Section 1.7 The Reliability of a Measurement

Numbers are usually written so that…

the uncertainty is in the last reported digit. More digits, more certainty; fewer digits, fewer certainty. Scientific measurements are reported so that every digit is certain except the last, which is estimated. EX: 5.213, the first three digits are certain; the last digit is estimated.

The number of digits reported in a measurement depends on…

the measuring device.

The precision of a measurement…

depends on the instrument used to make the measurement, and must be preserved, not only when recording the measurement, but also when performing calculations that use the measurement.

Significant figures (significant digits)

the non-place holding digits, the ones that are not simply marking the decimal place. The greater the number of significant figures, the greater the certainty of the measurement.

Significant Figures

Tell you how well a number is known, how precise it is. This is called precision.

Exact Numbers

Have unlimited significant figures. This applies to numbers that can be counted (like eggs in a carton), some numbers in equations (2πr, the 2 is exact), and numbers by definition (1 inch. is EXACTLY 2.54 cm). Note that the WORDING in a problem can sometimes tell you if a number is exact or not. Look for words like EXACTLY (exact numbers), ABOUT and APPROXIMATELY (means that it is an estimate, therefore there is a numbered amount of sig figs, not unlimited).

Precision

The precision or uncertainty of the number is determined by the last significant digit of the number. Precision refers to how close a series of measurements are to one another or how reproducible they are. (In science, this is very important because science relies on measurements and measurements always have some uncertainty to them).

Accuracy

refers to how close the measured value is to the actual value.

A series of measurements can be precise, but,

not accurate (not close to the true value).

Random error

Error that has equal probability of being too high or too low. Random error, with enough trials, can average itself out.

Systemic error

Error that tends to be either too high or too low. Unlike random error, systemic error does not average itself out with repeated trials.

All Non Zero Digits

Are significant. (1234).

All Internal/Captive Zeroes Between Two Non-Zero Digits

Are significant. (1230450067).

Leading Zeroes, The Zeroes Infront of the First Non Zero Digit

Are NOT significant. (0000.004567, 007865).

Trailing Zeroes, The Zeroes After the Last Non-Zero Digit

Are ONLY significant when the number has a DECIMAL point AND are after the last non zero digit. (2.00, 000.34500, 0.000345000000).

Irrational Numbers

Have the same number of significant figures as are shown in the number. EX: π is the most commonly used irrational number in science. Any given value for π is an APPROXIMATION, therefore, its significant figures are determined by how the number is approximated.

For Numbers Written in Scientific Notation

The significant figures are determined by the coefficient (mantissa), the number in front of the power of 10.

When Two Different Unit Systems are Being Used to Convert

The answer is usually NOT exact (with some exceptions, some conversions are EXACT by definition). When you convert between two systems, like meters to miles, the answer is an APPROXIMATION.

Conversions WITHIN their own systems

Are exact, like cups to gallons.

Rules for Sig Figs AFTER Multi/Divid

The number of significant figures in your answer is determined by the term with the LEAST number of Sig Figs.

Rules For Sig Figs AFTER Add/Sub

The decimal places that are left in your final answer are determined by the term with the LEAST amount of Sig Fig Decimal Places. EX: 2.345 + 2.34 + 2.3 = 7.0 Why? Your term with the least amount of decimal places is 2.3, therefore your answer must have the same number of decimal places. When you input this equation on a calc, your answer would be 6.985, but, because of the 8 next to your last digit, 9, you would round up from 6.9 to 7.0. Now why wouldn’t your answer just be 7? As we went over before, 7.0 is more precise than 7, AND 7.0 tells us readers the VALUE of your number.

Rules For Rounding

For rounding the last Sig Digit, you have to look at the digit to the right of it (the first digit that will be dropped). If the first dropped digit is ABOVE 5, you round up. If the digit is smaller than 5, the value does not change. EX: Round 2.367 to 2 sig figs = 2.4. The 6 is the digit to the right of the 3 (the last significant digit) and is above 5, therefore you round your 3 to 4.

In Addition/Subtraction, your Powers of 10

Must be the same before solving. EX: 8.7531 × 10³ + 1.702 × 10^4, you must change the power of 10 in the first problem to 4 to solve. Your new number, according to your new power of ten, would be 0.87531 X 10^4 + 1.702 X 10^4 = 2.57731 X 10^4 (remember that you must round to the term with the LEAST amount of significant digits). Your final answer would be 2.577 X 10^4.

In Multiplication/Division, your Powers of 10

Don’t matter. Remember that in Multiplication and Division you are looking at SIGNIFICANT FIGURES, not digits.