This video is about recording and working with uncertainties.
We will discuss uncertaintinoetation and talk about how to work with uncertainties in raw and processed data.
Uncertaintinoetation looks like this.
In this expression, x represents a certain quantity, for example time, distance, or mass,
and delta x represents the absolute uncertainty in this quantity.
Let's see an example.
Imagine that we are dropping a rock into the sea from a cliff.
We use a stopwatch to measure the time taken by the rock to reach the water.
Our stopwatch shows 5.48 seconds.
However, this is clearly not the actual time it takes the rock to fall into the sea,
because we have to take into account our reaction time,
and probably even the time it takes for the sound to travel from the water to our ears.
For simplicity's sake, let's just take with the reaction time for now,
and say that our reaction time is 0.25 seconds.
Now, there are different conventions to decide about the uncertainty value for a measurement,
and we will see some of these conventions later in this video.
But for now, let's just assume that the uncertainty in this measurement is simply our reaction time.
So we can write that t, the time taken by the rock,
to reach the water is 5.48 seconds,
while the absolute uncertainty in this time, delta t, is 0.25 seconds.
A quick side note here, the symbol in front of t that looks like a triangle, is the Greek letter delta.
So if we want to express t our quantity together with its absolute uncertainty,
we would simply write t is equal to 5.48 plus minus 0.25 seconds.
This basically means that we are saying that we estimate the time taken by the rock to reach the water,
to be somewhere between 5.23 and 5.73 seconds.
It could be 5.24 seconds, 5.5 seconds, or 5.71 seconds.
After collecting raw data from an experiment, we usually display this data in a table.
In the table heading, we should indicate the uncertainty of our raw data.
Let's see the three conventions that are used to record raw data uncertainties.
These are the least count, a fraction of the least count,
and the combination of the least count and some other factors.
Let's see an example of each convention.
We will start with least count.
Imagine that the display of digital balance looks like this.
The least count convention is based on the smallest scale division.
The smallest scale division of this digital balance is 0.1 gram.
Therefore, the absolute uncertainty in the mass is 0.1 gram.
For a fraction of the least count convention, let's take a ruler.
The scale division on this ruler is 1 centimeter.
If we would use the least count convention here, the absolute uncertainty would be 1 centimeter.
However, even though the smallest scale is 1 centimeter,
we can be relatively confident that when we read the ruler,
we can estimate the length of the object that we are measuring,
to the nearest 0.5 centimeter.
Therefore, we would use a fraction of the smallest scale division here,
which as we said could be 0.5 centimeter.
So the absolute uncertainty in the length we are measuring, data L, would be 0.5 centimeter.
Of course, depending on how confident we are in our reading abilities,
this absolute uncertainty could be a bit less, for example 0.3 centimeter,
or a bit more, say 0.8 centimeter, as long as it is a fraction of the smallest scale division.
The third convention is a combination of the least count and some other factors.
Of course, there can be many different factors that influence uncertainty.
Here, we will consider reaction time when using a digital stopwatch.
The display of a stopwatch usually looks like this.
The least count or the smallest scale division for this stopwatch is 0.01 second.
In theory, we could use this as our absolute uncertainty,
but it makes sense to take into account the reaction time of the person operating the stopwatch.
So the other factor that will influence our decision about the absolute uncertainty is reaction time or RT.
Every human reaction time for visual clues is about 0.25 seconds.
So for the absolute uncertainty in the time measured, data T,
we will use 0.25 seconds.
You can use any of these three conventions in your work as long as you justify your choice.
Before moving on to uncertainties and process data,
let's see how uncertainties should be displayed in a raw data table.
As a simple example, imagine that we measured a length of five different pieces of robes
and get the following results.
In the table heading, we should include the name, the symbol, and the unit of the variable,
and also indicate the absolute uncertainty.
Let's say we decided to use the second convention, a fraction of the least count,
so we record the absolute uncertainty in L as 0.5 centimeter.
An important note is that all five measured values are rounded to the same number of decimal places
as the absolute uncertainty.
The next step after collecting raw data from an experiment is to process this data.
Let's see how we deal with uncertainties when processing data.
Imagine that we are carrying out a simple experiment to see how much time does it take for a toy car
to travel different distances.
Let's assume that for each distance we carry out three trials
and for a distance of 2.5 meters, we get this data.
You can see that for the distance we decided to work with an uncertainty of 0.01 meter
and for the time, based on the smallest scale division of our stopwatch,
the absolute uncertainty is 0.01 second.
Let's start by finding the average of our three trials.
You have most likely done this in path science experiments.
So for the average, we have to add the three values 9.22 plus 8.91 plus 9.13 and divide by three.
After rounding to two decimal places, we get 9.09.
Let's add this star table.
To find the uncertainty in the average time, we have to find a difference between the maximum and the minimum value of the three trials and divided by two.
As a side note, it doesn't matter how many trials you have.
You will always look for the difference between the maximum and the minimum value and divided by two.
Here the maximum value is 9.22.
The minimum value is 8.91.
So we get 9.22 minus 8.91 divided by two.
And after rounding to two decimal places, we get 0.16.
Let's finish up by adding this to our data table.
It's time to summarize what we have learned in this video.
We started with uncertainty notation.
Here x represents a certain quantity and delta x, the absolute uncertainty in this quantity.
An example for this would be t is equal to 6.71 plus minus 0.25 seconds.
Where 6.71 is our measured quantity and 0.25 is the absolute uncertainty.
We also discussed the three conventions for recording row data uncertainty.
And these are the least count convention where we use our smallest scale division as the absolute uncertainty.
A fraction of the least count where we use a fraction of the smallest scale division and the combination of the least count and some other factor.
Finally, we have seen how to calculate uncertainty in process data when we have more trials.
To do this, we have to find a range of our trials, so substitute the minimum value from the maximum value,
and divide this by two in order to get the uncertainty in the process data.
This completes our discussion about how to record and work with uncertainties.
In the next video, we will talk a bit more about absolute uncertainties,
and we will discuss fractional and percentage uncertainty.