# 1.3 Accuracy, precision, and significant figures  (Page 5/12)

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$A={\mathrm{\pi r}}^{2}=\left(3\text{.}\text{1415927}\text{.}\text{.}\text{.}\right)×{\left(1\text{.}2\phantom{\rule{0.25em}{0ex}}\text{m}\right)}^{2}=4\text{.}\text{5238934}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}$

is what you would get using a calculator that has an eight-digit output. But because the radius has only two significant figures, it limits the calculated quantity to two significant figures or

$A\text{=}4\text{.}5\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}\text{,}$

even though $\pi$ is good to at least eight digits.

2. For addition and subtraction: The answer can contain no more decimal places than the least precise measurement . Suppose that you buy 7.56-kg of potatoes in a grocery store as measured with a scale with precision 0.01 kg. Then you drop off 6.052-kg of potatoes at your laboratory as measured by a scale with precision 0.001 kg. Finally, you go home and add 13.7 kg of potatoes as measured by a bathroom scale with precision 0.1 kg. How many kilograms of potatoes do you now have, and how many significant figures are appropriate in the answer? The mass is found by simple addition and subtraction:

$\begin{array}{}\phantom{+1}7.56\phantom{0}\phantom{\rule{0.25em}{0ex}}\text{kg}\phantom{=15.2\phantom{\rule{0.25em}{0ex}}\text{kg}\text{.}}\\ -\phantom{1}6.052\phantom{\rule{0.25em}{0ex}}\text{kg}\phantom{=15.2\phantom{\rule{0.25em}{0ex}}\text{kg}\text{.}}\\ \frac{+13.7\phantom{00}\phantom{\rule{0.25em}{0ex}}\text{kg}}{\phantom{+}15.208\phantom{\rule{0.25em}{0ex}}\text{kg}}=15.2\phantom{\rule{0.25em}{0ex}}\text{kg}\text{.}\end{array}$

Next, we identify the least precise measurement: 13.7 kg. This measurement is expressed to the 0.1 decimal place, so our final answer must also be expressed to the 0.1 decimal place. Thus, the answer is rounded to the tenths place, giving us 15.2 kg.

## Significant figures in this text

In this text, most numbers are assumed to have three significant figures. Furthermore, consistent numbers of significant figures are used in all worked examples. You will note that an answer given to three digits is based on input good to at least three digits, for example. If the input has fewer significant figures, the answer will also have fewer significant figures. Care is also taken that the number of significant figures is reasonable for the situation posed. In some topics, particularly in optics, more accurate numbers are needed and more than three significant figures will be used. Finally, if a number is exact , such as the two in the formula for the circumference of a circle, $c=2\pi r$ , it does not affect the number of significant figures in a calculation.

Perform the following calculations and express your answer using the correct number of significant digits.

(a) A woman has two bags weighing 13.5 pounds and one bag with a weight of 10.2 pounds. What is the total weight of the bags?

(b) The force $F$ on an object is equal to its mass $m$ multiplied by its acceleration $a$ . If a wagon with mass 55 kg accelerates at a rate of , what is the force on the wagon? (The unit of force is called the newton, and it is expressed with the symbol N.)

(a) 37.2 pounds; Because the number of bags is an exact value, it is not considered in the significant figures.

(b) 1.4 N; Because the value 55 kg has only two significant figures, the final value must also contain two significant figures.

## Phet explorations: estimation

Explore size estimation in one, two, and three dimensions! Multiple levels of difficulty allow for progressive skill improvement.

## Summary

• Accuracy of a measured value refers to how close a measurement is to the correct value. The uncertainty in a measurement is an estimate of the amount by which the measurement result may differ from this value.
• Precision of measured values refers to how close the agreement is between repeated measurements.
• The precision of a measuring tool is related to the size of its measurement increments. The smaller the measurement increment, the more precise the tool.
• Significant figures express the precision of a measuring tool.
• When multiplying or dividing measured values, the final answer can contain only as many significant figures as the least precise value.
• When adding or subtracting measured values, the final answer cannot contain more decimal places than the least precise value.

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