Step by step
The question
Question: find the line of best fit for (1, 12), (2, 13), (3, 16), (4, 17), (5, 14).
Then predict y when x is 8.
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: 15 divided by 5 is 3.
The mean of the y numbers: 72 divided by 5 is 14.4.
The line of best fit always goes through the mean point, (3, 14.4).
2 Make the table
A table with the points.
Each x minus the mean x, 3.
Each y minus the mean y, 14.4.
For the first point: 1 minus 3 is −2, and 12 minus 14.4 is −2.4.
The product: the two distances multiplied.
−2 times −2.4 is 4.8.
And the x distance squared.
−2 squared is 4.
We add the last two columns: 8 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.
8/10 is 0.8.
4 Find b
The line goes through the mean point.
So b is the mean y, minus m times the mean x.
0.8 times 3 is 2.4.
14.4 minus 2.4 is 12.
So the line of best fit is y = 0.8x + 12.
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 = 0.8x + 12: it passes close to all the points.
6 Predict
To predict, we put x = 8 into the line.
0.8 times 8 is 6.4.
Plus 12 is 18.4.
On the graph: at x = 8, the line is at 18.4.
7 Check
Let's check: the mean point must be on the line.
0.8 times 3 plus 12 is 14.4.
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.