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# 3.2: Unit Conversions, Error, and Uncertainty

Difficulty Level: At Grade Created by: CK-12

## Lesson Objectives

• Differentiate between accuracy and precision as they relate to a given measurement.
• Describe the reliability of a measurement and how it can be expressed in terms of uncertainty.
• Distinguish between mass and weight and describe how mass and weight are determined.
• Understand the concept of volume and how it can be determined for various substances, including regularly shaped and irregularly shaped solids.
• Define density and perform density calculations.
• Describe how many significant figures there are in a given measurement, and be able to perform measurement calculations involving numbers with significant figures.

## Lesson Vocabulary

• meniscus: The curved upper surface of a liquid in a tube.
• estimate: A process of referencing a physical quantity in terms of a calibration or reference point.
• uncertainty: All measurements have an uncertainty equal to one half of the smallest difference between reference marks.
• accuracy: Describes how close an estimate is to a known standard.
• precision: Describes how close estimates are to one another.
• calibration: A method of setting or correcting a measuring device by matching it to known measurement standards.
• percent uncertainty: The ratio of the uncertainty to the measured value, multiplied by one hundred.
• percent error: An expression of the accuracy of a measurement, standardized to how large the measurement is.
• significant figures: Consist of all the certain digits in that measurement plus one uncertain or estimated digit.
• density: An expression of the mass of substance in terms of the volume occupied by the substance.
• mass: The quantity of inertia possessed by an object.
• weight: The gravitational force acting on a mass, as measured on a scale.
• Fahrenheit scale: The most commonly used scale in the United States, it defines the normal freezing point and boiling point of water as 32°F and 212°F, respectively.
• Celsius scale: The most commonly used scale around the world, it defines the normal freezing point and boiling point of water as 0°C and 100°C, respectively.
• Kelvin scale: Referred to as the absolute temperature scale, it defines absolute zero as the lowest theoretically possible temperature.

1. What will it cost to carpet a room if the room is 10 feet wide and 20 feet long, and the price of carpet is \$2.36 per ft2?
2. Which of the following is the larger amount: 2.35 × 102 L or 3.46 × 105 mL?
3. List two SI units that would be appropriate for measuring each of the following quantities: volume, weight, and length.

## Introduction

In the last lesson, we studied the concept of measurement and how numbers are used to express various physical quantities. We studied scale and magnitude and investigated how to use scientific notation and SI units to report numbers in an efficient and consistent manner. However, we have not yet studied how measurement takes place. There are many different measurements we make in our investigations of the chemical world. Measurements such as distance, volume, and mass are important values that are frequently used to describe the characteristics and behavior of chemical species. To make measurements, we use instruments labeled with a known scale. However, it is impossible for measurements to be exact. In this lesson, we are going to study how measurements are made as well as the error and uncertainty involved in measurements.

## Uncertainty

Figure above shows a graduated cylinder, which is an instrument that is used to measure volume. The graduated cylinder gets its name because of the gradation or scaled lines drawn on its side. These serve as reference points that correspond to known volumes. When we make a measurement using a graduated cylinder, we look at the meniscus, or curved surface, of the liquid and estimate where the bottom of the meniscus is relative to the gradations. All measurement devices have reference marks of some kind. Can you think of another example of a measurement device with regular reference marks?

Example 3.9

Make an estimate of the volume that is shown in Figure above.

We see the bottom of the meniscus is at approximately 52.9 mL. We report this in mL because our cylinder is a 100 mL graduated cylinder, with mL reference marks. There are 9 equally spaced marks between the 50 mL and 60 mL lines, so each one must represent 1 mL. As we will see next, the space between these marks represents an area of uncertainty with regard to the estimate.

When the volume in our previous example was reported to be 52.9 mL, the uncertainty associated with this estimate also needed to be reported. For example, we know for certain that the true value for the volume must be between 52 mL and 53 mL. However, there is uncertainty regarding how close the value is to 52 or 53. We estimated the volume to be 52.9, but some students may have reported 52.8 or 53.0. These would be accurate estimates because they fall within the acceptable uncertainty of the device. All measurements have an uncertainty equal to one half of the smallest difference between reference marks. For our graduated cylinder, there is 1 mL between consecutive marks, so the uncertainty is one half of that value, 0.5 mL. To be rigorous about our certainty regarding this measured value, the estimate of 52.9 mL should be reported as 52.9 ± 0.5 mL.

### Accuracy

In measuring quantities we always aim for high accuracy. Estimates that fall within the range of uncertainty for a given instrument are said to be accurate. In our previous example, all of the values between and including 52.4 mL and 53.4 mL would be considered accurate. Estimates that fall outside this range are inaccurate. Accuracy describes how close an estimate is to a known standard.

### Precision

Precision describes how close estimates are to one another. Estimates that are relatively close to one another are precise. Let’s assume that ten different students made an estimate of the volume shown in Figure above, and the values were: 52.9, 52.8, 52.9, 52.9, 53.3, 52.0, 52.8, 52.9, 53.0, 52.8. We can determine how precise these data are by analyzing how close they are to an average. The average could be the mean, median or mode. The most common understanding of the average is the mean. This value is calculated by adding up all the numbers and then dividing by the total number of values. Other terms that can refer to the average are the median and the mode. The median is the middle value in a numerically ordered list of numbers. The mode is the value that occurs most often in a set of numbers. If no number is repeated, there is no mode for the list. Here are the calculated averages:

Mean=52.9+52.8+52.9+52.9+53.3+52.0+52.8+52.9+53.0+52.810=52.8\begin{align*}\text{Mean}=\dfrac{52.9+52.8+52.9+52.9+53.3+52.0+52.8+52.9+53.0+52.8}{10}=52.8\end{align*}

Median=52.0,52.8,52.8,52.8,[52.9],52.9,52.9,52.9,53.0,53.3=52.9\begin{align*}\text{Median}=52.0, 52.8, 52.8, 52.8, [52.9], 52.9, 52.9, 52.9, 53.0, 53.3=52.9\end{align*}

Mode=52.0,52.8,52.8,52.8,[52.9,52.9,52.9,52.9],53.0,53.3=52.9\begin{align*}\text{Mode}=52.0, 52.8, 52.8, 52.8, [52.9, 52.9, 52.9, 52.9], 53.0, 53.3=52.9\end{align*}

Based on this analysis, we see the value 52.8 was the mean, and 52.9 was the median and mode. Therefore, values that are relatively close to these averages would be considered precise. We can also calculate the standard deviation for these data, which is a more refined way of determining the precision of estimates. However, we will not concern ourselves with standard deviation at this point.

### Accuracy vs. Precision

As we just saw, accuracy describes how close a given set of data is to the “real” value, while precision describes how close the data points are to one another. These concepts are illustrated in Figure below.

Here we see three darts thrown at four different targets. Accurate shots would be those that were close to the bull’s-eye (the inner circle). Precision would be the shots that were close to one another.

Target A represents the best possible "data". All of the data point points are clustered in the center, close to the "actual" value and close to one another. This data is both accurate and precise. In target B, the set of data has good accuracy overall if the points are averaged together. However, the three points are not very close to each other making the imprecise. Target C, on the other hand, shows precise but inaccurate data. The three data points are close together, making them precise, but are far from the center of the target, giving low accuracy. Target D represents the worst possible "data". The data points are far from the center of the target lacking any accuracy, as well as being far apart from each other, lacking precision.

### Calibration

When using measuring devices, we often use a technique called calibration to increase the accuracy of our measurements. Calibration is a method of setting or correcting a measuring device by matching it to known measurement standards. To better understand calibration, we will look at the example of calibrating a thermometer. All thermometers are slightly different in their temperature readings. One way to calibrate a thermometer is by using the freezing point and boiling point of water. If we know that water freezes at 0°C and boils at 100°C, we can calibrate our thermometer by measuring the temperature of ice water and of boiling water. We place the thermometer in ice water and wait for the thermometer liquid to reach a stable height, then place a mark at this height which represents 0°C. Then we place the thermometer in boiling water, and after waiting for the thermometer liquid to reach a stable height, we place a mark at this height which represents 100°C. We can then place 100 equally spaced divisions between our 0 and 100°C marks to each represent 1°C. Our thermometer has now been calibrated using the known values for the freezing point and boiling point of water, and can be used to measure temperatures of objects between 0 and 100°C.

Calibration is used to standardize a variety of measuring devices, including meter sticks, graduated cylinders, scales, and thermometers. It is a good idea to calibrate any measuring equipment you use in an experiment to make sure the data you are collecting is measured as accurately as possible.

### Percent Uncertainty

To express the uncertainty in a measurement, we can calculate percent uncertainty. Percent uncertainty is the ratio of the uncertainty to the measured value, multiplied by one-hundred. For instance, the percent uncertainty associated with the measurement of (52.9 ± 0.5 mL), would be

%uncertainty=0.552.9×100=0.95%1%\begin{align*} \% \; \text{uncertainty} = \dfrac{0.5}{52.9} \times 100 = 0.95 \% \approx 1 \% \end{align*}

Example 3.10

Using our estimate of 52.9 mL, what would be the range of possible values for the true volume?

Upper estimate = 52.9 + 0.5 = 53.4 mL

Lower estimate = 52.9 - 0.5 = 52.4 mL

Assuming that our equipment is accurate, we can be confident that the true volume of the sample is somewhere in between these two values.'

### Percent Error

On the other hand, percent error is an expression of the accuracy of a measurement. There are various possible sources of error that arise in measurement. For example, there can be error associated with the observation, like misreading a graduated cylinder. There is error associated with the method or the procedure, like not drying a wet solid before weighing. Error can also arise from the object being measured. For example, a pure solid may have a residue fixed to it that affects its mass. There can also be errors that arise from the measurement instrument, like not zeroing a balance, or improper calibration. Percent error is calculated as follows:

%error=|MeasuredAccepted|Accepted×100\begin{align*}\% \; \text{error}=\dfrac{|\text{Measured} - \text{Accepted}|}{\text{Accepted}} \times 100\end{align*}

Example 3.11

Make an estimate of volume for the image shown in Figure below, and answer the questions below.

Which of the following estimates would be accurate? Report your answer in terms of uncertainty.

Which of the following sets of estimates would be most precise?

If a student reported an volume of 45.0 mL, calculate the percent error in his or her measurement if the actual volume is exactly 43.0 mL.

## Significant Figures

The significant figures in a measurement consist of all the certain digits in that measurement plus one uncertain or estimated digit. In the graduated cylinder example from the previous section, the measured value was reported to be 52.9 mL, which includes 3 significant figures. In a correctly reported measurement, the final digit is significant but not certain. Insignificant digits are not reported. It would not be inncorrect to report the volume as 52.923 mL, because even the tenths place (the 9) is uncertain, so no reasonable estimate could be made for any of the following digits.

When you look at a reported measurement, it is necessary to be able to count the number of significant figures. Table below details the rules for determining the number of significant figures in a reported measurement. For the examples in the table, assume that the quantities are correctly reported values of a measured quantity.

Significant Figure Rules
Rule Examples
1. All nonzero digits in a measurement are significant

A. 237 has three significant figures.

B. 1.897 has four significant figures.

2. Zeros that appear between other nonzero digits are always significant.

A. 39,004 has five significant figures.

B. 5.02 has three significant figures.

3. Zeros that appear in front of all of the nonzero digits are called left-end zeros. Left-end zeros are never significant.

A. 0.008 has one significant figure.

B. 0.000416 has three significant figures.

4. Zeros that appear after all nonzero digits are called right-end zeros. Right-end zeros in a number that lacks a decimal point are not significant.

A. 140 has two significant figures.

B. 75,210 has four significant figures.

5. Right-end zeros in a number with a decimal point are significant. This is true whether the zeros occur before or after the decimal point.

A. 620.0 has four significant figures.

B. 19,000. has five significant figures

It needs to be emphasized that just because a certain digit is not significant does not mean that it is not important or that it can be left out. Though the zero in a measurement of 140 may not be significant, the value cannot simply be reported as 14. An insignificant zero functions as a placeholder for the decimal point. When numbers are written in scientific notation, this becomes more apparent. The measurement 140 can be written as 1.4 × 102, with two significant figures in the coefficient. A number with left-end zeros, such as 0.000416, can be written as 4.16 × 10−4, which has 3 significant figures. In some cases, scientific notation is the only way to correctly indicate the correct number of significant figures. In order to report a value of 15,000,000 with four significant figures, it would need to be written as 1.500 × 107. The right-end zeros after the 5 are significant. The original number of 15,000,000 only has two significant figures.

### Exact Quantities

When numbers are known exactly, the significant figure rules do not apply. This occurs when objects are counted rather than measured. In your science classroom, there may be a total of 24 students. The actual value cannot be 23.8 students, as there is no such thing as 8 tenths of a student. So the 24 is an exact quantity. Exact quantities are considered to have an infinite number of significant figures; the importance of this concept will be seen later when we begin looking at how significant figures are dealt with during calculations. Numbers in many conversion factors, especially for simple unit conversions, are also exact quantities and have infinite significant figures. There are exactly 100 centimeters in 1 meter and exactly 60 seconds in 1 minute. Those values are definitions and are not the result of a measurement.

### Adding and Subtracting Significant Figures

The sum or difference is determined by the smallest number of significant figures to the right of the decimal point in any of the original numbers.

Example 3.13

89.332+1.1=90.432   round to 90.4\begin{align*}89.332+1.1=90.432\ \ \ \text{round to } 90.4\end{align*}

Example 3.14

2.0970.12=1.977   round to 1.98\begin{align*}2.097-0.12=1.977\ \ \ \text{round to } 1.98\end{align*}

### Multiplying and Dividing Significant Figures

The number of significant figures in the final product or quotient is equal to the number of significant figures in the starting value that has the fewest significant figures.

Example 3.15

2.8×4.5039=12.61092   round to 13\begin{align*}2.8 \times 4.5039=12.61092\ \ \ \text{round to }13\end{align*}

Example 3.16

6.85÷112.04=0.0611388789   round to 0.0611\begin{align*}6.85 \div 112.04=0.0611388789\ \ \ \text{round to } 0.0611\end{align*}

Example 3.17

For this example, the value 8 is known to be exact (so it has an infinite number of significant figures).

0.2786×8=2.229\begin{align*}0.2786 \times 8=2.229\end{align*}

## Calculating Density

Imagine holding a tennis ball in one hand and an orange in the other. Why does the orange feel heavier than the tennis ball, even though the two objects are about the same size? This can be explained with the concept of density. Density is an expression of the mass of a substance in terms of the volume occupied by the substance. The equation for density is:

Density=massvolume\begin{align*}\text{Density}=\dfrac{\text{mass}}{\text{volume}}\end{align*}

D=mV\begin{align*}D=\dfrac{m}{V}\end{align*}

So, even though a tennis ball and an orange may be about the same volume, the orange contains more mass within that volume than does the tennis ball. Therefore, the orange has a higher density. This is because the orange contains mostly water and the tennis ball contains mostly air; as you might imagine, water is much heavier than air.

Density is typically reported in terms of gram per milliliter (g/mL) or the equivalent value, grams per cubic centimeter (g/cm3). Oftentimes, scientists compare the density of an object to the density of water which is 1 g/mL at room temperature (25°C). The densities of some common materials are listed in the Table below.

Material

Density (g/mL)

hydrogen

0.00009

oxygen

0.0014

water

1.0

aluminum

2.7

iron

7.9

gold

19.3

### Mass vs. Weight

The terms mass and weight, while often used interchangeably, are technically different terms. Mass is the quantity of inertia possessed by an object. Weight refers to the gravitational force acting on a mass, as measured on a scale. On the surface of the earth, the numerical values of mass and the corresponding force of gravity (weight) are approximately equivalent. For now, we will use the terms mass and weight interchangeably although mass is the more appropriate scientific term.

### Determining the Volume of Regularly Shaped Objects

In order to calculate density, we must know the volume the object occupies. We can calculate the volumes of some regularly shaped objects using the following expressions in Table below.

Formulas for Calculating Volumes of Regularly Shaped Objects
Volume of a cube l×w×h\begin{align*}l \times w \times h\end{align*}
Volume of a sphere 43πr3\begin{align*}\dfrac{4}{3}\pi r^3\end{align*}
Volume of a cylinder πr2h\begin{align*}\pi r^2 h\end{align*}
Volume of a cone 13πr2h\begin{align*}\dfrac{1}{3}\pi r^2 h\end{align*}

### Determining the Volume of Irregularly Shaped Objects

If a solid is irregularly shaped, we can determine its volume by measuring the volume of water displaced by the solid. For example, say you want to measure the volume of the toy dinosaur in Figure below. After placing the dinosaur in the water, the volume measured in the container increases by an amount that is equal to the total volume of the dinosaur. Note that this method only works for solids that do not dissolve in water. If you tried to measure the volume occupied by a pile of salt, the salt would dissolve in the water and this method would not work very well.

Displacement of water by irregular solid.

## Temperature Scales

There are three temperature scales that are commonly used in measurement. Their units are °F (degrees Fahrenheit), °C (degrees Celsius), and K (Kelvin). The Fahrenheit scale, which is the most commonly used scale in the United States, defines the normal freezing point and boiling point of water as 32°F and 212°F, respectively. The Celsius scale defines the normal freezing point and boiling point of water as 0°C and 100°C, respectively. The Celsius scale is commonly used in most countries across the globe. The Kelvin scale, which is also referred to as the absolute temperature scale, defines absolute zero as the lowest theoretically possible temperature, which means that temperatures expressed in Kelvin cannot be negative numbers. We will further study the origins of this temperature scale in the chapter States of Matter.

### Converting Temperature Scales

Regardless of the temperature scale used, it is important to be able to convert from one scale to another. Here are the conversions we use.

°F to °C

TC=(TF32)×59\begin{align*}T_{^\circ C}=(T_{^\circ F}-32) \times \frac{5}{9}\end{align*}

°C to °F

TF=95×(TC)+32\begin{align*}T_{^\circ F}=\frac{9}{5} \times (T_{^\circ C})+32\end{align*}

°C to K

TK=TC+273.15\begin{align*}T_K=T_{^\circ C}+273.15\end{align*}

K to °C

TC=TK273.15\begin{align*}T_{^\circ C}=T_K-273.15\end{align*}

Example 3.19

The melting point of mercury is -38.84°C. Convert this value to degrees Fahrenheit and degrees Kelvin.

TFTFTKTK=95×(38.84C)+32=37.12F=38.84C+273.15=234.75 K\begin{align*} T_{^\circ F}&=\frac{9}{5} \times (-38.84^\circ \text{C})+32 \\ T_{^\circ F}&=-37.12 ^\circ \text{F} \\ T_K&=-38.84^\circ \text{C}+273.15 \\ T_K&=234.75 \ \text{K} \end{align*}

## Lesson Summary

• Accuracy describes how close an estimate is to a known standard.
• Precision describes how close estimates are to one another.
• The accuracy of an estimate cannot be improved through calculation.
• Calibration is a technique used to standardize a measuring instrument and increase the accuracy of measurements.
• Estimation, as used in measurement, is the process of referencing a physical quantity in terms of a calibration or reference point. All measurement devices have reference marks of some kind.
• All measurements have an associated uncertainty. It is expressed as one-half of the smallest difference between calibration marks. It can also be expressed as a percent.
• Percent error is an expression of the accuracy of a measurement, standardized to how large the measurement is.
• Sources of error can originate from observation errors, methods or procedural errors, as well as errors associated with object that are measured. They can also originate from the measurement instrument itself.
• Significant figures are figures associated with uncertainty of a measurement.
• Density is an expression of the mass of substance in terms of the volume occupied by the substance.
• Density is typically reported in terms of grams/mL (g/mL) or grams per cubic centimeter (g/cm3).
• If a solid is irregularly shaped, we can determine its volume by measuring the volume of water that the solid displaces.
• There are three temperature scales that are commonly used. Their units are °F (degrees Fahrenheit), °C (degrees Celsius), and K (Kelvin).

## Review Questions

Make an estimate of the length that is shown in Figure below and use this information to answer the following questions.

1. If the length in Figure above were estimated to be 11.65 cm ± 0.05, what would be the range of values that fall within the acceptable uncertainty for this instrument?
2. Which of the following length estimates would be accurate for figure shown above?
1. 11.59
2. 11.71
3. 11.64
4. 12
3. Which of the following length estimates would be precise for the Figure above?
1. 11.64, 11.65, 11.65
2. 11.60. 11.56, 11.45
3. 10.9, 12.2, 12
4. 11, 11.23, 11.234
4. A student measures the density of gold and finds it to be 18.3 g/mL. The accepted value from the Handbook of Chemistry and Physics is 19.3 g/mL (Lide 1992-1993). What is the percent error of the student’s results?
5. How would you report 40.889 m3 to three significant figures using scientific notation?
6. Complete the Table below.
7. °F °C K
A 57
B 37
C -40
8. What is the average mass of three objects whose individual masses are 10.3 g, 9.334 g, and 9.25 g?
9. Complete the following calculation and report the answer with the correct number of significant figures: (1.68)(7.874)(1.0000/55.85).
10. Solve the following equation for n and report the answer with the correct number of significant figures: (11.2/760.0)(123.4) = n(0.0821)(298.3)

## Points to Consider

• Compare and contrast the differences between a number and a measurement? What would be an example of a number and an example of a measurement?
• One way to remember the formula for density, as well as how to rearrange variables within the density equation is with the following formula triangle
• In this lesson we have seen that all measurements have an associated uncertainty. Yet, this does not imply there are flaws in the process of measurement. How might you explain to someone the concept of uncertainty, and how reconciling uncertainty in measurement actually makes the estimate more trustworthy, not less?

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