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This video is about recording and working with uncertainties.

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We will discuss uncertaintinoetation and talk about how to work with uncertainties in raw and processed data.

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Uncertaintinoetation looks like this.

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In this expression, x represents a certain quantity, for example time, distance, or mass,

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and delta x represents the absolute uncertainty in this quantity.

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Let's see an example.

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Imagine that we are dropping a rock into the sea from a cliff.

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We use a stopwatch to measure the time taken by the rock to reach the water.

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Our stopwatch shows 5.48 seconds.

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However, this is clearly not the actual time it takes the rock to fall into the sea,

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because we have to take into account our reaction time,

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and probably even the time it takes for the sound to travel from the water to our ears.

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For simplicity's sake, let's just take with the reaction time for now,

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and say that our reaction time is 0.25 seconds.

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Now, there are different conventions to decide about the uncertainty value for a measurement,

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and we will see some of these conventions later in this video.

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But for now, let's just assume that the uncertainty in this measurement is simply our reaction time.

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So we can write that t, the time taken by the rock,

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to reach the water is 5.48 seconds,

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while the absolute uncertainty in this time, delta t, is 0.25 seconds.

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A quick side note here, the symbol in front of t that looks like a triangle, is the Greek letter delta.

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So if we want to express t our quantity together with its absolute uncertainty,

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we would simply write t is equal to 5.48 plus minus 0.25 seconds.

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This basically means that we are saying that we estimate the time taken by the rock to reach the water,

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to be somewhere between 5.23 and 5.73 seconds.

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It could be 5.24 seconds, 5.5 seconds, or 5.71 seconds.

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After collecting raw data from an experiment, we usually display this data in a table.

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In the table heading, we should indicate the uncertainty of our raw data.

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Let's see the three conventions that are used to record raw data uncertainties.

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These are the least count, a fraction of the least count,

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and the combination of the least count and some other factors.

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Let's see an example of each convention.

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We will start with least count.

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Imagine that the display of digital balance looks like this.

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The least count convention is based on the smallest scale division.

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The smallest scale division of this digital balance is 0.1 gram.

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Therefore, the absolute uncertainty in the mass is 0.1 gram.

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For a fraction of the least count convention, let's take a ruler.

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The scale division on this ruler is 1 centimeter.

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If we would use the least count convention here, the absolute uncertainty would be 1 centimeter.

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However, even though the smallest scale is 1 centimeter,

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we can be relatively confident that when we read the ruler,

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we can estimate the length of the object that we are measuring,

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to the nearest 0.5 centimeter.

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Therefore, we would use a fraction of the smallest scale division here,

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which as we said could be 0.5 centimeter.

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So the absolute uncertainty in the length we are measuring, data L, would be 0.5 centimeter.

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Of course, depending on how confident we are in our reading abilities,

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this absolute uncertainty could be a bit less, for example 0.3 centimeter,

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or a bit more, say 0.8 centimeter, as long as it is a fraction of the smallest scale division.

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The third convention is a combination of the least count and some other factors.

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Of course, there can be many different factors that influence uncertainty.

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Here, we will consider reaction time when using a digital stopwatch.

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The display of a stopwatch usually looks like this.

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The least count or the smallest scale division for this stopwatch is 0.01 second.

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In theory, we could use this as our absolute uncertainty,

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but it makes sense to take into account the reaction time of the person operating the stopwatch.

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So the other factor that will influence our decision about the absolute uncertainty is reaction time or RT.

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Every human reaction time for visual clues is about 0.25 seconds.

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So for the absolute uncertainty in the time measured, data T,

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we will use 0.25 seconds.

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You can use any of these three conventions in your work as long as you justify your choice.

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Before moving on to uncertainties and process data,

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let's see how uncertainties should be displayed in a raw data table.

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As a simple example, imagine that we measured a length of five different pieces of robes

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and get the following results.

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In the table heading, we should include the name, the symbol, and the unit of the variable,

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and also indicate the absolute uncertainty.

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Let's say we decided to use the second convention, a fraction of the least count,

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so we record the absolute uncertainty in L as 0.5 centimeter.

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An important note is that all five measured values are rounded to the same number of decimal places

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as the absolute uncertainty.

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The next step after collecting raw data from an experiment is to process this data.

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Let's see how we deal with uncertainties when processing data.

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Imagine that we are carrying out a simple experiment to see how much time does it take for a toy car

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to travel different distances.

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Let's assume that for each distance we carry out three trials

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and for a distance of 2.5 meters, we get this data.

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You can see that for the distance we decided to work with an uncertainty of 0.01 meter

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and for the time, based on the smallest scale division of our stopwatch,

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the absolute uncertainty is 0.01 second.

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Let's start by finding the average of our three trials.

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You have most likely done this in path science experiments.

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So for the average, we have to add the three values 9.22 plus 8.91 plus 9.13 and divide by three.

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After rounding to two decimal places, we get 9.09.

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Let's add this star table.

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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.

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As a side note, it doesn't matter how many trials you have.

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You will always look for the difference between the maximum and the minimum value and divided by two.

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Here the maximum value is 9.22.

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The minimum value is 8.91.

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So we get 9.22 minus 8.91 divided by two.

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And after rounding to two decimal places, we get 0.16.

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Let's finish up by adding this to our data table.

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It's time to summarize what we have learned in this video.

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We started with uncertainty notation.

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Here x represents a certain quantity and delta x, the absolute uncertainty in this quantity.

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An example for this would be t is equal to 6.71 plus minus 0.25 seconds.

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Where 6.71 is our measured quantity and 0.25 is the absolute uncertainty.

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We also discussed the three conventions for recording row data uncertainty.

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And these are the least count convention where we use our smallest scale division as the absolute uncertainty.

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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.

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Finally, we have seen how to calculate uncertainty in process data when we have more trials.

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To do this, we have to find a range of our trials, so substitute the minimum value from the maximum value,

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and divide this by two in order to get the uncertainty in the process data.

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This completes our discussion about how to record and work with uncertainties.

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In the next video, we will talk a bit more about absolute uncertainties,

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and we will discuss fractional and percentage uncertainty.