# Computing Uncertainties and errors in Laboratory Data and Result

DEFINITIONS:

Accuracy is the closeness of agreement between a measured value and the true value.

Precision is the closeness of agreement between independent measurements of a quantity under the same conditions.

Uncertainty is the component of a reported value that characterizes the range of values within which the true value is asserted to lie.

Error is the difference between a measurement and the true value of the object being measured. Error does not include mistakes.

We will consider the error and uncertainty in experimental measurements and calculated results. You wil be asked to show the measurements and errors in your labs/exams.

These are some important tips you will need.

• Every measurement has an uncertainty associated with it, unless it is an exact, counted integer,

The two following examples exemplify these situations:

 MEASURED VALUE COUNTED INTEGER  volume: 25.68 cm3 quantity: 4 ELEPHANTS
• The numerical value "plus or minus" (±) the uncertainty value tells you the range of the result. For example a result reported as (5.35 ± 0.05) cm3means that the experimenter has some degree of confidence that the true value falls in between

5.35 - 0.05 = 5.30 cm3               and              5.35 + 0.05 = 5.40 cm3

• Hence the true value is contained in the whole area between 5.30 and 5.40
• Every calculated result also has an uncertainty, related to the uncertainty in the measured data used to calculate it.
• This uncertainty should be reported using the appropriate number of significant figures.

To consider error and uncertainty in more detail, we begin with definitions of accuracy and precision.

Accuracy and Precision WATCH THIS VIDEO

Accuracy and Precision

• The accuracy of a set of measurements is how close the measurement is to the true value of the quantity.
• The precision of a set of measurements is how close the measurements are to each other. (if an instrument is not properly calibrated,  you may have high precision but not accuracy)

The relationship of accuracy and precision may be illustrated by using the following example: You can see that good precision does not necessarily imply good accuracy.

Types of Error

The error of an observation is the difference between the observation and the actual or true value of the quantity observed.

Errors are often classified into two types:

Systematic :

Systematic errors may be caused by problems in either the equipment, the observer, or the use of the equipment. Systematic errors can result in high precision, but poor accuracy. They are difficult to discover. Examples:

♦ A student may overshoot the endpoint of a titration over and over again.

♦ A balance may always read 0.001 g more because it was zeroed incorrectly. .

Random:

Random errors vary in a completely nonreproducible way from measurement to measurement. but can be treated statistically, and so relate the precision of a calculated result to the precision with of the measurements taken.

Take, for example, the simple task of measuring the volume in the following picture: You would probably , align your eyes to be able to see a imaginary horizontal line just underneath the meniscus. You might think that the errors arose from only two sources,

(1) Instrumental error
(How "well calibrated" is the graduated cylinder? How thin and how closely spaced are the cylinder's graduations?)

(2) Uncertainties in the thing being measured
(How thin are the lines? Is the liquid subject to temperature and humidity changes?)

A third source of error exists, related to how any measuring device is used.

Example

When you perform a titration, you fill the burette to the top mark and record 0.00 mL as your starting volume. This is INCORRECT. The correct way of using it, is: Add enough solution so that the burette is nearly full, but then simply read the starting value to whatever precision the burette allows and record that value. It will be subtracted from your final burette reading to yield the most unbiased measurement of the delivered volume

IGCSE AND HONORS, NOTES UP TO HERE

Absolute Error and % Error:

In your Cambridge Examination you will be asked to express the error in your measurements  as absolute or %.

Although the detail in your Chemistry examination is much less than the required in Physics, you cannot express the sources of error as "I did not rinse the burette properly" this is not considered error but incompetence. Be sure not to mention HUMAN ERROR in the exam.

Examples:

Consider weighing 1g of solid. If you use a two decimal place balance, the mass recorded will be to the nearest 0.01g.
We should express this measurement as (1.00 ± 0.01) g, where 0.01 is the absolute error.
In this example, the % error will be:

0.01 g x 100 = 1%

1.00 g

Error Propagation and Precision in Calculations

You will not asked to calculate the ERROR PROPAGATION in your exam, but you need to express the results with the correct amount of significant figures:

Let work it out through an example:

We need to calculate the density of an object. You measure the mass using a 2 decimal place balance. You read 30.05 g, but you should record this as:

mass: 30.05 g ± 0.05 g  - 4 SIGNIFICANT FIGURES   % error: 0.166%

Then you use a 50 mL graduated cylinder and the volume measured is 25.5 mL, you should record this as:

volume: 25.5 mL ± 0.5 mL - 3 SIGNIFICANT FIGURES  % error: 1.96%

Calculating the density:

DENSITY = MASS/VOLUME = 30.05 g / 25.5 mL = 1.17843 g/mL => this answer is WRONG!

The correct answer should show only 3 SIG FIG. so the correct answer will be

DENSITY: 1.18 g/mL