fit the main cluster of points, without letting the one outlier determine the slope. Common errors include drawing through the outlier only or using a curved or horizontal line. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.
A scatter plot shows a positive linear association between the number of books checked out (x) and the total minutes spent reading (y). A student drew a line that goes above almost every point (most points are below the line). What is the best way to improve the line of best fit?
Move the line downward so it goes through the middle of the points, with a more balanced number of points above and below.
Rotate the line so it slopes downward, because lines of best fit should always slope down.
Ignore most points and draw the line through only the two highest points.
Make the line pass through every single point, even if it changes direction.
Explanation
This question tests informally fitting a straight line to a scatter plot showing linear association and assessing fit by judging point closeness to the line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of books checked out vs minutes reading showing upward linear trend, if a line is above most points, adjust it downward through the center for balance, improving fit by reducing vertical distances. The best improvement is to move the line downward so it goes through the middle of the points, with a more balanced number of points above and below. Common errors include forcing the line through all points or ignoring most points for extremes. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.
A student plotted the number of push-ups completed (x) and the time in seconds to finish (y). The points show a negative linear trend (more push-ups means more time). Two lines are suggested. Which line is the better fit?
Line B, because it is as steep as possible and passes through an extreme point.
Line A is worse, because a best-fit line must pass through every point exactly.
Line A, because it goes through the middle of the point cloud and leaves about the same number of points above and below.
Line B, because a best-fit line should make all points fall above the line.
Explanation
This question tests informally fitting a straight line to a scatter plot showing linear association and assessing fit by judging point closeness to the line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of push-ups vs time showing downward linear trend, fitting line through approximate center sloping downward, assessing fit by observing most points within ±5 seconds of line vertically (good fit), or comparing two possible lines where one passes closer to majority of points (better fit). The correct choice is Line A, because it goes through the middle of the point cloud and leaves about the same number of points above and below, capturing the negative trend. Common errors include prioritizing extreme points or expecting the line to pass through every point. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.
A scatter plot shows (x = number of chores completed, y = allowance earned). Two students drew different lines. Student 1’s line has many points about 1–2 dollars away from it. Student 2’s line has many points about 5–6 dollars away from it. Which line is a better fit, and why?
Both lines are equally good, because a best-fit line must have all points on one side of it.
Student 1’s line, because the vertical distances (residuals) from the points to the line are smaller for most points.
Student 2’s line, because it is steeper and steeper lines are always better.
Student 2’s line, because being farther from the points means the line is more accurate.
Explanation
This question tests informally fitting a straight line to a scatter plot showing linear association and assessing fit by judging point closeness to the line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of chores vs allowance, comparing two lines where one has smaller vertical distances (1-2 dollars away) vs larger (5-6 away), the one with smaller residuals fits better. Student 1’s line is better, because the vertical distances (residuals) from the points to the line are smaller for most points. Common errors include preferring steeper lines or thinking farther distances are more accurate. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.
A scatter plot shows the relationship between the number of text messages sent in a day ($x$) and the number of minutes spent on the phone ($y$). A student drew the line $y=-2x+120$.
The points on the scatter plot clearly trend upward from left to right.
What is the best critique of the student’s line?
The line is wrong because it crosses the $y$-axis.
The slope is the wrong direction; the line should have a positive slope to match the upward trend.
The line is correct because negative slopes always balance points above and below.
The line is wrong because a best-fit line must go through every point.
Explanation
Tests informally fitting straight line to scatter plot showing linear association and assessing fit by judging point closeness to line. Linear association (points trending along straight direction) modeled by straight line: fit line informally by drawing through middle of point cloud, balancing points above and below (roughly equal numbers each side). Assess fit: good fit has most points close to line (small vertical distances from points to line—predictions accurate), poor fit has points far from line (large vertical deviations—predictions less reliable). Line captures linear trend, allows prediction (use line to estimate y for new x). For example, in a scatter plot of text messages vs phone minutes with upward trend, a downward-sloping line fails to capture the positive association. The best critique is that the slope is the wrong direction; the line should have a positive slope to match the upward trend (choice A). A common error is accepting a negative slope because it balances points or crosses the y-axis, but the slope must match the trend direction. Fitting: (1) observe trend direction (upward→positive slope, downward→negative), (2) estimate center of points (where is middle of cloud?), (3) draw line through center following trend (balance points above/below), (4) verify reasonable (does line capture pattern? points fairly close?). Assessing: (1) observe vertical distances from points to line (how far off?), (2) count how many close (within 1-2 grid squares) vs far (5+ squares away), (3) judge: most close=good fit, many far=poor fit, (4) compare alternatives (if multiple lines, which has points closer on average?). Real data: rarely perfect (scatter means variability, line approximates), outliers exist (don't force line to hit outlier—fit majority), linear adequate if points roughly straight trend (even if imperfect). Mistakes: line through outlier ignoring majority, all points above or below (unbalanced), fit by horizontal distance (wrong—vertical distance for y-prediction matters), claiming poor fit is good or vice versa.