Step by step
The question
Question: find the line of best fit for (1, 8), (2, 7), (3, 6), (4, 9), (5, 10).
Then predict y when x is 10.
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: 40 divided by 5 is 8.
The line of best fit always goes through the mean point, (3, 8).
2 Make the table
A table with the points.
Each x minus the mean x, 3.
Each y minus the mean y, 8.
For the first point: 1 minus 3 is −2, and 8 minus 8 is 0.
The product: the two distances multiplied.
−2 times 0 is 0.
And the x distance squared.
−2 squared is 4.
We add the last two columns: 6 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.
6/10 is 0.6.
4 Find b
The line goes through the mean point.
So b is the mean y, minus m times the mean x.
0.6 times 3 is 1.8.
8 minus 1.8 is 6.2.
So the line of best fit is y = 0.6x + 6.2.
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 = 0.6x + 6.2: it passes close to all the points.
6 Predict
To predict, we put x = 10 into the line.
0.6 times 10 is 6.
Plus 6.2 is 12.2.
On the graph: at x = 10, the line is at 12.2.
7 Check
Let's check: the mean point must be on the line.
0.6 times 3 plus 6.2 is 8.
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.