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This video is about error bars and the uncertainty of the gradient and the intersect of the
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best fit line. We will discuss how to draw error bars after you have plotted points based
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on your process data table. We will see how to correctly add the line of best fit and we
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will wrap up by finding the uncertainty in the gradient and the intercept of this line of best
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fit. We will begin with error bars. Let's assume that we carried out an experiment where
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we measure the speed of a ball rolling down a slope at different points in time. After
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processing the roll data from the experiment, so calculating averages for our trials
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and finding uncertainties based on the range of these trials, we get this process data
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table. Let's plot this data on a speed versus time graph. The next step is to add
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error bars to our data points. These error bars are based on the uncertainty values in
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the process data table. The uncertainty in time is plus minus 0.2 seconds. Therefore,
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the length of the horizontal error bars will be twice this value so 0.4 seconds. Vertical
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error bars confirm these uncertainty values. The length of each vertical error bar is the
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double of the given uncertainty and unlike horizontal error bars which all have the same length.
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These vertical error bars might be different for different data points. Let's draw these
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error bars. Let's see a final detail about error bars before we move on. We will zoom in
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on our last data point and magnify it a bit. Next, we will draw a rectangle based on the
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error bars like this. This rectangle is called the zone of uncertainty, meaning that the value
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of the given data point could lie anywhere inside this rectangle. Next, let's see how to draw
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the line of best fit. Depending on the situation, you might have to draw the line of best
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fit by hand or you might be able to use a computer program such as Logger Pro or Excel. The
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graphs in this video were drawn in Logger Pro and I can strongly recommend this program for
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plotting data. Let's stop for a moment here and talk about the expression line of best fit.
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It is common to call the line of best fit trend line. In most cases, this is not a problem.
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However, doing this is not fully correct. Trend lines are all lines that can be drawn
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so that they pass through all error bars. There are usually infinitely many trend lines that can be drawn.
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Only one of these lines is the line of best fit, which can be found by using the least squares
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method, a statistical approach that we will not discuss here in detail. If you are using a program,
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it can usually draw the line of best fit automatically and if you are drawing the line of best fit
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by hand, just make sure that it passes through all error bars and as close to the data points
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as possible. A very important point and probably the key takeaway from this video is that in order
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for it to be correct, the trend lines that you draw, including your line of best fit, must
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pass through all error bars. People often make a mistake here by drawing the trend line so that
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it passes through the error bars of the first and the last data point, but doesn't pass through
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some of the error bars in between. The line of best fit for our data here, drawn in logger pro,
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looks like this. Finally, let's see how to find the uncertainty in the gradient and the
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intercept of this line of best fit. To do this, in addition to the line of best fit,
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we have to add two more trend lines, one with the steepest possible and the other with the
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shallowest possible gradient. So the line of best fit and these two additional trend lines look
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like this. As you can see, all three lines pass through all error bars. Next, let's write down the
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equation of these three lines. Most programs will automatically calculate these equations.
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For the line of best fit, so the black line, we get V is equal to 1.9T plus 2.4.
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For the trend line with the shallowest gradient, so the red line, we get V is equal to 1.8T plus 2.7
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and for the trend line with the steepest gradient, so the blue line, we get V is equal to 2.1T plus
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1.9. We are looking for the uncertainty in the gradient and the intercept of the line of best fit.
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In the equations, these numbers represent the gradient, I will denote the gradient by m,
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and these numbers represent the vertical axis intercept. I will denote this by c.
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Let's find the uncertainty in the gradient. To do this, we will find the difference between the
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gradient of the shallowest and the steepest lines, so 2.1 minus 1.8, and we will divide this by 2.
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This gives us 0.15, which we will round to 0.2 to 1 decimal place.
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For the uncertainty in the intercept, we will follow a similar method and get 2.7 minus 1.9 divided by 2.
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This is equal to 0.4. So we can conclude that the uncertainty in the gradient is 0.2
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and the uncertainty in the vertical axis intercept is 0.4. The reason for calculating these uncertainties
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is that in physics, the gradient and the vertical axis intercept
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often have some sort of meaning in the context of the situation. Here, for example, since we have a
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speed time graph, the gradient represents acceleration and the vertical axis intercept represents
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the speed at t equals 0. Let's summarize what we have learned in this video. We have seen that error
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bars are based on uncertainty values in the process data table. In fact, their length is double,
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the uncertainty values associated with the data point. We mentioned that the rectangle drawn
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around error bars is called the zone of uncertainty and that the given data value could lie
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anywhere inside this rectangle. We have mentioned a very important point about trend lines,
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and this is that trend lines must always pass through all error bars. We have seen that the
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line of best fit is the trend line that best fits the data based on the least squares method.
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In order to find the uncertainty in the gradient and the intercept of the line of best fit,
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we have drawn two additional trend lines, one with the steepest and one with the shallowest gradient
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and using the equations of these lines, we managed to find the uncertainties that we were
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looking for and we finished with a reminder that when we graph a line in physics, the gradient
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and the intercept of this line often represent meaningful quantities in the given real-life situation.
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This wraps up our discussion about error bars and the uncertainty of the gradient and the
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intercept of the best fit line. This is the final video in subtopic 1.2. In the first video of subtopic
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1.3, we will learn about vectors and scalars.