1.5 Measurement Uncertainty, Accuracy, and Precision

A second important principle of uncertainty is

A second important principle of uncertainty is that results calculated from a measurement are at least as uncertain as the measurement itself.

significant figures or significant digits.

All of the digits in a measurement, including the uncertain last digit zero may be a measured value; for example, if you stand on a scale that shows weight to the nearest pound and it shows "120," then the 1 (hundreds), 2 (tens) and 0 (ones) are all significant (measured) values.

Are captive zeros significant? How about leading zeros?

Captive zeros result from measurement and are therefore always significant. Leading zeros, however, are never significant—they merely tell us where the decimal point is located.

uncertainty

Every measurement has some uncertainty, which depends on the device used (and the user's ability)

all nonzero digits are significant, and it is only zeros that require some thought. What terms will we use for the zeros?

We will use the terms "leading," "trailing," and "captive" for the zeros and will consider how to deal with them.

Exact Number

number derived by counting or by definition, defining a unit The result of such a counting measurement is an example of an exact number.

Meniscus

the lowest point on the curved surface of the liquid.

Not always significant

-Leading zeros -Trailing zeros: When the measurement does not contain a decimal point

significant

-Non zero digits -Imbedded zeros -trailiing zeros: When the number contains a decimal point, when in scientific notation

rounding numbers

1. When adding or subtracting numbers, round the result to the same number of decimal places as the number with the least number of decimal places (the least certain value in terms of addition and subtraction). 2. When multiplying or dividing numbers, round the result to the same number of digits as the number with the least number of significant figures (the least certain value in terms of multiplication and division). 3. If the digit to be dropped (the one immediately to the right of the digit to be retained) is less than 5, "round down" and leave the retained digit unchanged; if it is more than 5, "round up" and increase the retained digit by 1; if the dropped digit is 5, round up or down, whichever yields an even value for the retained digit. (The last part of this rule may strike you as a bit odd, but it's based on reliable statistics and is aimed at avoiding any bias when dropping the digit "5," since it is equally close to both possible values of the retained digit.)

Measurements are said to be precise if they yield? A measurement is considered accurate if it yields? Precise values agree with? Value?

Measurements are said to be precise if they yield very similar results when repeated in the same manner. A measurement is considered accurate if it yields a result that is very close to the true or accepted value. Precise values agree with each other; accurate values agree with a true value.

scientists typically make repeated measurements of a quantity to ensure?

Scientists typically make repeated measurements of a quantity to ensure the quality of their findings and to evaluate both the precision and the accuracy of their results.

Process:

Starting with the first nonzero digit on the left, count this digit and all remaining digits to the right. This is the number of significant figures in the measurement unless the last digit is a trailing zero lying to the left of the decimal point.