ACS Student Magazine

## Know Your Techniques: Accuracy, Precision, and Using the Right Instrument

You’re in the lab and need to prepare a sample. Do you use a volumetric flask or an Erlenmeyer? The analytical balance or a top-loader? What about pipettes? Choosing the right equipment is the first step in a successful experiment. You need precise measurements at each step because small errors can add up quickly. But sometimes you don’t actually need the most accurate values, and spending a lot of time using more precise options might not provide any benefit.

How can you decide if (and when) accuracy and precision are worth the extra effort? How do you select the right measuring instrument? Read on to find out.

## Understanding Accuracy and Precision

In experimentation, it’s important to remember what we mean by accuracy and precision . Accuracy is how close your value is to the true value. The problem is you can’t always know what the actual value is. If you’re in a lab for a chemistry course, chances are the instructor or TA knows the value you should get and can let you know how accurate you are. But for research projects, often you won’t know the value. In fact, you’re usually trying to find it.

This is where precision comes in. Precision is how reproducible your results are, or how close a set of measurements are to each other. If you measure the volume of a liquid multiple times with the same piece of glassware, do you get the same value each time? If you do, you know you’re being precise, even if you can’t determine your accuracy.

Striving for high precision ensures you’re doing your best to eliminate errors from measurements and calculations. The more precise you are, the better your chances are of getting an accurate result because high-precision equipment is usually calibrated to a high degree of accuracy. If you are struggling with precision, it is usually a sign that you are using the instrument incorrectly or there is a problem with the equipment itself.

It is possible to have precise measurements and still get the wrong result. Your hypothesis or calculations might be wrong, your equipment might need to be recalibrated, or your reagents might be contaminated. But for now, let’s focus just on precision.

Another way to achieve better precision (and therefore, hopefully, high accuracy) is to be able to report your values to a larger number of significant figures. Significant figures are all the digits in a measured or calculated value you know for certain, plus a final digit that contains some uncertainty. All measurements have an inherent amount of uncertainty, which is reflected in the number of digits you can report in a measured or calculated value.

For example, 800 mL beakers are marked to every 100 mL. You can know the hundreds place with high certainty, but you have to make your best guess at the tens place. Similarly, a 100 mL graduated cylinder is marked to every 1 mL. Therefore, you can know the ones place and higher values with certainty, but you have to make the best guess at the tens place.

Digital readouts are simpler: the more values given, the more precise your measurement is. A top-loading balance may only give you two decimal places in your mass reading, whereas an analytical balance may give you three or four decimal places. If your sample is more than 10 g, a top-loading balance will give you at least four significant figures. However, samples smaller than 1 g need to be measured on an analytical balance to get the same precision.

More significant figures mean higher precision, but do we always need to be very precise? Sometimes we need a very precise and accurate value, while other times close enough is good enough.

You need to consider why you are preparing the sample. Analytical work typically requires precision. A solution to use in a titration will probably require high accuracy, as you are using it to determine an unknown. If you’re making a solution you’re going to standardize later, you might just need to know the approximate concentration of the solution, since you’ll be determining the precise value later.

You can also look to your procedure to find out how precise you need to be. For example, does your procedure call for you to make a 1 M solution or a 1.00 M solution? A 1 M solution just needs to be around 1 M, whereas a 1.00 M solution would require much more precision in its preparation so you can more accurately know its concentration.

## Balances: Top Loading versus Analytical

When you’re working with balances you may have the option of a top-loading balance or an analytical balance. Top-loading balances are good for larger samples but don’t give as many decimal values in your mass measurements, so you may not get as many significant figures in your reading. They are useful for mass measurements that don’t need to be precise and can be more approximate. An example is if you’re making a buffer or a saturated solution and need to add a mass within a wide range.

Analytical balances are best used for weighing small sample amounts where you need to know the mass with a higher accuracy. These types of balances tend to give more decimal places but can be overloaded with larger sample sizes. Analytical balances also tend to have draft shields on them because even an air draft can impact the mass reading.

In general, if you need to know a small mass (typically less than a gram) to the nearest 0.0001 g, use an analytical balance. If you need a less exact mass or have a large amount of material (> 10 g) use the top-loading balance.

Choosing glassware to make a solution can be more daunting than choosing a balance. There are so many types, and they all have different purposes. Beakers and Erlenmeyer flasks have volume markings for reference but aren’t really meant for measuring. They are great for preparing or holding a solution while you are measuring out a more accurate amount. For example, if you are doing a titration several times, you might prepare a large amount of stock solution, pour some into a beaker, then pour solution from the beaker to the buret for a precise measurement. The exact amount in the beaker doesn’t matter because you are using the buret for precision.

If you want a very precise volume and are dealing with variable amounts, burets are a good choice. They are marked to every tenth place so you can get two decimal places in your volume measurements. Burets are commonly used in titrations because they give very accurate measurements.

You can also measure out smaller volumes precisely and accurately with a volumetric pipette or tiny volumes with a micropipette. Pipettes are sometimes called “transfer pipettes” because they are used to transfer a precise amount of liquid from one container to another. When used correctly, pipettes tend to have errors of +/- 0.1% of the volume they measure. They are even calibrated to account for the tiny amounts of fluid left behind when they are emptied.

Volumetric flasks are good for preparing a specific volume of solution. Typically, you will put a precise amount of solute into a volumetric flask and then dilute with your solvent to the etched mark to get a very accurate total volume. Note that unlike volumetric pipettes, volumetric flasks do not account for liquid that remains after you pour out the contents. If you need to measure out 10.00 mL of a solution for a reaction, you should use a 10.00 mL pipette instead of simply pouring out the contents of a 10.00 mL volumetric flask.

## Other Equipment

A pH meter is a tricky instrument. Most pH meters only give 1–2 decimal places in their readings and because pH values are log functions, the decimals are the only significant figures in the value. (See Significant Figures Refresher box.) This is fine if you want to have a rough idea of the pH; however, if you need to determine the precise H + concentration or a p K a value of a weak acid, you will need to do a titration using volumetric glassware.

When it comes to temperature readings, the type of thermometer you use can also determine the accuracy and precision of your reading. Analog thermometers, which you have to manually read, may not have many digits in them depending on how many markings they have (every one degree versus a tenth of a degree), whereas a digital thermometer may provide additional significant figures. This would allow for greater precision and accuracy in your temperature reading.

If you’re measuring the absorbance of a sample, the number of significant figures is determined by the spectrophotometer. This absorbance then can be used to calculate the concentration of an analyte in solution with Beer’s law, but the number of significant figures in your concentration is limited by the absorbance value’s significant figures.

Now you’re armed with the knowledge and tools needed to determine when accuracy and precision are important, and what equipment will best provide you with the information you need. Your time is valuable and sometimes you need to expend the extra effort to have the most precise measurements to get the most accurate values, but other times close enough is good enough.

## Significant Figures Refresher

Significant figures are the primary way to indicate the precision of your work. Understanding them may seem daunting when you first start, but it gets easier as you use them. Here’s a refresher on the rules:

- Digits to the left of the decimal place in a logarithm are not significant because they indicate an order of magnitude.
- Digits in exact definitions are not relevant. For example, there are 100 cm in a meter, by definition. This value does not change no matter what is used to measure length.
- Counted numbers are also exact numbers and have no error associated with them, so they do not limit the number of significant figures in your reported value.
- Leading zeros are not significant, as they are placeholders. You might use a small graduated cylinder to measure 5.26 mL or 0.00526 L of solution, but you would still have three significant figures.
- Zeros in between significant figures are significant because they are part of the measurement.
- Zeros at the end of the number are only significant if they are part of the measurement. If you filled a 500 mL beaker to the line marked “200,” you can say you have 200 mL +/- 10 mL. So, that last zero cannot be verified, and you have two significant figures. Using scientific notation can clarify this: 2.0 x 10 2 has two significant figures.
- Your answer will have the same number of decimal places as the least precise measurement in the calculation. Performing addition and subtraction operations can result in gaining or losing significant figures in the final value (See the addition step of the example below). Consider filling a 1 L beaker half-full with water so that you have roughly 500 mL +/-50 mL. Now imagine pipetting in 1.00 mL. Do you really know your volume any more precisely?
- Your answer has the same number of significant figures as the least precise number in your calculation. For example, in the solved problem 52.450 x 0.0023 / 1.06 = 0.113806604, the answer has two significant figures. You report it as 0.11.
- Do the calculations in steps (make sure you pay attention to the order of operations!) and evaluate the significant figures you can keep at each step. Then use these to determine the significant figures you report your final result to.

Pro tip: Do not round your values until the very last step to avoid introducing truncation errors into your calculated values.

Pro tip : Calculators are great at math but terrible at significant figures. Watch out for “sneaky” zeros that calculators remove when doing calculations, especially for addition and subtraction operations. You can always add zeros if you need additional significant figures beyond what the calculator gives you. (See the mixed-operation example below.)

Mixed Operation:

(0.761 + 0.429) / (214.081 – 2.58) = 0.005626451

(1.190) / (211.501) = 0.005626451

(three decimal places: [CA1] four sig figs) / (two decimal places: [CA2] five sig figs) = four sig figs in reported value

Report as 0.005626 or 5.626 x 10 -3

## About the Author

Amanda J. Carroll is a Senior Lecturer and student chapter faculty advisor at Tennessee Tech University. She is also a member of the ACS Undergraduate Student Advisory Board.

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Chapter 1: Essential Ideas

## 1.5 Measurement Uncertainty, Accuracy, and Precision

Learning outcomes.

- Define accuracy and precision
- Differentiate between measured and exact numbers
- Determine the number of significant figures in a number and perform the calculation to the correct number of significant figures. Employ metric prefixes.
- Perform calculations using conversion factors to change from one unit to another (e.g. use density to convert between mass and volume).
- Apply proper rounding rules to computed quantities

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.

## Significant Figures in Measurement

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 1.5.1 . The bottom of the meniscus in this case clearly lies between the 21 and 22 mL 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 g. The quarter weighs about 6.72 g, with a nominal uncertainty in the measurement of ± 0.01 g. If we weigh the quarter on a more sensitive balance, we may find that its mass is 6.723 g. This means its mass lies between 6.722 and 6.724 g, an uncertainty of 0.001 g. 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 figures or significant digits . 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.

A measurement result is properly reported when its significant digits accurately represent the certainty of the measurement process. 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 figures 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 Essential Mathematics ) and express the number as 8.32407 × 10 [latex]-[/latex]3 ; then the number 8.32407 contains all of the significant figures, and 10[latex]^{-3}[/latex] locates the decimal point.

The number of significant figures 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 × 10 3 (two significant figures), 1.30 × 10 3 (three significant figures, if the tens place was measured), or 1.300 × 10 3 (four significant figures, 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 figures, be sure to pay attention to reported values and think about the measurement and significant figures in terms of what is reasonable or likely when evaluating 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 figures, 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 3.17 × 10 8 people.

## Significant Figures 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 figures, 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 figures (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 followed only by zeros or nothing, we round up or down, whichever yields an even value for the retained digit. If any nonzero digits follow the dropped 5, round up. (The part of this rule about the 5 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 figures:

- 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 lesser than 5)
- 6.8752 rounds “up” to 6.88 (the dropped digit is 5, and a nonzero digit follows it)
- 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 1.5.1: Rounding Numbers

Round the following to the indicated number of significant figures:

- 31.57 (to two significant figures)
- 8.1649 (to three significant figures)
- 0.051065 (to four significant figures)
- 0.90275 (to four significant figures)
- 31.57 rounds “up” to 32 (the dropped digit is 5, and a nonzero digit follows it)
- 8.1649 rounds “down” to 8.16 (the dropped digit, 4, is lesser 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)

## Check Your Learning

Example 1.5.2: addition and subtraction with significant figures.

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.

Show Solutions

- 1.0023 + 4.383 = 5.3853 Answer is 5.385 g (round to the thousandths place; three decimal places)
- 486 [latex]-[/latex] 421.23 = 64.77 Answer is 65 g (round to the ones place; no decimal places)

## Example 1.5.3: Multiplication and Division with Significant Figures

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 figures (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.
- [latex]\begin{array}{l}\begin{array}{l}\text{0.6238 cm}\times 6.6\text{cm}=4.11708{\text{cm}}^{2}\rightarrow\text{result is }4.1{\text{cm}}^{2}\left(\text{round to two significant figures}\right)\hfill \\ \text{four significant figures}\times \text{two significant figures}\rightarrow\text{two significant figures answer}\hfill \end{array}\hfill \end{array}[/latex]
- [latex]\begin{array}{l}\dfrac{\text{421.23 g}}{\text{486 mL}}=\text{0.86728 g/mL}\rightarrow\text{result is 0.867 g/mL}\left(\text{round to three significant figures}\right)\\ \dfrac{\text{five significant figures}}{\text{three significant figures}}\rightarrow\text{three significant figures answer}\end{array}[/latex]

In the midst of all these technicalities, it is important to keep in mind the reason why we use significant figures 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 1.5.4: Calculation with Significant Figures

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.

[latex]\begin{array}{lll}V\hfill & =\hfill & l\times w\times d\hfill \\ & =\hfill & \text{13.44 dm}\times \text{5.920 dm}\times \text{2.54 dm}\hfill \\ & =\hfill & \text{202.09459} {\text{dm}}^{3}\left(\text{value from calculator}\right)\hfill \\ & =\hfill & {\text{202 dm}}^{3}\text{, or 202 L}\left(\text{answer rounded to three significant figures}\right)\hfill \end{array}[/latex]

## Example 1.5.5: Experimental Determination of Density Using 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 number 1 support this statement? How?

The volume of the piece of rebar is equal to the volume of the water displaced:

[latex]\text{volume}=\text{22.4 mL}-\text{13.5 mL}=\text{8.9 mL}={\text{8.9 cm}}^{3}[/latex]

(rounded to the nearest 0.1 mL, per the rule for addition and subtraction)

The density is the mass-to-volume ratio:

[latex]\text{density}=\dfrac{\text{mass}}{\text{volume}}=\dfrac{\text{69.658 g}}{{\text{8.9 cm}}^{3}}={\text{7.8 g/cm}}^{3}[/latex]

(rounded to two significant figures, 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.

An irregularly shaped piece of a shiny yellowish material is weighed and then submerged in a graduated cylinder, with results as shown.

## 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 true or accepted value. Precise values agree with each other; accurate values agree with a true value. These characterizations can be extended to other contexts, such as the results of an archery competition ( Figure 1.5.4 ).

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 1.5.1 .

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).

## Key Concepts and Summary

Quantities can be exact or measured. Measured quantities have an associated uncertainty that is represented by the number of significant figures 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 true value) and/or precise (showing little variation when measured repeatedly).

- the number of seconds in an hour
- the number of pages in this book
- the number of grams in your weight
- 2 × 10 18 m × 2 × 10 18 m
- 3,486,002 kg
- 6.382 × 10 3
- (a) 7.04 × 10 2 ; (b) 3.344 × 10 –2 ; (c) 5.479 × 10 2
- (a) three; (b) one; (c) seven
- (a) 0.44; (b) 9.0; (c) 27

accuracy: how closely a measurement aligns with a correct value

exact number: number derived by counting or by definition

precision: how closely a measurement matches the same measurement when repeated

rounding: procedure used to ensure that calculated results properly reflect the uncertainty in the measurements used in the calculation

significant figures: (also, significant digits) all of the measured digits in a determination, including the uncertain last digit

uncertainty: estimate of amount by which measurement differs from true value

CC licensed content, Shared previously

- Chemistry 2e. Provided by : OpenStax. Located at : https://openstax.org/ . License : CC BY: Attribution . License Terms : Access for free at https://openstax.org/books/chemistry-2e/pages/1-introduction

number derived by counting or by definition

estimate of amount by which measurement differs from true value

(also, significant digits) all of the measured digits in a determination, including the uncertain last digit

procedure used to ensure that calculated results properly reflect the uncertainty in the measurements used in the calculation

how closely a measurement matches the same measurement when repeated

how closely a measurement aligns with a correct value

Chemistry Fundamentals Copyright © by Dr. Julie Donnelly, Dr. Nicole Lapeyrouse, and Dr. Matthew Rex is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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## COMMENTS

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