Step by step
The question
Question: find the line of best fit for (1, 3), (3, 6), (5, 10), (7, 10), (9, 12).
Then predict y when x is 11.
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: 25 divided by 5 is 5.
The mean of the y numbers: 41 divided by 5 is 8.2.
The line of best fit always goes through the mean point, (5, 8.2).
2 Make the table
A table with the points.
Each x minus the mean x, 5.
Each y minus the mean y, 8.2.
For the first point: 1 minus 5 is −4, and 3 minus 8.2 is −5.2.
The product: the two distances multiplied.
−4 times −5.2 is 20.8.
And the x distance squared.
−4 squared is 16.
We add the last two columns: 44 and 40.
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.
44/40 is 1.1.
4 Find b
The line goes through the mean point.
So b is the mean y, minus m times the mean x.
1.1 times 5 is 5.5.
8.2 minus 5.5 is 2.7.
So the line of best fit is y = 1.1x + 2.7.
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 = 1.1x + 2.7: it passes close to all the points.
6 Predict
To predict, we put x = 11 into the line.
1.1 times 11 is 12.1.
Plus 2.7 is 14.8.
On the graph: at x = 11, the line is at 14.8.
7 Check
Let's check: the mean point must be on the line.
1.1 times 5 plus 2.7 is 8.2.
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.