[philonossis]

"Sulfur oxides are air pollutants that arise primarily from the burning of coal.  A student measures the sulfur oxide concentration in the atmosphere on five successive days with the following results:

Tues 24 ppb (parts per billlion)
Wed 21 ppb
Thur 21  ppb
Fri   24  ppb
Sat  15  ppb

What is the average sulfur oxide concentration in the air over this time period based on the student's measurements?

Solution: Let us solve this problem using two different methods.  The first method involves...the 'normal' way of taking an average.  By this method, the average is found by dividing the sum of the nubmer by the number of values summed.

24 + 21 + 21 + 24 + 15 ppb = 21 ppb (calculator answer)
              5

                                         = 21.0 ppb (correct answer)

My question: what does the book add the decimal and the zero?

My guess is that when you add the numerator figures (ppb), you get 105, =  three sig figs.

So you are doing a division by a whole number which does not count for sig figs, into a 3 figure number. In division the number with the fewest sig figs determines how many sig figs you use. You then have to have 3 sig figs in the answer, so that is why the book adds to decimal and zero.  

But the book does not show this in the calculation.

I am checking to see if this is correct, or if there is another reason for the decimal.

[RandoFlyer]

The decimal place is shown because that .0 is significant

[MOTOBALL]

An answer of 21.0 ppb means that it is not 20.9, nor ia 21.1---it is measured to 1 part in 210.

However, the 5 values from which this mean are calculated are NOT measured to 1 in 210----only 1 in 21, 1 in 24 etc.

The mean should be quoted as 21 ppb.

[CuLater]

Tues 24 ppb (parts per billlion)
Wed 21 ppb
Thur 21  ppb
Fri   24  ppb
Sat  15  ppb

Don't forget why there are sig fig rules to begin with.  The whole point of sig figs is to understand and report reasonable error in a calculation.  If your book is reporting "21.0" as the correct answer, then your book is (probably) applying the sig fig rules to a calculation where they are not appropriate.  Your logic is perfect for that, when you add this series of numbers, the result does indeed have 3 sig figs at 105, and if you divide a 3-sig-fig number by an integer, the result should have 3 sig figs, 21.0.

BUT this is not just a bunch of numbers being added together, these are repeat measurements of the same thing, so when you take the average you should use the actual values to determine the error.  The average is 21, the maximum value is 24 and the minimum is 15.  That means that the error in this data set appears in the "ones" digit, so the result should be rounded to the "ones" digit as well.  This result could also be reported as 21 +/- 5, which maybe shows that the error is in the "ones" digit a little more explicitly.

[MOTOBALL]

"BUT this is not just a bunch of numbers being added together, these are repeat measurements of the same thing,"

Actually, this IS just a bunch of numbers being added together--only if the 5 determinations are made on the same day (at the same time e.g. with sampling tubes) are they repeat measurements of the same entity.

As it is, they are 5 different measurements of a variable entity.  For this reason, you cannot apply "error" to this set of measurements---only mean and range.

[CuLater]

Actually, this IS just a bunch of numbers being added together--only if the 5 determinations are made on the same day (at the same time e.g. with sampling tubes) are they repeat measurements of the same entity.

As it is, they are 5 different measurements of a variable entity.  For this reason, you cannot apply "error" to this set of measurements---only mean and range.

If they're not repeat measurements of the same entity, then "mean" is a meaningless quantity.  Is this a definitional disagreement?  When I use the term "error" I mean it in the most general sense, "error" is the variability in a measurement regardless of the source of that variability; therefore, the "range" that you mention is an indication of the "error" in your mean.  I also don't understand your example... by collecting multiple samples at the same time with sampling tubes, you can certainly assess the error in your sampling method or in your instrumentation, but those are not the only sources of error (variability) in your measurements.  Perhaps I should use the word "variability" more often... but that's so much harder to type than error...

[AWK]

http://calculator.sig-figs.com/

http://books.google.pl/books?id=ZOKIRTFkGTsC&pg=PA23&lpg=PA23&dq=significant+digits+in+arithmetic+mean&source=bl&ots=tXCc7htK4k&sig=EZanY1eDcbwl0pcfpXLf39it-2c&hl=pl&ei=xAU_TILOF-eXOOWQ0cMH&sa=X&oi=book_result&ct=result&resnum=8&ved=0CEsQ6AEwBw#v=onepage&q=significant%20digits%20in%20arithmetic%20mean&f=false

Read from: Ocasionally..

[MOTOBALL]

If they're not repeat measurements of the same entity, then "mean" is a meaningless quantity.  Is this a definitional disagreement?  When I use the term "error" I mean it in the most general sense, "error" is the variability in a measurement regardless of the source of that variability; therefore, the "range" that you mention is an indication of the "error" in your mean.

There may well be some semantics at play here, but I nevertheless believe that you have some fundamental mistakes in your statements.

(1) "If they're not repeat measurements of the same entity, then "mean" is a meaningless quantity." 

If the height of each student in a class is measured, then we cannot calculate a mean height for the class

(2) "When I use the term "error" I mean it in the most general sense, "error" is the variability in a measurement regardless of the source of that variability; therefore, the "range" that you mention is an indication of the "error" in your mean."

Error is indeed a measure of the variability of REPLICATE determinations of the SAME entity that is being measured (e.g. speed of light) that has an absolute value (same yesterday as today and tomorrow).

In the sulfur oxides case, there is no expectation that the values should be the same day-to-day---such an event is purely coincidental.  Therefore there can be a "mean" value determined over 5 days/ 5 months/ 5 years, but no error can be calculated, because there is no fixed, absolute value.  The range of observations of inherently variable quantities is just that, not an error.  There may well be an error associated with any individual observation taken on  any given day, but one  would need replicate measurements on the SAME day to calculate the error (usually the std. dev./variance).

[philonossis]

The decimal place is shown because that .0 is significant

Yes, this what my book says, but I am asking why it is significant.

after you add the five two-digit numbers, you get a three digit number.  this is the figure that is divided by 5 to get the average.  After you do the division, you get a two-digit figure again.

I am guessing the book adds the .0 to bring the result to three sig figs because of the three digit number that is divided.

{(two digit) + (two digit) + (two digit) + (two digit) + (two digit)} = (three digit)

[MOTOBALL]

 
 
Tutorial on the Use of Significant Figures, page 5

--------------------------------------------------------------------------------

Rules for mathematical operations
In carrying out calculations, the general rule is that the accuracy of a calculated result is limited by the least accurate measurement involved in the calculation.

(1) In addition and subtraction, the result is rounded off to the last common digit occurring furthest to the right in all components. Another way to state this rules, it that,in addition and subtraction, the result is rounded off so that it has the same number of decimal places as the measurement having the fewest decimal places. For example,

100 (assume 3 significant figures) + 23.643 (5 significant figures) = 123.643,
which should be rounded to 124 (3 significant figures).

(2) In multiplication and division, the result should be rounded off so as to have the same number of significant figures as in the component with the least number of significant figures. For example,

3.0 (2 significant figures ) × 12.60 (4 significant figures) = 37.8000
which should be rounded off to 38 (2 significant figures).
 
This text is extracted from an-on-line totorial on the use of significant figures, by Prof. Stephen L. Morgan (he of "Morgan & Deming") of Univ. S. Carolina.

Please note the first sentence !!!!

The correct answer to the mean is 21 ppb

[Jorriss]

Don't forget why there are sig fig rules to begin with.  The whole point of sig figs is to understand and report reasonable error in a calculation.  If your book is reporting "21.0" as the correct answer, then your book is (probably) applying the sig fig rules to a calculation where they are not appropriate.  Your logic is perfect for that, when you add this series of numbers, the result does indeed have 3 sig figs at 105, and if you divide a 3-sig-fig number by an integer, the result should have 3 sig figs, 21.0.

You can definitely apply sig fig rules here.

Finding a reasonable error has to do with error in measurement, which applies even if you are measuring different objects. All you are doing i reporting error in how accurately each days measurements were taken.


To the OP, you answered your own question I think, you add them up you get 105. Divided by 5 is 3 sig figs, 21.0. I don't see why you gained a decimal place of accuracy honestly but according to the rules, it makes sense?

[nigel433]

I think the chief difficulty here is that the author of the textbook has made up a terrible example because he is in a muddle himself.

(1) Ten-fold-increase is just too large a magnitude for use as "orders of magnitude"
but we are unfortunately culturally stuck with digits notation (although information
theory uses binary - yay). Dividing by 5 is just on the cusp of changing significant figures
- so you are stuck between a rock and a hard place.

(2) There are two main, different, USES of the purely MATHEMATICAL concept of "arithmetric mean" and that implies (at least) two different PHYSICAL concepts.
This problem does not specify which physical concept but uses language which could make you think it goes either way - when only one is viable in this instance.

 A mean can be a representative value in the sense that it tells you what to expect
 on different occasions. If you PLAN for 21 ppb each day you will not have too many
 surprises. Obviously significant figures is a pointless refinement in this context
 but a statement of the variance would be useful.

 The second use is where the averaged quantity can be used directly for some
 purpose and therefore the mean itself connotes something of a phenomenon.
 That is not the case in this example, and therefore it is a terrible example.

One commentator said it would be unfortunate if you could not take the average
height of a class of students- well, in the second sense above, YOU CAN'T.

Weight (mass) is a different matter. Imagine you are putting the class in a lift.
The average mass x class size will tell you if the lift is overloaded (because
the sum of the masses is a relevant phenomenon; but the average height
will not tell you anything about fitting in without bumping heads, because the
sum of the heights is not a relevant phenomenon.

"ppb" is a sort of average. You can't average averages (at least not without
a scheme of statistical weights). That is why there is no such concept as
average temperature - despite what you may read in a million places.
 
The essential point is that numbers as numbers do not have significant figures;
they are what they are. If somebody refers to significant figures you have to
examine the relation betwen his mathematical model and his physical problem
to see what he might be doing. If it is not crystal clear, that is the fault of the
author not the reader.