"to within"
What does it mean when used in mathematical context, e.g., in the following sentence?

"...the statistic is to within a scale factor the square of the third moment."

Thank you in advance.

Question

"to within"
What does it mean when used in mathematical context, e.g., in the following sentence?

"...the statistic is to within a scale factor the square of the third moment."

Thank you in advance.

Answer

I just remembered an idiomatic expression using "to within" that is somewhat common.
Sometimes people will say something like this:

"I came TO WITHIN  a  hair's breadth  of  _(doing just about anything)__________."

like:   "I came TO WITHIN  a  hair's breadth  of  hitting a home run in baseball!"

Answer

But you are right to ask about this Dolokhov, because  it is  a  strange construction in English.    "To within"   just means, basically;  Something-----comes----THAT-----close.

Something comes    to within  THAT value range.

I do not think I have ever seen the phrase used in any other context but measurement.
In general English,  which is to say outside  of actual Mathematics, you might see that  phrase in an expression like this:

"I came to within  an inch of getting killed by that car!"

"I came to within an inch  of  persuading that girl to go to the movies with me."

"I came to within an inch of getting rich in the stock market."

.

Answer

(Correction on last sentence)

"...the statistic  APPROXIMATES a scale factor the square of the third moment."

Answer

Hi Dolokhov:

"To within"   as a phrase, means this.    The statistic (or any mathematical value)  shows us something.  The mathematical value is a statistic which shows a  tendencies.  There is therefore, a   scale of value.  Normally,  such scales can be reflected by  Place Value.

It is common to see decimal values of Tenths,  Hundredths, and Thousandths.
   So, where the precision or accuracy is concerned,  the phrase "to within"  refers to the degree of accuracy.    So sentences are written this way:

The machine  could cut the steel to within   a   Tenth of an   Millimeter...
(or)   Hundredth of a Millimeter  (or)   a  Ten Thousandth of a Millimeter.

Your statistical data might be referred to this way in your sentence:

"...the statistic is APPROXIMATES   a scale factor the square of the third moment."

(I hope that helps)

Answer

You're breaking that sentence down incorrectly. Think of it this way:

to 'within a scale factor' the square of the third moment

I'd like to see the rest of the context if you can. I can explain it better if I know what the numbers are talking about, but 'within a scale factor' likely means 'almost exactly the same'.