1.3: Mathematical Treatment of Measurement Results
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- Scott Van Bramer
- Widener University
Learning Objectives
- Explain the dimensional analysis (factor label) approach to mathematical calculations involving quantities.
- Describe how to use dimensional analysis to carry out unit conversions for a given property and computations involving two or more properties.
- Convert between the three main temperature units: Fahrenheit, Celsius, and Kelvin.
Math Support
If you have any concerns about your math background or just want to make sure you have all the math tools needed to succeed in this course, review the short "Math Minutes" videos from Dr. Martin in the Appendix: 11.1: Dr Martin's Math Minutes . You may also want to review Appendix: 11.3: Essential Mathematics .
Introduction
It is often the case that a quantity of interest may not be easy (or even possible) to measure directly but instead must be calculated from other directly measured properties and appropriate mathematical relationships. For example, consider measuring the average speed of an athlete running sprints. This is typically accomplished by measuring the time required for the athlete to run from the starting line to the finish line, and the distance between these two lines, and then computing speed from the equation that relates these three properties:
\[\mathrm{speed=\dfrac{distance}{time}}\]
An Olympic-quality sprinter can run 100 m in approximately 10 s, corresponding to an average speed of
\[\mathrm{\dfrac{100\: m}{10\: s}=10\: m/s}\]
Note that this simple arithmetic involves dividing the numbers of each measured quantity to yield the number of the computed quantity (100/10 = 10) and likewise dividing the units of each measured quantity to yield the unit of the computed quantity (m/s = m/s). Now, consider using this same relation to predict the time required for a person running at this speed to travel a distance of 25 m. The same relation between the three properties is used, but in this case, the two quantities provided are a speed (10 m/s) and a distance (25 m). To yield the sought property, time, the equation must be rearranged appropriately:
\[\mathrm{time=\dfrac{distance}{speed}}\]
The time can then be computed as:
\[\mathrm{\dfrac{25\: m}{10\: m/s}=2.5\: s}\]
Again, arithmetic on the numbers (25/10 = 2.5) was accompanied by the same arithmetic on the units (m/m/s = s) to yield the number and unit of the result, 2.5 s. Note that, just as for numbers, when a unit is divided by an identical unit (in this case, m/m), the result is “1”—or, as commonly phrased, the units “cancel.”
These calculations are examples of a versatile mathematical approach known as dimensional analysis, or the factor-label method . Dimensional analysis is based on this premise: the units of quantities must be subjected to the same mathematical operations as their associated numbers . This method can be applied to computations ranging from simple unit conversions to more complex, multi-step calculations involving several different quantities.
Conversion Factors
A ratio of two equivalent quantities expressed with different measurement units can be used as a unit conversion factor . For example, the lengths of 2.54 cm and 1 in. are equivalent (by definition), and so a unit conversion factor may be derived from the ratio,
\[\mathrm{\dfrac{2.54\: cm}{1\: in.}\:(2.54\: cm=1\: in.)\: or\: 2.54\:\dfrac{cm}{in.}}\]
Several other commonly used conversion factors are given in Table \(\PageIndex{1}\). A more extensive list is in Appendix 11.6: Units and Conversion Factors .
| Length | Volume | Mass |
|---|---|---|
| 1 m = 1.0936 yd | 1 L = 1.0567 qt | 1 kg = 2.2046 lb |
| 1 in. = 2.54 cm (exact) | 1 qt = 0.94635 L | 1 lb = 453.59 g |
| 1 km = 0.62137 mi | 1 ft 3 = 28.317 L | 1 (avoirdupois) oz = 28.349 g |
| 1 mi = 1609.3 m | 1 tbsp = 14.787 mL | 1 (troy) oz = 31.103 g |
When we multiply a quantity (such as distance given in inches) by an appropriate unit conversion factor, we convert the quantity to an equivalent value with different units (such as distance in centimeters). For example, a basketball player’s vertical jump of 34 inches can be converted to centimeters by:
\[\mathrm{34\: \cancel{in.} \times \dfrac{2.54\: cm}{1\:\cancel{in.}}=86\: cm}\]
Since this simple arithmetic involves quantities , the premise of dimensional analysis requires that we multiply both numbers and units . The numbers of these two quantities are multiplied to yield the number of the product quantity, 86, whereas the units are multiplied to yield
\[\mathrm{\dfrac{in.\times cm}{in.}}.\]
Just as for numbers, a ratio of identical units is also numerically equal to one,
\[\mathrm{\dfrac{in.}{in.}=1}\]
and the unit product thus simplifies to cm . (When identical units divide to yield a factor of 1, they are said to “cancel.”) Using dimensional analysis, we can determine that a unit conversion factor has been set up correctly by checking to confirm that the original unit will cancel, and the result will contain the sought (converted) unit.
Example \(\PageIndex{1}\): Using a Unit Conversion Factor
The mass of a competition Frisbee is 125 g. Convert its mass to ounces using the unit conversion factor derived from the relationship 1 oz = 28.349 g (Table \(\PageIndex{1}\)).
Solution
Since we have the conversion factor, we can determine the mass in ounces using an equation similar the one used for converting length from inches to centimeters.
\[x\:\mathrm{g=125\: g\times unit\: conversion\: factor}\nonumber\]
We write the unit conversion factor in its two forms:
\[\mathrm{\dfrac{1\: oz}{28.349\: g}\:and\:\dfrac{28.349\: g}{1\: oz}}\nonumber\]
The correct unit conversion factor is the ratio that cancels the units of grams and leaves ounces.
\[\begin{align*}
x\:\ce{oz}&=\mathrm{125\:\cancel{g}\times \dfrac{1\: oz}{28.349\:\cancel{g}}}\\
&=\mathrm{\left(\dfrac{125}{28.349}\right)\:oz}\\
&=\mathrm{4.41\: oz\: (three\: significant\: figures)}
\end{align*}\]
Exercise \(\PageIndex{1}\)
Convert a volume of 9.345 qt to liters.
- Answer
-
8.844 L
Conversion Factors and SI Prefixes
The same problem solving strategy can be used to convert units within the SI system using SI prefixes. The prefix is a conversion factor that allows conversion between units. For example when we multiply a quantity (such as distance given in cm) by an appropriate unit conversion factor, we convert the quantity to an equivalent value with different units (such as distance in meters). For example, since centi = 10 -2 , a basketball player’s vertical jump of 86 cm can be converted to meters by:
\[\mathrm{86\: \cancel{cm.} \times \dfrac{1\: cm}{10^{-2}\:\cancel{m.}}=0.86\: m}\]
Example \(\PageIndex{2}\): Using SI Prefixes for Unit Conversion
The mass of a competition Frisbee is 125 g. Convert its mass to milligrams using the unit conversion factor derived from the relationship milli = 10 -3 .
Solution
If we have the conversion factor, we can determine the mass in milligrams using an equation similar the one used for converting length from inches to centimeters.
\[x\:\mathrm{g =125\: g\times unit\: conversion\: factor}\nonumber\]
We write the unit conversion factor in its two forms:
\[\mathrm{\dfrac{1\: mg}{10^{-3}\: g}\:and\:\dfrac{10^{-3}\: g}{1\: mg}}\nonumber\]
The correct unit conversion factor is the ratio that cancels the units of grams and leaves ounces.
\[\begin{align*}
x\:\ce{g}&=\mathrm{125\:\cancel{g}\times \dfrac{1\: mg}{10^{-3}\:\cancel{g}}}\\
&=\mathrm{\left(\dfrac{125}{10^{-3}}\right)\:mg}\\
&=\mathrm{1.25 \times 10^5\: mg\: (three\: significant\: figures)}
\end{align*}\]
Exercise \(\PageIndex{2}\)
Convert a volume of 25.4 milliliters to liters.
- Answer
-
0.0254 L
Conversion Factors and Dimensional Analysis
Beyond simple unit conversions, the factor-label method can be used to solve more complex problems involving computations. Regardless of the details, the basic approach is the same—all the factors involved in the calculation must be appropriately oriented to insure that their labels (units) will appropriately cancel and/or combine to yield the desired unit in the result. This is why it is referred to as the factor-label method. As your study of chemistry continues, you will encounter many opportunities to apply this approach.
Example \(\PageIndex{3}\): Computing Quantities from Measurement Results
What is the density of common antifreeze in units of g/mL? A 4.00-qt sample of the antifreeze weighs 9.26 lb.
Solution
Since \(\mathrm{density=\dfrac{mass}{volume}}\), we need to divide the mass in grams by the volume in milliliters. In general: the number of units of B = the number of units of A \(\times\) unit conversion factor. The necessary conversion factors are given in Table 1.7.1: 1 lb = 453.59 g; 1 L = 1.0567 qt; 1 L = 1,000 mL. We can convert mass from pounds to grams in one step:
\[\mathrm{9.26\:\cancel{lb}\times \dfrac{453.59\: g}{1\:\cancel{lb}}=4.20\times 10^3\:g}\nonumber \]
We need to use two steps to convert volume from quarts to milliliters.
- Convert quarts to liters.
\[\mathrm{4.00\:\cancel{qt}\times\dfrac{1\: L}{1.0567\:\cancel{qt}}=3.78\: L}\nonumber\]
- Convert liters to milliliters.
\[\mathrm{3.78\:\cancel{L}\times\dfrac{1000\: mL}{1\:\cancel{L}}=3.78\times10^3\:mL}\nonumber\]
Then,
\[\mathrm{density=\dfrac{4.20\times10^3\:g}{3.78\times10^3\:mL}=1.11\: g/mL}\nonumber\]
Alternatively, the calculation could be set up in a way that uses three unit conversion factors sequentially as follows:
\[\mathrm{\dfrac{9.26\:\cancel{lb}}{4.00\:\cancel{qt}}\times\dfrac{453.59\: g}{1\:\cancel{lb}}\times\dfrac{1.0567\:\cancel{qt}}{1\:\cancel{L}}\times\dfrac{1\:\cancel{L}}{1000\: mL}=1.11\: g/mL}\nonumber\]
Exercise \(\PageIndex{3}\)
What is the volume in liters of 1.000 oz of antifreeze, given that 1 L = 1.0567 qt and 1 qt = 32 oz (exactly)?
- Answer
-
\(\mathrm{2.956\times10^{-2}\:L}\)
Example \(\PageIndex{4}\): Computing Quantities from Measurement Results
While being driven from Philadelphia to Atlanta, a distance of about 1250 km, a 2014 Lamborghini Aventador Roadster uses 213 L gasoline.
- What (average) fuel economy, in miles per gallon, did the Roadster get during this trip?
- If gasoline costs $3.80 per gallon, what was the fuel cost for this trip?
Solution
(a) We first convert distance from kilometers to miles:
\[\mathrm{1250\: km\times\dfrac{0.62137\: mi}{1\: km}=777\: mi}\nonumber\]
and then convert volume from liters to gallons:
\[\mathrm{213\:\cancel{L}\times\dfrac{1.0567\:\cancel{qt}}{1\:\cancel{L}}\times\dfrac{1\: gal}{4\:\cancel{qt}}=56.3\: gal}\nonumber\]
Then,
\[\mathrm{(average)\: mileage=\dfrac{777\: mi}{56.3\: gal}=13.8\: miles/gallon=13.8\: mpg}\nonumber\]
Alternatively, the calculation could be set up in a way that uses all the conversion factors sequentially, as follows:
\[\mathrm{\dfrac{1250\:\cancel{km}}{213\:\cancel{L}}\times\dfrac{0.62137\: mi}{1\:\cancel{km}}\times\dfrac{1\:\cancel{L}}{1.0567\:\cancel{qt}}\times\dfrac{4\:\cancel{qt}}{1\: gal}=13.8\: mpg}\nonumber \]
(b) Using the previously calculated volume in gallons, we find:
\[\mathrm{56.3\: gal\times\dfrac{$3.80}{1\: gal}=$214}\nonumber \]
Exercise \(\PageIndex{4}\)
A Toyota Prius Hybrid uses 59.7 L gasoline to drive from San Francisco to Seattle, a distance of 1300 km (two significant digits).
- What (average) fuel economy, in miles per gallon, did the Prius get during this trip?
- If gasoline costs $3.90 per gallon, what was the fuel cost for this trip?
- Answer a
-
51 mpg
- Answer b
-
$62
Unit Conversion Video Example
This project was preformed to supply Libretext authors with videos on General Chemistry topics which can be used to enhance their projects. Also, these videos are meant to act as a learning resource for all General Chemistry students .
Video Topics
Dimensional analysis: Units can be multiplied, divided, and canceled just like an algebraic quantity. A conversion factor is a fractional quantity used to convert one unit to another.
Link to Video
C onverting Between Units: https://youtu.be/wSaOh48k8Wg
Significant Digits in Measurement
Counting is the only type of measurement that is free from uncertainty, provided the number of objects being counted does not change while the counting process is underway. The result of such a counting measurement is an example of an exact number . If we count eggs in a carton, we know exactly how many eggs the carton contains. The numbers of defined quantities are also exact. By definition, 1 foot is exactly 12 inches, 1 inch is exactly 2.54 centimeters, and 1 gram is exactly 0.001 kilogram. Quantities derived from measurements other than counting, however, are uncertain to varying extents due to practical limitations of the measurement process used.
The numbers of measured quantities, unlike defined or directly counted quantities, are not exact. To measure the volume of liquid in a graduated cylinder, you should make a reading at the bottom of the meniscus, the lowest point on the curved surface of the liquid.
Refer to the illustration in Figure \(\PageIndex{1}\). The bottom of the meniscus in this case clearly lies between the 21 and 22 markings, meaning the liquid volume is certainly greater than 21 mL but less than 22 mL. The meniscus appears to be a bit closer to the 22-mL mark than to the 21-mL mark, and so a reasonable estimate of the liquid’s volume would be 21.6 mL. In the number 21.6, then, the digits 2 and 1 are certain, but the 6 is an estimate. Some people might estimate the meniscus position to be equally distant from each of the markings and estimate the tenth-place digit as 5, while others may think it to be even closer to the 22-mL mark and estimate this digit to be 7. Note that it would be pointless to attempt to estimate a digit for the hundredths place, given that the tenths-place digit is uncertain. In general, numerical scales such as the one on this graduated cylinder will permit measurements to one-tenth of the smallest scale division. The scale in this case has 1-mL divisions, and so volumes may be measured to the nearest 0.1 mL.
This concept holds true for all measurements, even if you do not actively make an estimate. If you place a quarter on a standard electronic balance, you may obtain a reading of 6.72 g. The digits 6 and 7 are certain, and the 2 indicates that the mass of the quarter is likely between 6.71 and 6.73 grams. The quarter weighs about 6.72 grams, with a nominal uncertainty in the measurement of ± 0.01 gram. If we weigh the quarter on a more precise balance, we may find that its mass is 6.723 g. This means its mass lies between 6.722 and 6.724 grams, an uncertainty of 0.001 gram. Every measurement has some uncertainty , which depends on the device used (and the user’s ability). All of the digits in a measurement, including the uncertain last digit, are called significant digits or significant figures . Note that 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.
Whenever you make a measurement properly, all the digits in the result are significant. But what if you were analyzing a reported value and trying to determine what is significant and what is not? Well, for starters, all nonzero digits are significant, and it is only zeros that require some thought. We will use the terms “leading,” “trailing,” and “captive” for the zeros and will consider how to deal with them.
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 digits in the measurement unless the last digit is a trailing zero lying to the left of the decimal point.
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.
The leading zeros in this example are not significant. We could use exponential notation (as described in Appendix B) and express the number as 8.32407 \(\times\) 10 −3 ; then the number 8.32407 contains all of the significant digits, and 10 −3 locates the decimal point.
The number of significant digits is uncertain in a number that ends with a zero to the left of the decimal point location. The zeros in the measurement 1,300 grams could be significant or they could simply indicate where the decimal point is located. The ambiguity can be resolved with the use of exponential notation: 1.3 \(\times\) 10 3 (two significant digits), 1.30 \(\times\) 10 3 (three significant digits, if the tens place was measured), or 1.300 \(\times\) 10 3 (four significant digits, if the ones place was also measured). In cases where only the decimal-formatted number is available, it is prudent to assume that all trailing zeros are not significant.
When determining significant digits, be sure to pay attention to reported values and think about the measurement and significant digits in terms of what is reasonable or likely—that is, whether the value makes sense. For example, the official January 2014 census reported the resident population of the US as 317,297,725. Do you think the US population was correctly determined to the reported nine significant digits, that is, to the exact number of people? People are constantly being born, dying, or moving into or out of the country, and assumptions are made to account for the large number of people who are not actually counted. Because of these uncertainties, it might be more reasonable to expect that we know the population to within perhaps a million or so, in which case the population should be reported as 317 million, or \(3.17 \times 10^8 \) people.
Significant Digits in Calculations
A second important principle of uncertainty is that results calculated from a measurement are at least as uncertain as the measurement itself. We must take the uncertainty in our measurements into account to avoid misrepresenting the uncertainty in calculated results. One way to do this is to report the result of a calculation with the correct number of significant digits, which is determined by the following three rules for rounding numbers:
- When we add or subtract numbers, we should round the result to the same number of decimal places as the number with the least number of decimal places (the least precise value in terms of addition and subtraction).
- When we multiply or divide numbers, we should round the result to the same number of digits as the number with the least number of significant digits (the least precise value in terms of multiplication and division).
- If the digit to be dropped (the one immediately to the right of the digit to be retained) is less than 5, we “round down” and leave the retained digit unchanged; if it is more than 5, we “round up” and increase the retained digit by 1; if the dropped digit is 5, we 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.)
The following examples illustrate the application of this rule in rounding a few different numbers to three significant digits:
- 0.028675 rounds “up” to 0.0287 (the dropped digit, 7, is greater than 5)
- 18.3384 rounds “down” to 18.3 (the dropped digit, 3, is less than 5)
- 6.8752 rounds “up” to 6.88 (the dropped digit is 5, and the retained digit is even)
- 92.85 rounds “down” to 92.8 (the dropped digit is 5, and the retained digit is even)
Let’s work through these rules with a few examples.
Example \(\PageIndex{5}\): Rounding Numbers
Round the following to the indicated number of significant digits:
- 31.57 (to two significant digits)
- 8.1649 (to three significant digits)
- 0.051065 (to four significant digits)
- 0.90275 (to four significant digits)
Solution
- 31.57 rounds “up” to 32 (the dropped digit is 5, and the retained digit is even)
- 8.1649 rounds “down” to 8.16 (the dropped digit, 4, is less than 5)
- 0.051065 rounds “down” to 0.05106 (the dropped digit is 5, and the retained digit is even)
- 0.90275 rounds “up” to 0.9028 (the dropped digit is 5, and the retained digit is even)
Exercise \(\PageIndex{5}\)
Round the following to the indicated number of significant digits:
- 0.424 (to two significant digits)
- 0.0038661 (to three significant digits)
- 421.25 (to four significant digits)
- 28,683.5 (to five significant digits)
- Answer a
-
0.42
- Answer b
-
0.00387
- Answer c
-
421.2
- Answer d
-
28,684
Example \(\PageIndex{6}\): Addition and Subtraction with Significant Digits Rule:
When we add or subtract numbers, we should round the result to the same number of decimal places as the number with the least number of decimal places (i.e., the least precise value in terms of addition and subtraction).
- Add 1.0023 g and 4.383 g.
- Subtract 421.23 g from 486 g.
Solution
(a)
\[\begin{align*}
&\mathrm{1.0023\: g}\\ +\: &\underline{\mathrm{4.383\: g}\:\:}\\ &\mathrm{5.3853\: g}
\end{align*}\]
Answer is 5.385 g (round to the thousandths place; three decimal places)
(b)
\[\begin{align*}
&\mathrm{486\: g}\\ -\: &\underline{\mathrm{421.23\: g}}\\ &\mathrm{\:\:64.77\: g}
\end{align*}\]
Answer is 65 g (round to the ones place; no decimal places)
Exercise \(\PageIndex{6}\)
- Add 2.334 mL and 0.31 mL.
- Subtract 55.8752 m from 56.533 m.
- Answer a
-
2.64 mL
- Answer b
-
0.658 m
Example \(\PageIndex{7}\): Multiplication and Division with Significant Digits
Rule: When we multiply or divide numbers, we should round the result to the same number of digits as the number with the least number of significant digits (the least precise value in terms of multiplication and division).
- Multiply 0.6238 cm by 6.6 cm.
- Divide 421.23 g by 486 mL.
Solution
(a)
\[\mathrm{0.6238\: cm\times6.6\:cm=4.11708\:cm^2\rightarrow result\: is\:4.1\:cm^2}\:\textrm{(round to two significant digits)}\]
\[\textrm{four significant digits}\times \textrm{two significant digits}\rightarrow \textrm{two significant digits answer}\]
(b)
\[\mathrm{\dfrac{421.23\: g}{486\: mL}=0.86728...\: g/mL\rightarrow result\: is\: 0.867\: g/mL} \: \textrm{(round to three significant digits)}\]
\[\mathrm{\dfrac{five\: significant\: digits}{three\: significant\: digits}\rightarrow three\: significant\: digits\: answer}\]
Exercise \(\PageIndex{7}\)
- Multiply 2.334 cm and 0.320 cm.
- Divide 55.8752 m by 56.53 s.
- Answer a
-
0.747 cm 2
- Answer b
-
0.9884 m/s
In the midst of all these technicalities, it is important to keep in mind the reason why we use significant digits and rounding rules—to correctly represent the certainty of the values we report and to ensure that a calculated result is not represented as being more certain than the least certain value used in the calculation.
Example \(\PageIndex{8}\): Calculation with Significant Digits
One common bathtub is 13.44 dm long, 5.920 dm wide, and 2.54 dm deep. Assume that the tub is rectangular and calculate its approximate volume in liters. Give your answer with the appropriate number of significant digits.
Solution
\[\begin{align*}
V&=l\times w\times d\\ &=\mathrm{13.44\: dm\times 5.920\: dm\times 2.54\: dm}\\ &=\mathrm{202.09459...dm^3}\:\textrm{(value from calculator)}\\ &=\mathrm{202\: dm^3,}\textrm{ or 202 L (answer rounded to three significant digits)}
\end{align*}\]
Exercise \(\PageIndex{8}\):
What is the density of a liquid with a mass of 31.1415 g and a volume of 30.13 cm 3 ? Give your answer with the appropriate number of significant digits.
- Answer
-
1.034 g/mL
Example \(\PageIndex{9}\): Determining Density by Water Displacement
A piece of rebar is weighed and then submerged in a graduated cylinder partially filled with water, with results as shown.
- Use these values to determine the density of this piece of rebar.
- Rebar is mostly iron. Does your result in (a) support this statement? How?
Solution
The volume of the piece of rebar is equal to the volume of the water displaced:
\[\mathrm{volume=22.4\: mL-13.5\: mL=8.9\: mL=8.9\: cm^3}\nonumber\]
(rounded to the nearest 0.1 mL, per the rule for addition and subtraction)
The density is the mass-to-volume ratio:
\[\mathrm{density=\dfrac{mass}{volume}=\dfrac{69.658\: g}{8.9\: cm^3}=7.8\: g/cm^3}\nonumber\]
(rounded to two significant digits, per the rule for multiplication and division)
The density of iron is 7.9 g/cm 3 , very close to that of rebar, which lends some support to the fact that rebar is mostly iron.
Exercise \(\PageIndex{9}\)
An irregularly shaped piece of a shiny yellowish material is weighed and then submerged in a graduated cylinder, with results as shown.
- Use these values to determine the density of this material.
- Do you have any reasonable guesses as to the identity of this material? Explain your reasoning.
- Answer a
-
19 g/cm 3
- Answer b
-
It is likely gold; it has the right appearance for gold and very close to the density given for gold.
Accuracy and Precision
Scientists typically make repeated measurements of a quantity to ensure the quality of their findings and to know both the precision and the accuracy of their results. 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 accepted or true value. Precise values agree with each other; accurate values agree with an accepted value. These characterizations can be extended to other contexts, such as the results of an archery competition (Figure \(\PageIndex{2}\)).
Suppose a quality control chemist at a pharmaceutical company is tasked with checking the accuracy and precision of three different machines that are meant to dispense 10 ounces (296 mL) of cough syrup into storage bottles. She proceeds to use each machine to fill five bottles and then carefully determines the actual volume dispensed, obtaining the results tabulated in Table \(\PageIndex{2}\).
| Dispenser #1 | Dispenser #2 | Dispenser #3 |
|---|---|---|
| 283.3 | 298.3 | 296.1 |
| 284.1 | 294.2 | 295.9 |
| 283.9 | 296.0 | 296.1 |
| 284.0 | 297.8 | 296.0 |
| 284.1 | 293.9 | 296.1 |
Considering these results, she will report that dispenser #1 is precise (values all close to one another, within a few tenths of a milliliter) but not accurate (none of the values are close to the target value of 296 mL, each being more than 10 mL too low). Results for dispenser #2 represent improved accuracy (each volume is less than 3 mL away from 296 mL) but worse precision (volumes vary by more than 4 mL). Finally, she can report that dispenser #3 is working well, dispensing cough syrup both accurately (all volumes within 0.1 mL of the target volume) and precisely (volumes differing from each other by no more than 0.2 mL).
Summary
Measurements are made using a variety of units. It is often useful or necessary to convert a measured quantity from one unit into another. These conversions are accomplished using unit conversion factors, which are derived by simple applications of a mathematical approach called the factor-label method or dimensional analysis. This strategy is also employed to calculate sought quantities using measured quantities and appropriate mathematical relations.
Quantities can be exact or measured. Measured quantities have an associated uncertainty that is represented by the number of significant digits in the measurement. The uncertainty of a calculated value depends on the uncertainties in the values used in the calculation and is reflected in how the value is rounded. Measured values can be accurate (close to the accepted value) and/or precise (showing little variation when measured repeatedly).
Key Equations
- \(T_\mathrm{^\circ C}=\dfrac{5}{9}\times T_\mathrm{^\circ F}-32\)
- \(T_\mathrm{^\circ F}=\dfrac{9}{5}\times T_\mathrm{^\circ C}+32\)
- \(T_\ce{K}={^\circ \ce C}+273.15\)
- \(T_\mathrm{^\circ C}=\ce K-273.15\)
Glossary
- dimensional analysis
- (also, factor-label method) versatile mathematical approach that can be applied to computations ranging from simple unit conversions to more complex, multi-step calculations involving several different quantities
- Fahrenheit
- unit of temperature; water freezes at 32 °F and boils at 212 °F on this scale
- unit conversion factor
- ratio of equivalent quantities expressed with different units; used to convert from one unit to a different unit
- uncertainty
- estimate of amount by which measurement differs from accepted value
- significant digits
- (also, significant figures) all of the measured digits in a determination, including the uncertain last digit
- rounding
- procedure used to ensure that calculated results properly reflect the uncertainty in the measurements used in the calculation
- precision
- how closely a measurement matches the same measurement when repeated
- exact number
- number derived by counting or by definition
- accuracy
- how closely a measurement aligns with a correct value
Contributors and Attributions
-
Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/85abf193-2bd...a7ac8df6@9.110 ).
-
Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/85abf193-2bd...a7ac8df6@9.110 ).