0:00

Today I

Â want to

Â talk to you about how science

Â answers questions. We all want the right answers.

Â Should I prepare for rain today? How can I deal with my

Â boyfriend or girlfriend?

Â Should I go to the ball game or the concert tonight?

Â Every day we face myriads of circumstances

Â for which we need answers, the right answer.

Â 0:39

Well, sorry gang.

Â But science never can tell you the right answer.

Â All we can do is give you a probability that an answer is correct or not.

Â What this means is, is that we live in a statistical

Â universe. If the universe is statistical by

Â nature and we now have quantum mechanics to bla, back up that claim.

Â It may be that fundamentally a 100% crew deterministic

Â answer does not even exist. In this very real sense, the universe

Â with its constituents is not a machine, but an indeterminate

Â process. However, even if it is indeterminate,

Â it is not a free for all, but constrained by certain boundaries.

Â It is these boundaries that physics wishes to explore, quantify, and refine.

Â So we have a problem on

Â our hands.

Â If there are three kinds of lies; lies, damned lies and statistics.

Â How are we to proceed?

Â Incidentally, that phrase is often attributed to several

Â people, among who are Mark Twain and Benjamin Disraeli.

Â Well, the fact of the matter is, that the only way

Â we can find out whether we are dealing with lies or

Â damned lies, is through statistics. The way it works is as follows.

Â 2:13

If a theory predicts a phenomenon that is not observed, we can rule out the theory.

Â But if it does accord with the data we have, all we can say is that the theory is

Â consistent with the data on hand, not that the theory has been proven correct.

Â In fact, we can never prove a theory correct.

Â What this means in practice is that the most important part of any scientific

Â experiment is the probable error associated with the measurement.

Â It is more or less the wiggle room that we give a measurement, by

Â how much our measurement might be different

Â if we did the experiment over and

Â over again. As a concrete example, let's

Â image we are trying to measure the length of a dining room table.

Â We get out our trusty old stone-age tape measure, and do an experiment.

Â 3:13

Measurement one, 260 centimeters. Now,

Â what is the experimental error that, that we can estimate from this measurement?

Â You might think nothing, since we have nothing to compare it to.

Â But wait.

Â If we look at our stone-age device, we see that

Â it is very crudely made with big unmarked segments and

Â thick lines marking the intervals. So we can estimate something

Â called a systematic error, at say, ten centimeters.

Â But we are not satisfied with that. So we make more measurements.

Â 250 centimeters. 260 centimeters.

Â 250, 270, 260, 100. 100 centimeters?

Â 4:09

Whoa, what happened?

Â Do we really think that the table length is variable?

Â It might be.

Â But at first glance it appears we have made what we call a blunder.

Â If we use that 100 centimeter measurement in our computation

Â of, say, an average length, we will throw off everything.

Â But if we suspect a blunder, we need to track down

Â its source, if possible. So, using the

Â six supposedly valid measurements, we can obtain a sample

Â average of 258 centimeters. Now

Â we can ask, what is the experimental error associated with this

Â determination? In other words, how close to 258

Â centimeters would you expect each measurement to be?

Â Clearly, we must compare our sample average

Â with the individual measurements we already have.

Â Also, we sense that the measurement that's smaller than the

Â average, should count equally with a measurement that is larger.

Â So, we better square things first and then take the

Â square root, in order to avoid minus signs that would

Â be associated with measurements that might be smaller than average.

Â Thus, we expect that we need to take the following quantity.

Â In other words, for each of our N measurements, we compare the actual

Â measurement, x of i with the mean of all the measurements,

Â x bar, and square it. Then, add all N

Â results together and take the square root of the whole shebang.

Â 6:02

But clearly something is missing here because larger samples of measurements,

Â in other words, N larger, should not imply a larger error.

Â Measurements have to be worth something.

Â So we sense that our estimate of standard

Â deviation should include some factor of 1 over N.

Â In fact, it turns out that this quantity, usually designated as sigma,

Â is equal to our previous sum, but divided by the square root of N.

Â In essence, we are taking the average value of the square deviations

Â from the mean.

Â Now, remembering that you need a minimum of two measurements to get any average

Â value, we lead, this leads to the refinement of our equation as follows.

Â 6:56

It's the same as before in, except it has N minus 1, instead

Â of N under the square root. The value of this quantity,

Â sigma, associated with a mean value, can be shown to have an astonishing property.

Â 7:15

That 68% or about 2 3rds of all measurements you can possibly make,

Â even into the future, will fall within plus or minus 1 sigma of the mean.

Â As long as the properties of the phenomenon haven't been altered.

Â To summarize,

Â 7:49

And plus or minus 3 sigma contains 99.7%

Â of all data measurements. That's

Â it. It doesn't matter what you are measuring.

Â You could be interested in the height of 25 year old women in Borneo.

Â A comparison of daily maximum temperatures of two cities anywhere in the world.

Â Measurement of returns on investment versus risks in

Â financial markets. Analysis of statistics of scoring

Â in sports.

Â All of these basically use the same ideas presented here.

Â 8:31

However, if the phenomenon has changed, or a new phenomenon is somehow buried in the

Â data, a smaller standard deviation will enable you to detect it more easily.

Â Let's get back to our table and fast forward to the 21st century.

Â New devices now allow us to obtain much better precision in our measurements.

Â Now, using the same table, we may obtain the following results.

Â Now, you look you at these numbers closely and say, hmm.

Â Are we seeing something significant in the fact that the numbers seem to cluster

Â around 258.65 and

Â 258.85? Maybe, maybe not.

Â But we pay attention to this detail,

Â and then find out with additional measurements,

Â that we have obtained the higher numbers

Â when the temperature of the room is significantly

Â higher than when we look at the lower values.

Â We have discovered something.

Â The table is changing its length in

Â response to a temperature change in its environment.

Â Our measurements have revealed the thermal expansion of the table.

Â Something we had not anticipated, perhaps. And X-ray astronomy, as we

Â shall see, is filled with surprises of this sort.

Â And I'm sure your life is filled with them, as well.

Â When have you thought that you were exploring or answering one question,

Â when in reality, you were finding out something quite different instead?

Â Why not discuss this on the forum?

Â 10:30

We now shift gears and look at, look at a hypothetical astronomical example.

Â In this case, our determination of the standard deviation, or uncertainty in our

Â measurements, are even easier to obtain than our result for the table length.

Â This is because in certain situations,

Â which fortunately include most astronomical observations,

Â a very simple result ensues concerning what we might expect

Â from a measurement of say, the brightness of a cosmic X-ray source

Â as a function of time. The idea is as follows.

Â Let's suppose you have a random process, such

Â as the emission of light from an object.

Â We know that when an electron changes

Â its energy from within an atom by jumping from one level to

Â another, it is accompanied by the emission or absorption of a photon.

Â And we know that it is random in

Â the mathematical sense, because we can never know

Â exactly when this will happen, but it will

Â probably happen in a certain given time period.

Â And if it happens to lots and lots

Â of electrons, lots and lots of times, we will get lots and

Â lots of photons into our cameras or detectors.

Â Let's suppose, to make this concrete, we observe a source for ten

Â minutes and count 21,262 photons.

Â We sense that if we were to do this measurement

Â again and again, we would not get exactly 21,262

Â photons again and again, even if the source were unchanging.

Â The randomness of the process ensures this.

Â 12:29

Well, in these circumstances, there is a simple way to

Â estimate what the probability is of getting another result similar

Â to but not identical with our first trial, if we were to repeat our measurement.

Â We simply take the square root of the number of photons

Â observed, and that represents the range, plus and minus from our

Â observation, that we would expect to see 2 3rds of the

Â time, if we were to do the observation over and over again.

Â Thus, if we consider our original observation, we would expect to

Â observe 21,262 photons

Â plus or minus 146, about 2 3rds

Â of the time if we were to repeat the experiment over and over.

Â Why? Because 146 is

Â approximately the square root of 21,262.

Â The number 146, once again, is the

Â standard deviation of our observation. In astronomy

Â however, just raw numbers of photons are not particularly interesting.

Â We are more interested in rates. How much energy

Â is emitted per second or how many photons

Â are detected per second during any given observation?

Â So, let's see how this plays out in practice.

Â 14:10

Let's imagine that we have 100 photons

Â in ten seconds. Our expected range,

Â or in statistical language, our standard deviation, will then be

Â 100 plus or minus 10, because

Â 10 is the square root of hundred, of 100 over the ten seconds.

Â This translates into a rate of 100 counts over

Â 10 seconds, plus or minus 10 counts

Â over 10 seconds. Or, 10 plus

Â or minus 1 count per second.

Â 15:02

Let's imagine the same source, which is assumed unchanging, but now we

Â observe it for 1,000 seconds. In other words, our observation

Â is 100 times as long. Since we get

Â 100 counts in 10 seconds, we would expect to

Â get 10,000 counts in 1,000 seconds.

Â And therefore, we would expect to have 10,000,

Â plus or minus 100 counts, in our observation.

Â 15:41

Our rate then would be 10,000

Â divided by 1,000 seconds,

Â plus or minus 100 counts in

Â 1,000 seconds for 10,

Â plus or minus 0.1 counts

Â per second. Notice, that we

Â 16:12

needed 100 times more data to get our standard

Â deviation down by only a factor of 10.

Â What a bummer.

Â So, as you see, it can be slow going and sometimes very

Â expensive to get better and better results.

Â But it is the reason why scientists are always

Â asking for more data, better detection instruments, and bigger telescopes.

Â We will explore this important issue in greater depth in

Â week three, when we talk about clocks in the sky.

Â But for now, we will just state that

Â the size of the error bar, or standard deviation,

Â may play a decisive role in what we can legitimately say

Â about an astronomical source. Consider the following hypothetical

Â data points measuring the brightness of an object versus time.

Â So what we're going to do is we're going

Â to plot the brightness of a source versus time.

Â And let's imagine that we have the following

Â points on our graph, something that looks like this.

Â 17:52

Now we ask a simple question, is this source varying?

Â Well, it depends on the size of the error bars.

Â With a small sigma, we are more or

Â less forced to connect the observations with some

Â kind of variable curve. Let's look at that.

Â Okay? Let's just look at what

Â 18:18

would happen if we have very small error bars

Â attached to which of these points. You can

Â see without even drawing anything, that

Â it's impossible to just fit an ordinary,

Â non-varying line through the data that

Â would meet our requirement that 2 3rds of our data points be

Â within one sigma of that particular line. We are almost

Â forced into drawing something like that.

Â 19:08

Let's imagine that we have exactly the same data points.

Â They're the same, but now we have associated

Â with each measurement, maybe because we're using a smaller

Â telescope, or our detectors aren't as good or whatever.

Â Now, we imagine that associated with the same data are really, really big

Â error bars. Now, you can see that it's quite

Â easy to meet our requirement, more or less, of 2 3rds of

Â all of the data being within plus or minus 1 sigma of our

Â mean, by fitting a straight unvarying line through the data.

Â 19:59

So you can see that the standard deviation sigma is

Â critical to our observation and it will determine

Â whether our scientific estimate of variability

Â is a lie, a damned lie, or a legitimate

Â statement

Â of probable

Â fact.

Â