What’s the BEST line that describes the pattern? Scatter Plots and Best Fitting Lines
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- Опубліковано 6 лют 2025
- How to find the best fitting line in a scatter plot. Learn more math at TCMathAcademy.....
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When we take the points (-5, -3) and (3, 7) it is easy to formulate the equation of the line that goes 'through' both points. First we need to formulate two equations: - 3 = m.-5 + b and 7 = m.3 + b. Then subtract the second equation from the first and we get y = - m.8 = - 10. So 8m = 10 => m = 10/8 = 5/4. Then b = 7 - (5/4 x 3) = 28/4 - 15/4 = 13/4. Thus the equation of the line is this linear function: y = 5/4x + 13/4.. Plug these numbers in the first equation and you get - 3 (y) = 5/4 (m) x - 5 (x) = - 25/4 + 13/4 (b). And that is correct too. :).
The line doesn't need to start or end at or pass through any points. What matters is that points are relatively evenly distributed either side of the line, allowing for outliers
Using Excel lineair regression I find these 2 points : (-3,0) and (5,4)
Slope m = dy / dx = ( 4 - 0 ) / ( 5 - ( -3 ) ) = 4 / 8 = 1/2
Crossing y-axle in starting from (-3,0) so 3 to the right then y = 0 + 1/2 . 3 = 3/2
The best estimate for the trendline is : y = 1/2 x + 3/2 (or y = 0.5 x + 1.5 )
Oh yeah, regression lines, best mathematical fit for the points. This took me back to statistics for a minute.
Just one question: what is the best line that describes the pattern (and the pattern is based on all the points and not just on two of them)?
Why don't you answer your own question?
Here's an idea: add up all the x and y values. You get (-6, 13). Divide by 9. You get (-2/3, 13/9). It seems to me that that would be the center of the plot, and so it would be on the trend line. Then you would have to determine the slope, and I don't know how to do that.
I remembered the term "Least Squares Regression" while watching this. Based on your points, I came up with a slope of about .5 and none of the points were included on the line.
I think there is a fundamental fallacy, here, that the trend line needs to go through _any_ of the points on the graph. After choosing your two end points you counted the other points, three above the line and four below it (not "three or four"). But I noticed that the three above the line were collectively closer to the line than the four below it, so the trend line should either be a little lower or tilted a little bit down on the right end.
Since we're choosing our starting points just by gut feeling, let's use a little logic and choose our own end points which actually match the scatter a little better. Like maybe (-7, -1) and (8, 6).
Also, when you are demonstrating this, I really think you should use an actual graph. You didn't do a very good job of plotting those points. On the positive side, though, you make some of the best freehand parabolas I've ever seen!