They cross at the mean point, they are not the same line, and using the wrong one is a mark.
The same eight points, with both least squares lines drawn. Slide the value of y you want to predict x from, and watch the two answers separate.
Both lines are correct. They answer different questions.
y on x minimises the vertical distances. Use it to predict y from a known x.
x on y minimises the horizontal distances. Use it to predict x from a known y. It is a different line because it is minimising different gaps.
Multiply the two gradients together and you get r². Here 1.405 × 0.4917 = 0.691 = r². So when r = ±1 the product is 1, the lines are inverses of each other, and they coincide. The weaker the correlation, the wider they open. The gap between the two lines is a picture of how much r is missing.
Both lines pass through (x̅, y̅). So at the mean they agree exactly, and the disagreement grows the further out you go. That is the same reason extrapolation is unsafe, arriving from a different direction.
1. The mean point is (5.5, 10.5). Predict x when y = 10.5 using the x on y line.
2. The two gradients are 1.405 and 0.4917. Find r² to 3 decimal places.
3. A question gives you a height and asks for the predicted arm span. You should use:
Choosing the line by what you are predicting, not by what the question mentions first. The thing you want is on the left of the "on".
Not rearranging one line to do the other's job. It is the commonest error here and it produces a plausible, wrong number.
Saying when a prediction is unreliable: weak r, or a value well outside the data.
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