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3.3: Uncertainty in Measurements

Difficulty Level: At Grade Created by: CK-12

Lesson Objectives

• Distinguish between accuracy and precision in measurements.
• Calculate the percent error of a measured quantity.
• Report measured values to the correct number of significant figures based on the measuring tool.
• Perform calculations with measured quantities, rounding the answers to the correct number of significant figures.

Lesson Vocabulary

• accepted value
• accuracy
• error
• experimental value
• percent error
• precision
• significant figures

Check Your Understanding

Recalling Prior Knowledge

• Suppose that a baseball pitcher throws very accurately. What does that mean? Is it different from throwing very precisely?
• When you make a measurement with a specific measuring tool, how well can you read that measurement? Will individual measurements have an affect on quantities that are calculated from those measurements?

When making a measurement, there is always going to be some uncertainty. Some of that uncertainty is related to the reliability of the measuring tool, while some of it is related to the skill of the measurer. When you are performing measurements, you should always strive for the greatest accuracy and precision that you possibly can (Figure below).

Students in a chemistry lab are separating a solution into volumetric flasks, which are each calibrated to hold a specific volume that is known to a high level of accuracy. The long thin stems of these flasks are designed to increase the precision with which a sample of this known volume can be produced.

Accuracy and Precision

In everyday speech, the terms accuracy and precision are frequently used interchangeably. However, their scientific meanings are quite different. Accuracy is a measure of how close a measurement is to the correct or accepted value of the quantity being measured. Precision is a measure of how close a series of measurements are to one another. Precise measurements are highly reproducible, even if the measurements are not near the correct value.

Darts thrown at a dartboard are helpful in illustrating accuracy and precision (Figure below).

The distribution of darts on a dartboard shows the difference between accuracy and precision.

Assume that three darts are thrown at the dartboard, with the bulls-eye representing the true, or accepted, value of what is being measured. A dart that hits the bulls-eye is highly accurate, whereas a dart that lands far away from the bulls-eye displays poor accuracy. Pictured above are the four possible outcomes (Figure above).

(A) The darts have landed far from each other and far from the bulls-eye. This grouping demonstrates measurements that are neither accurate, nor precise.

(B) The darts are close to one another, but far from the bulls-eye. This grouping demonstrates measurements that are precise, but not accurate. In a laboratory situation, high precision with low accuracy often results from a systematic error. Either the measurer makes the same mistake repeatedly or the measuring tool is somehow flawed. A poorly calibrated balance may give the same mass reading every time, but it will be far from the true mass of the object.

(C) The darts are not grouped very near to each other, but they are generally centered around the bulls-eye. This demonstrates poor precision but fairly high accuracy. This situation is not desirable because in a lab situation, we do not know where the "bulls-eye" actually is. Continuing with this analogy, measurements are taken in order to find the bulls-eye. If we could only see the locations of the darts and not the bulls-eye, the large spread would make it difficult to be confident about where the exact center was, even if we knew that the darts were thrown accurately (which would correspond to having equipment that is calibrated and operated correctly).

(D) The darts are grouped together and have hit the bulls-eye. This demonstrates high precision and high accuracy. Scientists always strive to maximize both in their measurements. Turning back to our laboratory situation, where we can see the darts but not the bulls-eye, we have a much narrower range of possibilities for the exact center than in the less precise situation depicted in part C.

Percent Error

An individual measurement may be accurate or inaccurate, depending on how close it is to the true value. Suppose that you are doing an experiment to determine the density of a sample of aluminum metal. The accepted value of a measurement is the true or correct value based on general agreement with a reliable reference. For aluminum, the accepted density is 2.70 g/cm3. The experimental value of a measurement is the value that is measured during the experiment. Suppose that in your experiment you determine an experimental value of 2.42 g/cm3 for the density of aluminum. The error of an experiment is the difference between the experimental and accepted values.

Error=experimental valueaccepted value\begin{align*}\text{Error} = \text{experimental value} - \text{accepted value}\end{align*}

If the experimental value is less than the accepted value, the error is negative. If the experimental value is larger than the accepted value, the error is positive. Often, error is reported as the absolute value of the difference in order to avoid the confusion of a negative error. The percent error is the absolute value of the error divided by the accepted value and multiplied by 100%.

Percent Error=|experimental valueaccepted value|accepted value×100%\begin{align*}\text{Percent Error} = \dfrac{\vert \text{experimental value} - \text{accepted value} \vert}{\text{accepted value}} \times 100\%\end{align*}

To calculate the percent error for the aluminum density measurement, we can substitute the given values of 2.45 g/cm3 for the experimental value and 2.70 g/cm3 for the accepted value.

Percent Error=|2.45 g/cm32.70 g/cm3|2.70 g/cm3×100%=9.26%\begin{align*}\text{Percent Error} = \dfrac{\vert 2.45 \ \text{g/cm}^3 - 2.70 \ \text{g/cm}^3 \vert}{2.70 \ \text{g/cm}^3} \times 100\% = 9.26\%\end{align*}

If the experimental value is equal to the accepted value, the percent error is equal to 0. As the accuracy of a measurement decreases, the percent error of that measurement rises.

Significant Figures in Measurements

Uncertainty

Some error or uncertainty always exists in any measurement. The amount of uncertainty depends both upon the skill of the measurer and upon the quality of the measuring tool. While some balances are capable of measuring masses only to the nearest 0.1 g, other highly sensitive balances are capable of measuring to the nearest 0.001 g or even better. Many measuring tools such as rulers and graduated cylinders have small lines which need to be carefully read in order to make a measurement. Pictured below is an object (indicated by the blue arrow) whose length is being measured by two different rulers (Figure below).

With either ruler, it is clear that the length of the object is between 2 and 3 cm. The bottom ruler contains no millimeter markings, so the tenths digit can only be estimated, and the length may be reported by one observer as 2.5 cm. However, another person may judge that the measurement is 2.4 cm or perhaps 2.6 cm. While the 2 is known for certain, the value of the tenths digit is uncertain.

The top ruler contains marks for tenths of a centimeter (millimeters). Now, the same object may be measured as 2.55 cm. The measurer is capable of estimating the hundredths digit because he can be certain that the tenths digit is a 5. Again, another measurer may report the length to be 2.54 cm or 2.56 cm. In this case, there are two certain digits (the 2 and the 5), with the hundredths digit being uncertain. Clearly, the top ruler is a superior ruler for measuring lengths as precisely as possible.

Determining 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 ruler example, the bottom ruler gave a length with two significant figures, while the top ruler gave a length with three significant figures. In a correctly reported measurement, the final digit is significant but not certain. Insignificant digits are not reported. It would not be correct to report the length as 2.553 cm with either ruler because there is no possible way that the thousandths digit could be estimated. The 3 is not significant and would not be reported.

When you look at a reported measurement, it is necessary to be able to count the number of significant figures. The table below (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.

Sample Problem 3.8: Counting Significant Figures

How many significant figures are there in each of the following measurements?

1. 19.5 m
2. 0.0051 L
3. 204.80 g
4. 1.90 × 105 s
5. 14 beakers
6. 700 kg

Step 1: Plan the problem.

Follow the rules for counting the number of significant figures in a measurement, paying special attention to the location of zeros in each. Note each rule that applies according to the table above (Table above).

Step 2: Solve.

1. three (rule one)
2. two (rule three)
3. five (rules two & five)
4. three (rule five)
5. infinite
6. one (rule four)

The 14 beakers is a counted set of items and not a measurement, so it has an infinite number of significant figures.

Practice Problems
1. Count the number of significant figures in each measurement.
1. 0.00090 L
2. 255 baseballs
3. 435,210 m
4. 40.1 kg
5. 9.026 × 10−6 mm
6. 12.40°C

Significant Figures in Calculations

Many reported quantities in science are the result of calculations involving two or more measurements. Density involves mass and volume, both of which are measured quantities. As an example, say that you have a precise balance that gives the mass of a certain object as 21.513 g. However, the volume is measured very roughly by water displacement, using a graduated cylinder that can only be read to the nearest tenth of a milliliter. The volume of the object is determined to be 8.2 mL. On a calculator, the density (mass divided by volume) would come out as 2.623536585 g/mL. Hopefully, it should be apparent that the calculator is giving us far more digits than we actually can be certain of knowing. In fact, the density should be reported as 2.6 g/mL. This is because the result of a calculated answer can be no more precise than the least precise measurement from which it was calculated. Since the volume was known only to two significant figures, the resultant density needs to be rounded to two significant figures.

Rounding

Before we get to the specifics of the rules for determining the significant figures in a calculated result, we need to be able to round numbers correctly. To round a number, first decide how many significant figures the number should have. Once you know that, round to the correct number of digits, starting from the left. If the number immediately to the right of the last significant digit is less than 5, it is dropped, and the value of the last significant digit remains the same. If the number immediately to the right of the last significant digit is greater than or equal to 5, the last significant digit is increased by 1.

Consider the measurement 207.518 m. Right now, the measurement contains six significant figures. How would we successively round it to fewer and fewer significant figures? Follow the process listed below (Table below).

Significant Figure Rounding
Number of Significant Figures Rounded Value Reasoning
6 207.518 All digits are significant
5 207.52 8 rounds the 1 up to 2
4 207.5 2 is dropped
3 208 5 rounds the 7 up to 8
2 210 8 is replaced by a 0 and rounds the 0 up to 1
1 200 1 is replaced by a 0

Significant Figures in Addition and Subtraction

Consider two separate mass measurements: 16.7 g and 5.24 g. The first mass measurement (16.7 g) is known only to the tenths place, which is one digit after the decimal point. There is no information about its hundredths place, so that digit cannot be assumed to be zero. The second measurement (5.24 g) is known to the hundredths place, which is two digits after the decimal point.

When these masses are added together, the result on a calculator is 16.7 + 5.24 = 21.94 g. Reporting the answer as 21.94 g suggests that the sum is known all the way to the hundredths place. However that cannot be true because the hundredths place of the first mass was completely unknown. The calculated answer needs to be rounded in such a way as to reflect the certainty of each of the measured values that contributed to it. For addition and subtraction problems, the answer should be rounded to the same number of decimal places as the measurement with the lowest number of decimal places. The sum of the above masses would be properly rounded to a result of 21.9 g.

When working with whole numbers, pay attention to the last significant digit that is to the left of the decimal point and round your answer to that same point. For example, consider the subtraction problem 78,500 m – 362 m. The calculated result is 78,138 m. However, the first measurement is known only to the hundreds place, as the 5 is the last significant digit. Rounding the result to that same point means that the final calculated value should be reported as 78,100 m.

Significant Figures in Multiplication and Division

The density of a certain object is calculated by dividing the mass by the volume. Suppose that a mass of 37.46 g is divided by a volume of 12.7 cm3. The result on a calculator would be:

D=mV=37.46 g12.7 cm3=2.94960299 g/cm3\begin{align*}\text{D}=\frac{\text{m}}{\text{V}}=\frac{37.46 \ \text{g}}{12.7 \ \text{cm}^3}=2.94960299 \ \text{g/cm}^3\end{align*}

The value of the mass measurement has four significant figures, while the value of the volume measurement has only three significant figures. For multiplication and division problems, the answer should be rounded to the same number of significant figures as the measurement with the lowest number of significant figures. Applying this rule results in a density of 2.95 g/cm3, which has three significant figures—the same as the volume measurement.

Note that the rule for multiplication and division problems is different than the rule for addition and subtraction problems. For multiplication and division, it is the number of significant figures that must be considered. For addition and subtraction, it is the position of the decimal place that determines the correct rounding. Review the sample problem below, paying special attention to this distinction.

Sample Problem 3.9: Significant Figures in Calculations

Perform the following calculations, rounding the answers to the appropriate number of significant figures.

1. 0.048 m × 32.97 m
2. 21.9 g – 19.417 g
3. 14,570 kg ÷ 5.81 L
4. 71.2 cm + 90 cm

Step 1: Plan the problem.

Decide which calculation rule applies. Analyze each of the measured values to determine how many significant figures should be in the result. Perform the calculation and round appropriately. Apply the correct units to the answer. When adding or subtracting, the units in each measurement must be identical and then remain the same in the result. When multiplying or dividing, the units are also multiplied or divided.

Step 2: Calculate.

1. 0.048 m × 32.97 m = 1.6 m2 - Round to two significant figures because 0.048 has two.
2. 21.9 g - 19.417 g = 2.5 g - Answer ends at the tenths place because of 21.9.
3. 14,570 kg ÷ 5.81 L = 2510 kg/L - Round to three significant figures because 5.81 has three.
4. 71.2 cm + 90 cm = 160 cm - Answer ends at the tens place because of 90.
Practice Problems
1. Solve each problem, rounding each answer to the correct number of significant figures.
1. 132.3 g ÷ 29.600 mL
2. 3.27 g/cm3 × 0.086 cm3
3. 125 m + 61.3 m + 310 m
4. 3.0 × 104 L − 1244 L
2. A rectangular prism has dimensions of 3.7 cm by 4.81 cm by 1.90 cm. The mass of the prism is 49.72 g. Calculate its density. (Hint: Do the entire calculation without rounding until the final answer in order to reduce “rounding error.”)

Lesson Summary

• Accuracy refers to how close a measured value is to the accepted value, whereas precision indicates how close individual measurements within a set are to each other.
• Percent error is the difference between the experimental and accepted values divided by the accepted value and multiplied by 100.
• The measuring tool dictates how many significant figures can be reported in a measurement. Significant figures include all of the certain digits plus one uncertain digit. A set of rules is followed for determining the number of significant figures in numbers that contain zeros. Counted quantities have infinite significant figures.
• For addition and subtraction problems, the answer should be rounded to the same number of decimal places as the measurement with the lowest number of decimal places. For multiplication and division problems, the answer should be rounded to the same number of significant figures as the measurement with the lowest number of significant figures.

Lesson Review Concepts

Reviewing Concepts

1. The density of a sample of copper metal was determined by three different students (Table below). Each performed the measurement three times. Describe the accuracy and precision of each student’s measurements. The accepted value for the density of copper is 8.92 g/cm3.
Density of copper (g/cm
Student Trial 1 Trial 2 Trial 3
Jane 8.94 8.89 8.91
Justin 8.32 8.31 8.34
Julia 8.64 9.71 9.13
1. What is wrong with the following statement? “My measurement of 8.45 m for the width of the room is very precise.”
2. Consider the following 5 mL graduated cylinders, which contain identical quantities of liquid. Which cylinder yields a measurement with a greater number of significant figures? How many significant figures can be reported for each cylinder?

1. Report the length measurement of the pink bar to the correct number of significant figures. Which digits in your measurement are certain? Which are uncertain?

1. How many significant figures are in each of the following measurements?
1. 9 potatoes
2. 4.05 cm
3. 0.0061 kg
4. 50 mL
5. 8.00 × 109 μg
6. 720.00 s
2. Round the measured quantity of 31.0753 g to each of the following amounts of significant figures.
1. five
2. four
3. three
4. two
5. one

Problems

1. Kyle measures the mass of a solid sample to be 8.09 g. The accepted value for the mass is 8.42 g. Calculate Kyle’s percent error.
2. Jamelle performs three separate determinations of the density of a mineral sample. She gets values of 4.58 g/cm3, 4.79 g/cm3, and 4.55 g/cm3.
1. Calculate the average value of the density of the mineral.
2. The deviation of a measured value is defined as the absolute value of the difference between the measured value and the average value: Deviation=|measured valueaverage value|\begin{align*}\text{Deviation}=\vert \text{measured value} - \text{average value} \vert\end{align*}. Calculate the deviation for each of the three measurements. According to the deviations, which measurement appears to be poorest compared to the others?
3. The average deviation is the sum of all the deviations divided by the total number of measurements. Calculate the average deviation of the three measurements.
4. When average deviation is high, does that indicate good precision or poor precision in the measurements? Explain.
3. Convert a measurement of 2.75 hours to seconds. Are the required conversion factors measured or exact quantities?
4. Convert 466.84 cm to inches, given that 1 inch = 2.54 cm. The conversion factor between centimeters and inches is a measured quantity.
5. Perform the following calculations and round to the correct number of significant figures.
1. 78.2 g ÷ 32 cm3
2. 3.0 m/s × 9.21 s
3. 59 g + 4 g + 0.79 g
4. 34,000 km − 2430 km
5. (9.59 g + 1.098 g) ÷ 2.313 mL

Points to Consider

Measurements will be a constant consideration throughout your study of chemistry. Next you will begin a study of the atom, its component parts, and the evolution of the atomic model.

• Atoms are extremely small and extremely light. How do you think that the mass and size of an atom can be measured? Do you think the accuracy of these measurements has improved over time?

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