Accuracy and Precision (Math Lair)

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Physical measurements necessarily yield approximations. Although we can measure some lengths with astonishing accuracy, we can't measure any perfectly. Some inexactness, if only one part in thousands, remains. If we try to get too fine with determining length, say of an iron rod, our object-level notion of rigid boundaries fails. We reach a level of atomic structure where the "iron rod" is in flux without exact boundaries.

Precision

Refers to the number of significant digits and/or decimal places that can be reliably determined with a given instrument or technique.

Accuracy

Refers to how close a measured value is to the "true" or correct value.

Significant figures

Those figures in a measurement that are known with reasonable accuracy.

Some examples of significant digits:
327: three significant digits
2.09: three significant digits
0.000000382: three significant digits
16.83: four significant digits
3.2: two significant digits
3.20: three significant digits
150,000,000: not clear how many significant digits. Use scientific notation to clarify.
1.5 x 10⁸: two significant digits
1.50 x 10⁸: three significant digits

All digits other than zero are significant.
Zeros between non-zero digits are significant.
Leading zeros in a number are not significant.
Trailing zeros in a number may or may not be significant. Use standard form when appropriate to avoid confusion.

When rounding off measurements:

When the digit immediately to the right of the last digit to be rounded is less than five, the last digit is unchanged.
When the digit immediately to the right of the last digit to be retained is greater than five (or is equal to five, with non-zero digits following), the last digit retained is increased by one.
When there is only one digit to the right of the last digit to be retained, and it is 5, the last digit to be retained remains unchanged if even and is increased by one if odd.