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Let's imagine that I want to know if my tree gives more and more apples as time goes by. I have two variables: X (unit of time, say week), and Y (number of apples). I have 100 weeks on record.

When I create a scatter plot and add the straight trend line to it, I get a growing line, but I've heard that the $R^2$ value is important to determine if the line is valid or not, and my $R^2$ value is 0.1 which seems very low.

My question is: Does this mean that my trend line is misleading? Or can I say without fear that the number of apples per week is growing and even say how much?

EDIT: Thank you so much for all the answers, I will try to clarify because now I see the example wasn't really good. I will take the example of the light bulbs in a factory that was hinted by one of the answers. The factory produces light bulbs every week and there's a randomness to the number of light bulbs produced. After reviewing the data collected from 2 years, one can construct a dispersion graphic and add a linear trend line to it. This trend line (through the trend line equation) can tell us how much (in %) the light bulb production has increased, but: is this % reliable if the trend line has a $R^2$ value of 0.1?

Maybe I can ignore $R^2$ altogether? I don't want to predict or explain anything, only to show that the number of light bulbs/year has grown and tell how much has grown in %...

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    $\begingroup$ It's unclear what precisely you mean by "valid" and "misleading". You can be certain that the number of applies is growing in the observed data (in the particular sense that is measured by the line); whether it even makes sense to talk about it in some broader sense would require us to identify a population about which we're trying to perform inference and to establish the sense in which we can be said to be randomly sampling from it, along with a number of other issues. $\endgroup$ – Glen_b -Reinstate Monica May 21 '18 at 4:12
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    $\begingroup$ First we're imagining data and then you seem to be implying that you have $R^2$ to hand from real data. Which is it? How literally are we expected to take the example? In particular, 100 weeks is almost 2 years and I don't expect apple trees to ignore the seasons and show a linear trend over 2 years. $\endgroup$ – Nick Cox May 21 '18 at 18:37
  • $\begingroup$ It sounds like you are relatively new here, and you are asking a question in human-speak and not statistician-speak. They are, sadly enough, not mutually inclusive languages. It sounds like there is a good stats question behind your human-speak question. How wrong is this interpretation: "How do I understand what $R^2$ means in context of this grandma-friendly example"? What can and can't $R^2$ tell me? $\endgroup$ – EngrStudent - Reinstate Monica May 22 '18 at 12:14
  • $\begingroup$ Hi EngrStudent! Your reformulation of my question sounds excellent. Can you answer to those? $\endgroup$ – Oto Kage May 23 '18 at 14:55
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It would be necessary to analyze the residuals of the model and the significance of the parameters. If the parameters are significant and all the assumptions of the model are met, you may not have a good model to predict but you can accept the existence of a trend.

In the Anscombe's quartet you can see some examples where the adjustment would be misleading (a single outlier in the data can change the trend line (slope) of the whole series). To better understand this problem I recommend the book Exploratory Data Analysis by John Tukey.

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First, $R^2$ is valid regardless. The question is whether it tells you what you want to know. $R^2$ here is a measure of the linear relationship between week and number of apples.

But that's not what you want to know. Trees don't give fruit in a way that grows steadily over time. They give fruit in certain seasons. Apple trees in the northern hemisphere usually give fruit in the late summer or early fall. So, while you have 100 weeks of data, you really only have 2 repeats - year 1 and year 2. That's not enough data to tell you anything about change in harvest from year to year.

Now, let's suppose that was a fake example and you really have something that works week-by-week - let's say it was number of light bulbs produced in a factory. Then you have a more reasonable amount of data, but you still may have problems. This is time series data, which has its own set of procedures. There may be seasonality (even if it won't be as extreme as with apples).

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