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