# Notes on Uncertainty Analysis

The final result from a chemical experiment, such as the value of ΔH for a particular reaction or the average of several molarities obtained from an acid-base titration, is often calculated from several different measured values.  The uncertainty of the result is influenced by the uncertainty of each of the individual measurements. Suppose, for example, that one found the density (mass/volume) of a piece of metal by weighing it on an analytical balance (mass uncertainty ± 0.0001 g) and determined its volume by the water it displaced in a graduated cylinder (volume uncertainty ± 0.5 mL).  The error, or uncertainty, in the calculated density would be dominated by the uncertainty in the volume, because there is more uncertainty in the volume measurement.  Uncertainty analysis (also known as error propagation) is the process of calculating uncertainty of a value that has been calculated from several measured quantities. Uncertainty analysis is governed by a few simple rules. We will present the rules without their differential calculus-based derivations. A few practice problems are given at the end of this section. Before you get started, be sure to read the Significant Figures Summary in the Lab Manual Appendices.

### Uncertainty (a.k.a. Error)

Uncertainty is also known as "error." Any measured or calculated value has some uncertainty in the reported value. This does not refer to mistakes, but rather unavoidable error due to the nature of the experiment. For example, if you were measuring the width of a grape using a ruler, you might report a value of 12.3 mm but there would definitely be some error incorporated in that last digit.

All uncertainties are reported to 1 significant figure. The reported value should then be rounded to the same digit as the uncertainty. When you know uncertainties, the significant figures of the reported value should be determined by the uncertainty rather than by standard sig fig rules.

It is also vital that you use several significant figures throughout your uncertainty calculations, so as to get an accurate representation of your overall uncertainty.  If you are performing a series of calculations, keep all digits in your calculations until you complete ALL of your calculations.  Then go back and report intermediate and final uncertainties with one significant figure.

There are two ways to represent uncertainty:

1. Absolute uncertainty (AU) is a measure of uncertainty with the same units as the reported value. For example, the grape's width is 12.3 ± 0.2 mm, where 0.2 mm is the AU.
2. Relative uncertainty (RU) represents AU as a fraction or percentage.
3. For example, 0.2mm/12.3mm = 0.02 = 2%. The grape's width is 12.3 mm ± 2%, where 2% is the RU.

### Absolute uncertainty (AU)

A measured quantity is often reported with uncertainty. Absolute uncertainty is the uncertainty given in the same units as the measurement:

meas = (23.27 ± 0.01) g

where 0.01 g is the absolute uncertainty.

There are two primary contributions to absolute uncertainty: accuracy and precision.

#### Accuracy (systematic error)

Systematic error is sometimes reported for specific instruments.  For example, Vernier temperature probes claim accuracy to within 0.03 º C.  This means that there may be a systematic error of up to 0.03 º C for any specific temperature probe.  Similarly, analytical balances are accurate to within 0.0001 g.

#### Precision (reproducibility error)

Reproducibility error is primarily determined in two different ways:

1. Ability to read an instrument. For example, using a ruler that is divided into cm, you may be able to determine that a wire is between 9.2 and 9.6 cm long. This could be written 9.4 ± 0.2 cm. By estimating your ability to read the ruler, you can estimate the absolute uncertainty.  In this case, reproducibility error is ± 0.2 cm.  Alternatively, if you are using an analytical balance and the ten-thousandths digit is fluctuating between 1 and 5, reproducibility error would be ± 0.0002 g.

2. Multiple measurements.  When several measurements are averaged, the reproducibility error can be approximated by the standard deviation of the measurements.

Most of the time, we will only be dealing with reproducibility uncertainty.  However, if we know both, AU is calculated:

AU = systematic error + reproducibility uncertainty

In the case of the analytical balance mentioned above:

AU = 0.0001 g + 0.0002 g = 0.0003 g

Notes:

• AUs are positive values with one significant figure.
• AUs have units if the associated value has units.

### Relative uncertainty (RU)

Relative uncertainty is a fractional value. If you measure a pencil to be 10cm ± 1cm, then the relative uncertainty is one tenth of its length (RU = 0.1 or 10%). RU is simply absolute uncertainty divided by the measured value. It is reported as a fraction or percent:

For the example given under AU:

meas = (23.27 ± 0.01) g

AU = 0.01g

Notes:

• RUs can be represented as either a fractional value or a percent.
• RUs have no units.

### Propagation of Uncertainty

When you perform calculations on numbers whose uncertainties are known, you can determine the uncertainty in the calculated answer using two simple rules. This is known as propagation of uncertainty. Rules for uncertainty propagation are very different for addition/subtraction operations as compared to multiplication/division operations. These rules are not interchangeable. The rules presented here determine the maximum possible uncertainty.

• Addition and Subtraction: AU = ΣAU

When calculating uncertainty for the sum or difference of measured values, AU of the calculated value is the sum of the absolute uncertainties of the individual terms.
• Example:

A = 19 ± 4 (AU = 4; RU = 0.2)

B = 28 ± 3 (AU = 3; RU = 0.1)

A + B = 47 ± 7 (AU = 7; RU = 0.1)

A - B = -9 ± 7 (AU = 7; RU = 0.8)

Notes:

• RUA+B ≠ RUA + RUB.
• RU can be calculated using the equation RU = AU/|value|.
• Even if you are subtracting measured values, be sure to add AUs.

Example: (underlines are used to indicate significant digits)

Calculate qtotal and its associated AU and RU values, using the equation:

qtotal =  - (qsolution + qcal

where qsolution and qcal are measured values:

qsolution = 1450 ± 20 J

qcal = 320 ± 50 J

Solution:

1. Calculate qtotal, ignoring uncertainties:
2. qtotal =  - (1450 + 320) J = -1770 J

4. AUqtotal = AUqsolution + AUqcal = 20 + 50 J = 70 J

5. Calculate relative uncertainty from absolute uncertainty:
6. RUqtotal = AU/|(qtotal)| = 70J/|-1770J| = 0.04

8. qtotal =  -1770 ± 70 J (RU = 4%)

• Multiplication and Division: RU =ΣRU
• When calculating uncertainty for the product or ratio of measured values, RU of the calculated value is the sum of the relative uncertainties of the individual terms. However, AU cannot be calculated directly from AU or RU values of the measured values.

Example:

A = 19 ± 4 (AU = 4; RU = 0.2)

B = 28 ± 3 (AU = 3; RU = 0.1)

A x B = 532

RUAxB = RUA + RUB = 4/19 + 3/28 = 0.3177 = 0.3

AUAxB = RUx|532| = 0.3177|532| = 169 = 200

A x B = 532 ± 169 = 500 ± 200

Notes:

• To determine the correct number of significant figures in your answer, calculate the absolute uncertainty, round it to one significant figure, and then round the calculated value to the same digit.
• AUAxB ≠ AUA + AUB.
• AU can be calculated using the equation AU = RUx|value|.
• Relative uncertainties are always positive.  Be sure to calculate RU using absolute values.

Example:

Calculate qcal and its associated AU and RU values, using the equation:

qcal = CΔT

where C and ΔT are measured values:

C = (54 ± 7) J/°C

ΔT = 6.0 ± 0.1 °C

Solution:

1. Calculate qcal, ignoring uncertainties:
2. qcal = (54 J/°C) x (6.0 °C) = 324 J = 320 J

4. RUqcal = RUC + RUΔT = 7/54 + 0.1/6.0 = 0.14630 = 0.1

5. Calculate absolute uncertainty from relative uncertainty:
6. AUqcal = RUx|qcal| = 0.14630 x 324 J = 47.4 J = 50 J

8. qcal = 320 ± 50 J (RU = 10%)

• Combination of addition/subtraction with multiplication/division
• For a combination of these operations we follow the standard order of operations.

Example:

Calculate qtotal and its associated AU and RU values, using the equation:

qtotal =  - (qsolution+ qcal) = - (qsolution + CΔT)

where qsolution = 1450 ± 20 J

C = 54 ± 7 J/°C

ΔT = 6.0 ± 0.1 °C

Solution:

1. Calculate qtotal, ignoring uncertainties:
2. qtotal = - (1450 J + (54 J/°C)(6.0 °C)) = - (1450 J + 324 J) = -1774 J = -1770 J

Note that you did the multiplication first, then the addition. We will approach the uncertainty calculations in the same order.

3. Determine uncertainties for the multiplication:
4. RUqcal = RUC + RUΔT = 7/54 + 0.1/6.0 = 0.14630 = 0.1

AUqcal = RUx|qcal| = 0.14630 x 324 J = 47.4 J = 50 J

5. Determine uncertainties for the addition:

AUqtotal = AUqsolution + AUqcal = 20 + 47.4 = 67.4 J= 70 J

RUqtotal = AU/|(qtotal)| = 67.4J/|-1774J| = 0.04