Soft question - determining how the line of best fit is drawn Watch

Tolgarda
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Dear chemists of TSR,

I have a question that pertains to a discrepancy regarding the line of best fit's appearence when plotting data. I am aware that this is probably a KS2/3 skill that should really be second nature, but I have dropped marks because of errors when drawing the line of best fit to make predictions or even complete a scatter graph. I'll explain the discrepancy below.

For AQA's second AS paper in June 2017, question 4.4 asked students to draw a line of best fit, and the mark scheme specifies that a smooth curve merits credit (the underlining of the word 'curve' denoting its necessity in gaining the mark). In contrast, when Edexcel's first AS paper for the same year also asked for a line of best fit to be drawn in question 8(b)(i), the mark scheme stated that the line of best fit should be straight (even having a picture showing a straight line of best fit as additional guidance).

Why does this contradiction exist? Is there a way that I can determine whether the line of best fit should be a 'smooth curve' or 'straight'? I always draw a straight line, and it seems like I am not alone, as even AQA's examiners' report for the paper mentions how 'a very significant proportion of them [students] drew a best fit line that was straight rather than a curve', and Edexcel mentions for its question that '[i]n general...lines of best fit were appropriate'.

I know it seems like a silly question at this level, but these are easy marks that I don't really feel I can afford to lose.

Thanks in advance for any answers.
Last edited by Tolgarda; 4 months ago
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BobbJo
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For question 4.4 in June 2017, the values follow a curve when plotted. This is because energy is required to convert the substances burnt from their standard states to gaseous states. Since the points follow a curve, a curve of best fit is drawn. A line can be straight or curved.

For question 8(b)(i) in the second paper, the values clearly follow a line. This is to be expected since the volume is directly proportional to the mass. So a straight line is drawn.

You have to look at the trend which the points follow and draw a line which is appropriate. If the relation is linear, a straight line is drawn. If not, a smooth curve is drawn.

I made a scatter plot for question 4.4:
https://imgur.com/a/4FfK2Wp
A straight line is clearly not appropriate
A smooth curve needs to be drawn.
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Tolgarda
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(Original post by BobbJo)
For question 4.4 in June 2017, the values follow a curve when plotted. This is because energy is required to convert the substances burnt from their standard states to gaseous states. Since the points follow a curve, a curve of best fit is drawn. A line can be straight or curved.

For question 8(b)(i) in the second paper, the values clearly follow a line. This is to be expected since the volume is directly proportional to the mass. So a straight line is drawn.

You have to look at the trend which the points follow and draw a line which is appropriate. If the relation is linear, a straight line is drawn. If not, a smooth curve is drawn.

I made a scatter plot for question 4.4:
https://imgur.com/a/4FfK2Wp
A straight line is clearly not appropriate
A smooth curve needs to be drawn.
I see, but in mathematics, this is not the case. Even if there is a strong positive correlation, the line of best fit always seems to have been straight rather than curved. Why is it different here?
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BobbJo
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(Original post by Tolgarda)
I see, but in mathematics, this is not the case. Even if there is a strong positive correlation, the line of best fit always seems to have been straight rather than curved. Why is it different here?
What do you mean by "in mathematics, this is not the case."?

A line of best fit does not need to be a straight line. A line of best fit can be a straight line or a curved line.

The line of best fit is the line which fits the data the best.

In statistics, we can not only do linear regression, but also non-linear regression (curve fitting).

http://www.math-mate.com/chapter39_2.shtml
https://blog.minitab.com/blog/advent...ear-regression
https://en.wikipedia.org/wiki/Curve_fitting

You can draw a linear graph, quadratic, exponential, power law for any data set, as shown in the links above. But we are looking for the graph which will fit the data the best and this line should be chosen as the line of best fit.
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Tolgarda
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(Original post by BobbJo)
What do you mean by "in mathematics, this is not the case."?

A line of best fit does not need to be a straight line. A line of best fit can be a straight line or a curved line.

The line of best fit is the line which fits the data the best.

In statistics, we can not only do linear regression, but also non-linear regression (curve fitting).

http://www.math-mate.com/chapter39_2.shtml
https://blog.minitab.com/blog/advent...ear-regression
https://en.wikipedia.org/wiki/Curve_fitting

You can draw a linear graph, quadratic, exponential, power law for any data set, as shown in the links above. But we are looking for the graph which will fit the data the best and this line should be chosen as the line of best fit.
Well, I'm definitely enlightened. Thanks for educating me aha! I now know that I will (hopefully) not fall victim to this pitfall in the future.
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Reality Check
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(Original post by BobbJo)
What do you mean by "in mathematics, this is not the case."?

A line of best fit does not need to be a straight line. A line of best fit can be a straight line or a curved line.

The line of best fit is the line which fits the data the best.

In statistics, we can not only do linear regression, but also non-linear regression (curve fitting).

http://www.math-mate.com/chapter39_2.shtml
https://blog.minitab.com/blog/advent...ear-regression
https://en.wikipedia.org/wiki/Curve_fitting

You can draw a linear graph, quadratic, exponential, power law for any data set, as shown in the links above. But we are looking for the graph which will fit the data the best and this line should be chosen as the line of best fit.
Yes, exactly. There seems to be a pervasive notion that 'lines of best fit' don't actually need to 'fit' the data at all, but need to be based on some arbitrary rules.

Tolgarda - BobbJo makes a good post here.
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Tolgarda
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(Original post by Reality Check)
Yes, exactly. There seems to be a pervasive notion that 'lines of best fit' don't actually need to 'fit' the data at all, but need to be based on some arbitrary rules.

Tolgarda - BobbJo makes a good post here.
I agree with both points, but the second one definitely seems true for me. I don't know where or why I developed this notion, but it stuck long enough for me to be confused about the actual concept itself.

EDIT: I actually think it is because I was usually taught that they roughly go through the middle of all the points (data) on the graph.
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Reality Check
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(Original post by Tolgarda)
I agree with both points, but the second one definitely seems true for me. I don't know where or why I developed this notion, but it stuck long enough for me to be confused about the actual concept itself.
Don't be too hard on yourself - it's easy to do this. I think we've all done it with various topics over the years where we get an idea in our heads that 'this is how it works' and then it's very difficult to see outside that paradigm.
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