Rounding: wording and algorithm

[jdaw1]

Some posts moved from Apostrophe crimes.

> Each [thing] shall then be [maths in words], rounded to the nearest thousandth of a basis point (exact half thousandths of a basis point of yield being rounded up).

Should there be any hyphenation in the above, and if so, where? Can the parenthetical clause be both clearer and more concise?

[djewesbury]

1. No.
2. It's already clear and concise.

[DRT]

Should "half thousands" be "half thousandths"?

[jdaw1]

Should "half thousands" be "half thousandths"?

Thank you. Yes, it should: fixed.

[djewesbury]

Should "half thousands" be "half thousandths"?

Thank you. Yes, it should: fixed.

Should it be two-thousandths?

[DRT]

How can a number that is exact be rounded up?

Do you mean "rounded up to the nearest two-thousandth of a basis point"?

[djewesbury]

The exact number 7 can be rounded up to 10.

[PhilW]

Should "half thousands" be "half thousandths"?

Thank you. Yes, it should: fixed.

Should it be two-thousandths?

Should probably be "exact half of a thousandth being rounded up"; two-thousandths would be confusing. basis-point might need hyphenation, I do now know.

[DRT]

The exact number 7 can be rounded up to 10.

Stop chattering in class, Jewesbury, and pay attention.

Should "half thousands" be "half thousandths"?

Thank you. Yes, it should: fixed.

Should it be two-thousandths?

Should probably be "exact half of a thousandth being rounded up"; two-thousandths would be confusing.

That's why I asked the question. Is JDAW rounding up to the nearest thousandth or the nearest half of a thousandth?

[jdaw1]

What I want is:

• 50 → 50
  50.0004 → 50
  50.00049 → 50
  50.0005 → 50.001
  50.00051 → 50.001
  50.0006 → 50.001
  50.001 → 50.001

Please suggest concise and unambiguous^† phrasing that achieves this.

† Assume that ambiguity is being sought by evil lawyer-mathematician types. So “unambiguous” means “really utterly unambiguous”. And don’t assume that all numbers are positive.

[DRT]

"rounded to the nearest thousandth of a basis point" does it for me. If you need more precision then perhaps "rounded to the nearest thousandth of a basis point (where greater than or equal to n.nnn5 shall be rounded up and less than n.nnn5 shall be rounded down).

[jdaw1]

If you need more precision

evil lawyer-mathematician types

That is a yes, I do need more “precision”.

"rounded to the nearest thousandth of a basis point (where greater than or equal to n.nnn5 shall be rounded up and less than n.nnn5 shall be rounded down).

Is that really clearer than my earlier suggestion (because it is longer, and the repeated n’s introduce an ambiguity)?

rounded to the nearest thousandth of a basis point (exact half thousandths of a basis point of yield being rounded up).

[djewesbury]

This is the wrong place for this discussion and you have already crowded out a true-bill spelling mistake, by Derek, whose mistakes are always most fun.

[jdaw1]

This is the wrong place for this discussion

Prior precedent can be found in posts starting at 1350, 1256, 1225, 1212, and perhaps elsewhere. In some of these you have contributed to the discussion without complaint.

[djewesbury]

Now I am complaining. Precedent means nothing here.

[jdaw1]

Precedent means nothing here

This is the wrong place for this discussion (other than to mention the omission of the terminal full stop).

[PhilW]

Your original was clear. For additional evil-lawyer satisfaction:
"Each [thing] shall then be [maths in words], rounded to the nearest thousandth of a basis point; exact half of a thousandths of a basis point of yield being rounded up."

Note, you have given several examples, but then said "And don’t assume that all numbers are positive" without confirming your intention with regard to negative numbers. Your definition would mean that (first three as per your quote, second three added by me):
50.00049 → 50
50.0005 → 50.001
50.00051 → 50.001
-50.00049 → 50
-50.0005 → 50
-50.00051 → 50.001
If this is as you intend, all is good; otherwise you would need to state that exact half of thousandths are rounded to the next thousandth away from zero, instead of rounded up, or similar.

[djewesbury]

Battle lost. Quants have taken control here as elsewhere. Is nowhere safe from their meaningless ranting?

[jdaw1]

you have given several examples, but then said "And don’t assume that all numbers are positive" without confirming your intention with regard to negative numbers.

For negative numbers I don’t particularly care which happens, except that I don’t want to pay lawyers to squabble about it. So anything clear.

Current favourite:
> Each [thing] shall then be [maths in words], rounded to the nearest thousandth of a basis point; an exact half of a thousandth of a basis point of yield being rounded up.

[DRT]

I am a little unclear about what there is to squabble about. My maths qualification extends to a 98%ish score in the Scottish equivalent of an A-Level. My intuitive understanding of rounding would always result in me rounding 5 and above to 10 and below 5 to 0.

Do lawyers do it another way?

[jdaw1]

Do lawyers do it another way?

Sometimes^† — it depending on the instructions from the client.

† Except when billing the client, rounding then always being up.

[DRT]

Does the method of rounding you wish to employ meet with an international standard or specific, documented academic definition? If so, reference the source.

[PhilW]

I am a little unclear about what there is to squabble about.

Sometimes it matters in order to avoid cumulative errors over analysis of large data sets, or addition of bias. For example, consider rounding a serial of positive floating point numbers; if you were to always round up to nearest integer, you would add a bias of 0.5 to the mean of the rounded data compared with the original. Rounding to the nearest integer works much better, but the if the rounding of the mid-point value always occurs in the same direction, this can cause a similar cumulative error over a large enough data set. This matters in some data processing applications.

[PhilW]

Does the method of rounding you wish to employ meet with an international standard or specific, documented academic definition? If so, reference the source.

Yes, the IEEE defines standards for definition, format and rounding of floating point numbers stored in binary format (IEEE 754). Julian's case here is slightly different, since it is referring to the rounding to thousandths, though the same terminology could be used; in Julian's current definition the terminology would be "rounding to nearest, ties to +infinity", although ironically this is not one of the standard formats; "rounding to nearest, ties to even" is the default, while "rounding to nearest, ties away from zero" is simpler to explain.

[jdaw1]

There is no problem with large datasets; no standard need be followed; and I don't particularly care how rounding is done provided that it is clear, seems reasonable to a numerate lay observer, and is utterly unsquabblable.

[DRT]

Is there any way we could shoehorn the word exponential into the definition?

[DRT]

...and (wince) could someone please explain what a floating point number is?

I get (for the first time) Phil's description of bias above and why JDAWs intended method introduces a risk of bias. I also get that JDAW is confident that that bias does not compromise the integrity of the thing to which the method will be applied. Let's all hope he is correct in that assertion.

So, is the point floating in the number or in my glass of Rioja?

[jdaw1]

...and (wince) could someone please explain what a floating point number is?

For your purposes it is a real number (so a number not necessarily a whole number or a precise fraction), stored to some finite precision in a moderate amount of computer space.

There’s plenty of other stuff, but — trust me — you don’t care.

Anyway, the risk of bias is of slight concern to me. (OK, I don’t care about small bias in either direction, but I might care about the appearance of bias.) I will investigate the suggested sources.

[DRT]

...and (wince) could someone please explain what a floating point number is?

For your purposes it is a real number (so a number not necessarily a whole number or a precise fraction), stored to some finite precision in a moderate amount of computer space.

Thank you. Would (for an idiot) "a number with a defined number of decimal places" work as an imprecise definition?

the risk of bias is of slight concern to me. (OK, I don’t care about small bias in either direction, but I might care about the appearance of bias.) I will investigate the suggested sources.

I will work on the basis that the value my unintended challenge has added will be rounded down to zero.

[jdaw1]

Well, I am pleased I asked here: Phil and Wikipedia have changed my mind.

Current favourite:
> Each [thing] shall then be [maths in words], rounded to the nearest thousandth of a basis point; an exact half of a thousandth of a basis point of yield being rounded to the nearest even multiple of a thousandth of a basis point.

I am also coming closer to Daniel’s view: should these threads be split into a new thread?

[DRT]

Yes.

[jdaw1]

Above posts moved from Apostrophe crimes.

[jdaw1]

Note for those coming to this thread at a later date.
In Excel, if A1 is in percentage points rather than basis points, this is

• = IF( ABS(A1-ROUNDDOWN(A1,5)-5E-6)<1E-10, ROUND(A1/2,5)*2, ROUND(A1,5) )

(100bp = 1%.)

[jdaw1]

...and (wince) could someone please explain what a floating point number is?

For your purposes it is a real number (so a number not necessarily a whole number or a precise fraction), stored to some finite precision in a moderate amount of computer space.

Thank you. Would (for an idiot) "a number with a defined number of decimal places" work as an imprecise definition?

More accurately, “… of significant figures”.

[DRT]

I don't know what an insignificant figure is, so that doesn't help me.

[jdaw1]

1.23456789012345678 is to 17 decimal places and 18 significant figures.
12345678901234567.8 is to 1 decimal place and 18 significant figures.

When writing decimal, the figures to the left are more ‘significant’, those to the right are less so. So these are to the 18 most significant figures.

[djewesbury]

Hi, I have a question about port, am I in the wrong place?

[jdaw1]

Hi, I have a question about port

Ask away. Pray, what question?

Actually, I might know the question, so I’ll jump to the answer: Port is fortified wine from the Douro Valley in Portugal.

Did that help?

[DRT]

1.23456789012345678 is to 17 decimal places and 18 significant figures.
12345678901234567.8 is to 1 decimal place and 18 significant figures.

When writing decimal, the figures to the left are more ‘significant’, those to the right are less so. So these are to the 18 most significant figures.

That makes no sense. There are 18 "numbers" and one decimal point in each example, both of which are stated to have the same number of significant figures.

Please provide an example of an insignificant number.

[DRT]

Hi, I have a question about port, am I in the wrong place?

Yes.

[jdaw1]

Is the accuracy proportional to the size of a number (≈ significant figures), or is it absolute (the number of digits to the right of the decimal point)? E.g., a number bigger than a billion could be given to 1 part in a billion (9 significant figures), despite being given to zero decimal places. But if a number was of size about 1, and was given to 1 part in a billion, there would be several digits after the decimal point.

[DRT]

Is the accuracy proportional to the size of a number (≈ significant figures), or is it absolute (the number of digits to the right of the decimal point)? E.g., a number bigger than a billion could be given to 1 part in a billion (9 significant figures), despite being given to zero decimal places. But if a number was of size about 1, and was given to 1 part in a billion, there would be several digits after the decimal point.

{sigh}

[DRT]

I just got it.

1,000,000,000.000000001 does not have nineteen significant numbers, it has ten.

But how many significant numbers does 1,234,567.890123 have?

...and what is the threshold between significant and insignificant?

If it is subjective I will be very disappointed.

[jdaw1]

I just got it.

1,000,000,000.000000001 does not have nineteen significant numbers, it has ten.

But how many significant numbers does 1,234,567.890123 have?

You have’t got it.

1,000,000,000.000000001 has nineteen significant digits (it is given correct to about one part in 10^19), but only ten decimal places (because it is accurate to 10^-10).

The mass of the sun is 1.98855×10^30 kg. That is six significant figures: i.e., to about one part in just under a million. But it is to a negative number of decimal places — if to six decimal places it would have been correct to the milligram (difficult, as the sun loses mass by about 4.26 million metric tons per second, that number being given to three significant figures).

[DRT]

You have’t got it.

+1

Let's leave it there.

[Alex Bridgeman]

You have’t got it.

+1

Let's leave it there.

No. Please don't. This has been a fascinating way to lose 10 minutes of my life.

However, DRT did ask for an example of an I significant number. Here is such an example .

Happy to help.