Step by step
The question
Question: find the line of best fit for (0, 4), (1, 1), (2, 5), (3, 6), (4, 9).
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 5 is 2.
The mean of the y numbers: 25 divided by 5 is 5.
The line of best fit always goes through the mean point, (2, 5).
2 Make the table
A table with the points.
Each x minus the mean x, 2.
Each y minus the mean y, 5.
For the first point: 0 minus 2 is −2, and 4 minus 5 is −1.
The product: the two distances multiplied.
−2 times −1 is 2.
And the x distance squared.
−2 squared is 4.
We add the last two columns: 15 and 10.
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.
15/10 is 1.5.
4 Find b
The line goes through the mean point.
So b is the mean y, minus m times the mean x.
1.5 times 2 is 3.
5 minus 3 is 2.
So the line of best fit is y = 1.5x + 2.
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 = 1.5x + 2: it passes close to all the points.
6 Predict
To predict, we put x = 6 into the line.
1.5 times 6 is 9.
Plus 2 is 11.
On the graph: at x = 6, the line is at 11.
7 Check
Let's check: the mean point must be on the line.
1.5 times 2 plus 2 is 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.