Kermos:
Before I respond to your most recent posts, I think a short tutorial in the basics of uncertainties in science might help you. I know I already explained this to you briefly in an earlier post to this thread, but clearly you're still all at sea on the topic, so I'll try again. It seems you need more detail. Please make sure you understand the content of this post before you post again, because neither of us wants you to be banned again for making basic mistakes about error bounds, do we? So, let's begin.
Essentially all measurements of continuous scientific quantities are uncertain. Sources of uncertainty can be inherent in the apparatus used to make the measurement, or they can result from features of the method used to make the measurement, or - in some cases - they can result from inherent indeterminacy in nature.
I gave you a simple example earlier that I will use again here. Consider a ruler. Let's assume the smallest markings on the ruler are centimetres and the total length of the ruler is 100 cm. Suppose that a scientist uses the ruler to measure the length of an object. He lines up the "zero" marking of the ruler at one end of the object and reads off the length on the scale marked on the ruler. Let's say the other end of the object falls somewhere between the 78 cm and 79 cm marks on the ruler. This particular scientist - let us call him Scientist A - estimates by eye that the other end of the object is approximately half way between the 78 and 79 cm marks, so he records the length of the object as 78.5 cm.
Now, Scientist A asks himself "How certain am I that the actual length of the object is 78.5 cm?" He figures that he can probably judge by that object is not 78.1 cm long, and not 78.9 cm, but he can't be certain it's not 78.2 cm or 78.8 cm instead of 78.5 cm. What to do then?
Well, Scientist A is a good, careful scientist. So he decides to record his measurement of the length as (78.5 +/- 0.3) cm.
Here's what that notation means. It means that Scientist A admits that he doesn't know that the length is 78.5 cm. All he is confident about is that the length is between 78.2 cm and 78.8 cm. His "best guess" is 78.5 cm, but he admits he might be wrong. The length could be 78.4 cm, say, or even 78.8 cm.
The notation "78.5 +/- 0.3" here is to be read as "78.5 plus or minus 0.3". In other words, it includes all numbers in the range 78.5 - 0.3 = 78.2 cm, through to 78.5 + 0.3 = 78.8 cm.
Again, I emphasise that the scientist is not claiming that he knows for sure that the length is 75.5 cm. In fact, he's doing the opposite. He's saying that the actual length could be any value in the range 75.2 through to 78.8 cm. Also, just to be clear, he is not saying "I don't have any clue about what the length of the object is". After all, he is confident he knows the length, within a quantified uncertainty range.
Now, let us suppose that Scientist B uses the same ruler to measure the length of the same object. Like Scientist A, he also notices that the end of the object falls somewhere between the 78 and 79 cm marks on the ruler. His best guess is that the length is 78.3 cm. He's confident that the object is certainly longer than 78 cm, but he is not sure that he can judge the distance between marks to a greater accuracy than one-quarter of a division. So, he records the length as (78.3 +/- 0.3) cm. Arguably, he could have written +/- 0.25 cm, but since he is not at all certain about the second decimal place in the length (one-hundredth of a millimeter), he rounds this value to a conservative estimate of the error and writes +/- 0.3 cm. That allows for the possibility that the actual length is 78.07 cm, for instance.
Let us summarise the results of these scientists' measurements:
Scientist A: 78.5 +/- 0.3 cm
Scientist B: 78.3 +/- 0.3 cm
Now, let's assume that some guy called Kermos turns up and says to the scientists "I can see that the length of the object decreased between your two measurements Look! The length decreased from 78.5 cm to 78.3 cm between the time that scientist A measured it and the time when scientist B measured it. These results prove that the length decreased."
Scientist A says "No, Kermos, you can't conclude that! My measurement is consistent with a length of 78.3 cm, because that value is in my estimated uncertainty range, which includes all values between 78.2 cm and 78.8 cm. The length 78.3 cm is certainly in that range. I totally agree with Scientist B that the length might actually be 78.3 cm."
Scientist B says "No, Kermos, you can't conclude that the length decreased! My measurement is consistent with a length of 78.5 cm, because that value is in my estimated uncertainty range, which includes all values between 78.0 cm and 78.6 cm. The length 78.5 cm is certainly in that range. I totally agree with Scientist A that the length might actually be 78.5 cm."
In fact, it can be shown quite easily that both scientists' results are consistent with the length being somewhere between 78.2 cm and 78.6 cm. Neither of them knows the actual length. Neither of them is pretending to know the actual length. However, an unbiased observer will agree that these independent measurements provide evidence that the actual length is probably somewhere between 78.2 cm and 78.6 cm.
This assumes, of course, that the ruler was manufactured correctly, such that the centimetre marks on the ruler are accurate to within +/-0.3 cm.
The guy Kermos would be obviously wrong to argue, on the basis of the scientists' measurements alone, that the scientists' measurements here show that the length of the object actually decreased between the time scientist A measured it and when scientist B measured it. An actual decrease over that time period can't be absolutely ruled out, of course, but the data we have does nothing to support that it occurred; in fact, it suggests that no change of length occurred (at least, technically, no change greater than +/-0.2 cm).
Next, consider more complicated measurements made by Scientists A and B of the temperature of the cosmic microwave background radiation. The two scientists might be using different methodology and/or equipment and conducting their experiments at different times, possibly years apart. Suppose they report results:
Scientist A: (2.5 +/- 0.9) K
Scientist B: (3.1 +/- 1.0) K
Now, let us check what you have learned from this short tutorial. Please answer the following questions. Based on these data points alone:
Q1. Would it be reasonable for some guy, Kermos, to conclude that the actual temperature of the CMBR could be 2.7 K?
Q2. Would it be reasonable for some guy, Kermos, to conclude that the temperature of the CMBR increased by 0.6 K between the time that Scientist A made his measurement and the time when Scientist B made his measurement?
Do your homework. Once you think you understand uncertainties to the level explained in this post, you should check your answers here:
If you're still confused about any of this, Kermos, please ask questions. If you don't, I'm going to assume, in our future conversations, that you have understood this simple explanation of the basics of error analysis.
You'd better not keeping making the same mistake you've been making over and over with the CMBR temperature data, because after this it will look like you're deliberately telling more lies. And neither of us wants you to tell more lies, do we? So, make sure you have a good grasp on this stuff before you post again.
Before I respond to your most recent posts, I think a short tutorial in the basics of uncertainties in science might help you. I know I already explained this to you briefly in an earlier post to this thread, but clearly you're still all at sea on the topic, so I'll try again. It seems you need more detail. Please make sure you understand the content of this post before you post again, because neither of us wants you to be banned again for making basic mistakes about error bounds, do we? So, let's begin.
Essentially all measurements of continuous scientific quantities are uncertain. Sources of uncertainty can be inherent in the apparatus used to make the measurement, or they can result from features of the method used to make the measurement, or - in some cases - they can result from inherent indeterminacy in nature.
I gave you a simple example earlier that I will use again here. Consider a ruler. Let's assume the smallest markings on the ruler are centimetres and the total length of the ruler is 100 cm. Suppose that a scientist uses the ruler to measure the length of an object. He lines up the "zero" marking of the ruler at one end of the object and reads off the length on the scale marked on the ruler. Let's say the other end of the object falls somewhere between the 78 cm and 79 cm marks on the ruler. This particular scientist - let us call him Scientist A - estimates by eye that the other end of the object is approximately half way between the 78 and 79 cm marks, so he records the length of the object as 78.5 cm.
Now, Scientist A asks himself "How certain am I that the actual length of the object is 78.5 cm?" He figures that he can probably judge by that object is not 78.1 cm long, and not 78.9 cm, but he can't be certain it's not 78.2 cm or 78.8 cm instead of 78.5 cm. What to do then?
Well, Scientist A is a good, careful scientist. So he decides to record his measurement of the length as (78.5 +/- 0.3) cm.
Here's what that notation means. It means that Scientist A admits that he doesn't know that the length is 78.5 cm. All he is confident about is that the length is between 78.2 cm and 78.8 cm. His "best guess" is 78.5 cm, but he admits he might be wrong. The length could be 78.4 cm, say, or even 78.8 cm.
The notation "78.5 +/- 0.3" here is to be read as "78.5 plus or minus 0.3". In other words, it includes all numbers in the range 78.5 - 0.3 = 78.2 cm, through to 78.5 + 0.3 = 78.8 cm.
Again, I emphasise that the scientist is not claiming that he knows for sure that the length is 75.5 cm. In fact, he's doing the opposite. He's saying that the actual length could be any value in the range 75.2 through to 78.8 cm. Also, just to be clear, he is not saying "I don't have any clue about what the length of the object is". After all, he is confident he knows the length, within a quantified uncertainty range.
Now, let us suppose that Scientist B uses the same ruler to measure the length of the same object. Like Scientist A, he also notices that the end of the object falls somewhere between the 78 and 79 cm marks on the ruler. His best guess is that the length is 78.3 cm. He's confident that the object is certainly longer than 78 cm, but he is not sure that he can judge the distance between marks to a greater accuracy than one-quarter of a division. So, he records the length as (78.3 +/- 0.3) cm. Arguably, he could have written +/- 0.25 cm, but since he is not at all certain about the second decimal place in the length (one-hundredth of a millimeter), he rounds this value to a conservative estimate of the error and writes +/- 0.3 cm. That allows for the possibility that the actual length is 78.07 cm, for instance.
Let us summarise the results of these scientists' measurements:
Scientist A: 78.5 +/- 0.3 cm
Scientist B: 78.3 +/- 0.3 cm
Now, let's assume that some guy called Kermos turns up and says to the scientists "I can see that the length of the object decreased between your two measurements Look! The length decreased from 78.5 cm to 78.3 cm between the time that scientist A measured it and the time when scientist B measured it. These results prove that the length decreased."
Scientist A says "No, Kermos, you can't conclude that! My measurement is consistent with a length of 78.3 cm, because that value is in my estimated uncertainty range, which includes all values between 78.2 cm and 78.8 cm. The length 78.3 cm is certainly in that range. I totally agree with Scientist B that the length might actually be 78.3 cm."
Scientist B says "No, Kermos, you can't conclude that the length decreased! My measurement is consistent with a length of 78.5 cm, because that value is in my estimated uncertainty range, which includes all values between 78.0 cm and 78.6 cm. The length 78.5 cm is certainly in that range. I totally agree with Scientist A that the length might actually be 78.5 cm."
In fact, it can be shown quite easily that both scientists' results are consistent with the length being somewhere between 78.2 cm and 78.6 cm. Neither of them knows the actual length. Neither of them is pretending to know the actual length. However, an unbiased observer will agree that these independent measurements provide evidence that the actual length is probably somewhere between 78.2 cm and 78.6 cm.
This assumes, of course, that the ruler was manufactured correctly, such that the centimetre marks on the ruler are accurate to within +/-0.3 cm.
The guy Kermos would be obviously wrong to argue, on the basis of the scientists' measurements alone, that the scientists' measurements here show that the length of the object actually decreased between the time scientist A measured it and when scientist B measured it. An actual decrease over that time period can't be absolutely ruled out, of course, but the data we have does nothing to support that it occurred; in fact, it suggests that no change of length occurred (at least, technically, no change greater than +/-0.2 cm).
Next, consider more complicated measurements made by Scientists A and B of the temperature of the cosmic microwave background radiation. The two scientists might be using different methodology and/or equipment and conducting their experiments at different times, possibly years apart. Suppose they report results:
Scientist A: (2.5 +/- 0.9) K
Scientist B: (3.1 +/- 1.0) K
Now, let us check what you have learned from this short tutorial. Please answer the following questions. Based on these data points alone:
Q1. Would it be reasonable for some guy, Kermos, to conclude that the actual temperature of the CMBR could be 2.7 K?
Q2. Would it be reasonable for some guy, Kermos, to conclude that the temperature of the CMBR increased by 0.6 K between the time that Scientist A made his measurement and the time when Scientist B made his measurement?
Do your homework. Once you think you understand uncertainties to the level explained in this post, you should check your answers here:
Q1. Yes, that would be perfectly reasonable, because the value 2.7 K falls within the range of uncertainty of both measurements.
Q2. No.
For instance, Scientist A's measurement range includes 3.1 K as a possible temperature and Scientist B's range includes 2.5 K as possible temperature, which would imply - if those were the actual values - that the temperature decreased instead of increasing.
Equally, Scientist A's measurement includes 3.1 K as a possible temperature and so (obviously) does Scientist B's, which together would imply no change in the temperature, if 3.1 K was the actual value.
And these data can't tell us whether we ought to prefer the value 2.5 K or 3.1 K for the temperature, either, because each uncertainty range includes both of those values (along with a range of other possible values).
What we'd actually need to establish a trend in the temperature would be many more measurements over a period of time. We'd also want to see higher precision (meaning small uncertainty ranges) in the measurements, which might be achieved by improving the equipment and/or methods used to find the temperature, for instance.
Q2. No.
For instance, Scientist A's measurement range includes 3.1 K as a possible temperature and Scientist B's range includes 2.5 K as possible temperature, which would imply - if those were the actual values - that the temperature decreased instead of increasing.
Equally, Scientist A's measurement includes 3.1 K as a possible temperature and so (obviously) does Scientist B's, which together would imply no change in the temperature, if 3.1 K was the actual value.
And these data can't tell us whether we ought to prefer the value 2.5 K or 3.1 K for the temperature, either, because each uncertainty range includes both of those values (along with a range of other possible values).
What we'd actually need to establish a trend in the temperature would be many more measurements over a period of time. We'd also want to see higher precision (meaning small uncertainty ranges) in the measurements, which might be achieved by improving the equipment and/or methods used to find the temperature, for instance.
You'd better not keeping making the same mistake you've been making over and over with the CMBR temperature data, because after this it will look like you're deliberately telling more lies. And neither of us wants you to tell more lies, do we? So, make sure you have a good grasp on this stuff before you post again.
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