When we display bivariate data that appears to have a linear relationship, we often wish to find a line that best models the relationship so we can see the trend and make predictions. We call this the line of best fit.
Exploration
We want to draw a line of best fit for the following scatterplot:
Let's try drawing three lines across the data and consider which is most appropriate.
We can tell straight away that $A$A is not the right line. This data appears to have a positive linear relationship, but $A$A has a negative gradient. $B$B has the correct sign for its gradient, and it passes through three points! However, there are many more points above the line than below it, and we should try to make sure the line of best fit passes through the centre of all the points. The means that line $C$C is the best fit for this data out of the three lines.
Drawing a line of best fit by eye
- One method is to draw an oval around the points on the scatterplot, then cut the oval in half with a line.
- The line may pass exactly through all of the points, some of the points, or none of the points.
- It always represents the general trend of the of the data (increasing or decreasing).
- The number of points above the line should be the same as the number of points below the line.
- You should generally ignore outliers (points that fall very far from the rest of the data) as they can skew the line of best fit.
Below is an example of what a good line of best fit might look like.
Practice questions
Question 1
The following scatter plot shows the data for two variables, $x$x and $y$y.
A scatter plot with both the x- and y-axes ranging from $0$0 to $10$10. Both axes are marked at intervals of $1$1. Eight data points are plotted as solid black dots on the scatter plot. The first data point is at $\left(1,2\right)$(1,2). The second data point is at $\left(2,1\right)$(2,1). The third data point is at $\left(3,3\right)$(3,3).The fourth data point is at $\left(4,5\right)$(4,5). The fifth data point is at $\left(5,6\right)$(5,6). The sixth data point is at $\left(6,5\right)$(6,5). The seventh data point is at $\left(7,7\right)$(7,7). The eighth data point is at $\left(8,7\right)$(8,7). The coordinates are not explicitly labeled or given.
Determine which of the following graphs contains the line of best fit.
A scatter plot with both the x- and y-axes ranging from $0$0 to $10$10. Both axes are marked at intervals of $1$1. Eight data points are plotted as solid black dots on the scatter plot. The first data point is at $\left(1,2\right)$(1,2). The second data point is at $\left(2,1\right)$(2,1). The third data point is at $\left(3,3\right)$(3,3).The fourth data point is at $\left(4,5\right)$(4,5). The fifth data point is at $\left(5,6\right)$(5,6). The sixth data point is at $\left(6,5\right)$(6,5). The seventh data point is at $\left(7,7\right)$(7,7). The eighth data point is at $\left(8,7\right)$(8,7). A straight green line runs diagonally from the lower left of the plot, upwards to the upper right of the scatter plot. The line passes though one of the plotted points. The plotted points are positioned closely along this line. Five points are above the line. Two points are below the line. The coordinates of the plotted points are not explicitly labeled. A
A scatter plot with both the x- and y-axes ranging from $0$0 to $10$10. Both axes are marked at intervals of $1$1. Eight data points are plotted as solid black dots on the scatter plot. The first data point is at $\left(1,2\right)$(1,2). The second data point is at $\left(2,1\right)$(2,1). The third data point is at $\left(3,3\right)$(3,3).The fourth data point is at $\left(4,5\right)$(4,5). The fifth data point is at $\left(5,6\right)$(5,6). The sixth data point is at $\left(6,5\right)$(6,5). The seventh data point is at $\left(7,7\right)$(7,7). The eighth data point is at $\left(8,7\right)$(8,7). A straight green line runs diagonally from the lower left of the plot, upwards to the upper right of the scatter plot. Four points are above the line. Four points are below the line. Only few points are plotted closely to the line. The coordinates of the plotted points are not explicitly labeled. B
A scatter plot with both the x- and y-axes ranging from $0$0 to $10$10. Both axes are marked at intervals of $1$1. Eight data points are plotted as solid black dots on the scatter plot. The data first point is at $\left(1,2\right)$(1,2). The second data point is at $\left(2,1\right)$(2,1). The third data point is at $\left(3,3\right)$(3,3).The fourth data point is at $\left(4,5\right)$(4,5). The fifth data point is at $\left(5,6\right)$(5,6). The sixth data point is at $\left(6,5\right)$(6,5). The seventh data point is at $\left(7,7\right)$(7,7). The eighth data point is at $\left(8,7\right)$(8,7). A straight green line runs diagonally from the lower left of the plot, upwards to the upper right of the plot. The line passes though one of the plotted points. The plotted points are positioned closely along the line. Two points are above the line. Five points are below the line. The coordinates of the plotted points are not explicitly labeled.
C
A scatter plot with both the x- and y-axes ranging from $0$0 to $10$10. Both axes are marked at intervals of $1$1. Eight data points are plotted as solid black dots on the scatter plot. The first data point is at $\left(1,2\right)$(1,2). The second data point is at $\left(2,1\right)$(2,1). The third data point is at $\left(3,3\right)$(3,3).The fourth data point is at $\left(4,5\right)$(4,5). The fifth data point is at $\left(5,6\right)$(5,6). The sixth data point is at $\left(6,5\right)$(6,5). The seventh data point is at $\left(7,7\right)$(7,7). The eighth data point is at $\left(8,7\right)$(8,7). A straight green line runs diagonally from the lower left of the plot, upwards to the upper right of the scatter plot. The plotted points are positioned very closely along this line. Four data points are above the line. Four data points are below this line. The coordinates of the plotted points are not explicitly labeled.
D
Question 2
The following scatter plot shows the data for two variables, $x$x and $y$y.
Determine which of the following graphs contains the line of best fit.
A scatter plot with an $x$x-axis labeled from $0$0 to $10$10 and a $y$y-axis labeled from $0$0 to $10$10. Both axes are in increments of $1$1 unit. Gray gridlines divide the plane into square units. Nine points are plotted on the grid: $\left(1,2\right)$(1,2), $\left(2,1\right)$(2,1), $\left(3,3\right)$(3,3), $\left(4,5\right)$(4,5), $\left(5,6\right)$(5,6), $\left(6,5\right)$(6,5), $\left(7,7\right)$(7,7), and $\left(8,7\right)$(8,7). A green line passes through the graph at an upward diagonal slope that follows the trend of the points, starting at point approximately $coord(0,0.7)$coord(0,0.7) and extending near the top-right corner at point approximately $coord(10,9.2)$coord(10,9.2). Points $\left(1,2\right)$(1,2),$\left(4,5\right)$(4,5),$\left(5,6\right)$(5,6), and $\left(7,7\right)$(7,7) are above the green line, while points $\left(2,1\right)$(2,1), $\left(3,3\right)$(3,3), $\left(6,5\right)$(6,5) and $\left(8,7\right)$(8,7) are below the green line. The points are plotted as solid blacks dots but the coordinates are not explicitly labeled nor stated in this problem. A
A scatter plot with an $x$x-axis labeled from $0$0 to $10$10 and a $y$y-axis labeled from $0$0 to $10$10. Both axes are in increments of $1$1 unit. Gray gridlines divide the plane into square units. Nine points are plotted on the grid: $\left(1,2\right)$(1,2), $\left(2,1\right)$(2,1), $\left(3,3\right)$(3,3), $\left(4,5\right)$(4,5), $\left(5,6\right)$(5,6), $\left(6,5\right)$(6,5), $\left(7,7\right)$(7,7), and $\left(8,7\right)$(8,7). A green line passes through the graph at an upward diagonal slope, starting at point approximately $coord(0,1.1)$coord(0,1.1) and extending near the top-right corner at point approximately $coord(10,9.7)$coord(10,9.7). Point $\left(1,2\right)$(1,2) lies on the green line, while points $\left(4,5\right)$(4,5), and $\left(5,6\right)$(5,6) are above the green line, and points $\left(2,1\right)$(2,1), $\left(3,3\right)$(3,3), $\left(6,5\right)$(6,5), $\left(7,7\right)$(7,7) and $\left(8,7\right)$(8,7) are below the green line. The points are plotted as solid blacks dots but the coordinates are not explicitly labeled nor stated in this problem. B
A scatter plot with an $x$x-axis labeled from $0$0 to $10$10 and a $y$y-axis labeled from $0$0 to $10$10. Both axes are in increments of $1$1 unit. Gray gridlines divide the plane into square units. Nine points are plotted on the grid: $\left(1,2\right)$(1,2), $\left(2,1\right)$(2,1), $\left(3,3\right)$(3,3), $\left(4,5\right)$(4,5), $\left(5,6\right)$(5,6), $\left(6,5\right)$(6,5), $\left(7,7\right)$(7,7), and $\left(8,7\right)$(8,7). A green line passes through the graph at an upward diagonal slope, starting at point approximately $coord(0,0.1)$coord(0,0.1) and extending near the top-right corner at point approximately $coord(10,8.8)$coord(10,8.8). Point $\left(8,7\right)$(8,7) lies on the green line, while points $\left(1,2\right)$(1,2), $\left(3,3\right)$(3,3), $\left(4,5\right)$(4,5), $\left(5,6\right)$(5,6), and $\left(7,7\right)$(7,7) are above the green line, and points $\left(2,1\right)$(2,1), and $\left(6,5\right)$(6,5) are below the green line. The points are plotted as solid blacks dots but the coordinates are not explicitly labeled nor stated in this problem. C
A scatter plot with an $x$x-axis labeled from $0$0 to $10$10 and a $y$y-axis labeled from $0$0 to $10$10. Both axes are in increments of $1$1 unit. Gray gridlines divide the plane into square units. Nine points are plotted on the grid: $\left(1,2\right)$(1,2), $\left(2,1\right)$(2,1), $\left(3,3\right)$(3,3), $\left(4,5\right)$(4,5), $\left(5,6\right)$(5,6), $\left(6,5\right)$(6,5), $\left(7,7\right)$(7,7), and $\left(8,7\right)$(8,7). A green line passes through the graph at an upward diagonal slope but does not follow the trend of the points, starting at point approximately $coord(0,1.5)$coord(0,1.5) and extending near the top-right corner at point approximately $coord(10,8.2)$coord(10,8.2). Points $\left(4,5\right)$(4,5),$\left(5,6\right)$(5,6), $\left(7,7\right)$(7,7) and $\left(8,7\right)$(8,7) are above the green line, while points $\left(1,2\right)$(1,2), $\left(2,1\right)$(2,1), $\left(3,3\right)$(3,3) and $\left(6,5\right)$(6,5) are below the green line. The points are plotted as solid blacks dots but the coordinates are not explicitly labeled nor stated in this problem. DUse the line of best fit to estimate the value of $y$y when $x=4.5$x=4.5.
A scatter plot with an $x$x-axis labeled from $0$0 to $10$10 and a $y$y-axis labeled from $0$0 to $10$10. Both axes are in increments of $1$1 unit. Gray gridlines divide the plane into square units. Nine points are plotted on the grid: $\left(1,2\right)$(1,2), $\left(2,1\right)$(2,1), $\left(3,3\right)$(3,3), $\left(4,5\right)$(4,5), $\left(5,6\right)$(5,6), $\left(6,5\right)$(6,5), $\left(7,7\right)$(7,7), and $\left(8,7\right)$(8,7). A green line with an equation of $y=(78/90)*x+(54/90)$y=(78/90)*x+(54/90) passes through the graph at an upward diagonal slope that follows the trend of the points. The equation of the line is not explicitly given nor stated in the graph and problem. Points $\left(1,2\right)$(1,2),$\left(4,5\right)$(4,5),$\left(5,6\right)$(5,6), and $\left(7,7\right)$(7,7) are above the green line, while points $\left(2,1\right)$(2,1), $\left(3,3\right)$(3,3), $\left(6,5\right)$(6,5) and $\left(8,7\right)$(8,7) are below the green line. The points are plotted as solid blacks dots but the coordinates are not explicitly labeled nor stated in this problem. $4.5$4.5
A$5$5
B$5.5$5.5
C$6$6
DUse the line of best fit to estimate the value of $y$y when $x=9$x=9.
$6.5$6.5
A$7$7
B$8.4$8.4
C$9.5$9.5
D