Step by step
The question
Question: find the line of best fit for (1, 3), (2, 5), (3, 6), (4, 8).
Then predict y when x is 6.
The line of best fit is the straight line that comes closest to all the points.
1 Find the means
The mean of the x numbers: 10 divided by 4 is 2.5.
The mean of the y numbers: 22 divided by 4 is 5.5.
The line of best fit always goes through the mean point, (2.5, 5.5).
2 Make the table
A table with the points.
Each x minus the mean x, 2.5.
Each y minus the mean y, 5.5.
For the first point: 1 minus 2.5 is −1.5, and 3 minus 5.5 is −2.5.
The product: the two distances multiplied.
−1.5 times −2.5 is 3.75.
And the x distance squared.
−1.5 squared is 2.25.
We add the last two columns: 8 and 5.
3 Find the slope
The slope is the total of the products, over the total of the squares.
Careful!
The bottom uses only the x distances, squared.
Not the y distances.
8/5 is 1.6.
4 Find b
The line goes through the mean point.
So b is the mean y, minus m times the mean x.
1.6 times 2.5 is 4.
5.5 minus 4 is 1.5.
So the line of best fit is y = 1.6x + 1.5.
5 Draw the line
Now the points on a graph.
Each point is a dot.
The line goes through the mean point.
The line y = 1.6x + 1.5: it passes close to all the points.
6 Predict
To predict, we put x = 6 into the line.
1.6 times 6 is 9.6.
Plus 1.5 is 11.1.
On the graph: at x = 6, the line is at 11.1.
7 Check
Let's check: the mean point must be on the line.
1.6 times 2.5 plus 1.5 is 5.5.
That is the mean y.
So it is right!
Let's look back.
First: find the means.
Then: make the table.
Then: find the slope.
Then: find b.
Then: draw the line.
Then: predict.
Last: check.