Topic 11: Measurement and Data Processing

Honors ChemistryMrs. Peters

Fall 2014

11.1 Uncertainty and Error in Measurement

• Measurement is important in chemistry.• Many different measurement apparatus are

used, some are more appropriate than others.

11.1 Uncertainty and Error in Measurement

• Example: You want to measure 25 cm3 (25 ml) of water, what can you use?– Beaker, volumetric flask, graduated cylinder,

pipette, buret, or a balance– All of these can be used, but will have different

levels of uncertainty. Which will be the best?

A & S 1. Systematic Errors

• Systematic Error: occur as a result of poor experimental design or procedure.– Cannot be reduced by

repeating experiment– Can be reduced by careful

experimental design

A & S 1. Systematic Errors

• Systematic Error Example: measuring the volume of water using the top of the meniscus rather than the bottom

• Measurement will be off every time, repeated trials will not change the error

A & S 1. Random Error

• Random Error: imprecision of measurements, leads to value being above or below the “true” value.

• Causes: – Readability of measuring instrument– Effects of changes in surroundings (temperature, air currents)– Insufficient data– Observer misinterpreting the reading

• Can be reduced by repeating measurements

A & S 1: Random and Systematic Error

Systematic and Random Error ExampleRandom: estimating the mass of Magnesium ribbon rather than

measuring it several times (then report average and uncertainty)

0.1234 g, 0.1232 g, 0.1235 g, 0.1234 g, 0.1235 g, 0.1236 g Avg Mass= 0.1234 + 0.0002 g

A & S 1: Random and Systematic Error

Systematic and Random Error ExampleSystematic: The balance was zeroed incorrectly with each

measurement, all previous measurements are off by 0.0002 g0.1236 g, 0.1234 g, 0.1237 g, 0.1236 g, 0.1237 g, 0.1238gAvg Mass = 0.1236

A & S 8. Distinguish between precision and accuracy in evaluating results

Precision: how close several experimental measurements of the same quantity are to each other – how many sig figs are in the measurement.– Smaller random error = greater precision

A & S 8. Distinguish between precision and accuracy in evaluating results

Accuracy: how close a measured value is to the correct value– Smaller systematic error = greater accuracy

•Example: masses of Mg had same precision, 1st set was more accurate.

U 5. Reduction of Random Error

• Random errors can be reduced by– Use more precise measuring equipment– Repeat trials and measurements (at least 3,

usually more)

A & S 2. Uncertainty Range (±)

• Random uncertainty can be estimated as half of the smallest division on a scale

• Always state uncertainty as a ± number

A & S 2. Uncertainty Range (±)

• Example:– A graduated cylinder has increments of 1 mL– The uncertainty or random error is

1mL / 2 = ± 0.5 mL

A & S 2. Uncertainty Range (±)

Uncertainty of Electronic Devises On an electronic devices the last digit is rounded up or down by the instrument and will have a random error of ± the last digit.Example: –Our balances measure ± 0.01 g–Digital Thermometers measure ± 0.1 oC

State uncertainties as absolute and percentage uncertainties

• Absolute uncertainty– The uncertainty of the apparatus– Most instruments will provide the uncertainty– If it is not given, the uncertainty is half of a

measurement– Ex: a glass thermometer measures in 1oC increments,

uncertainty is ±0.5oC; absolute uncertainty is 0.5oC

State uncertainties as absolute and percentage uncertainties

• Percentage uncertainty= (absolute uncertainty/measured value) x 100%

Determine the uncertainties in results

Calculate uncertaintyUsing a 50cm3 (mL) pipette, measure 25.0cm3.

The pipette uncertainty is ± 0.1cm3.What is the absolute uncertainty?0.1cm3

What is the percent uncertainty?0.1/25.0 x 100= 0.4%

Determine the uncertainties in results

Calculate uncertaintyUsing a 150 mL (cm3) beaker, measure 75.0 ml

(cm3). The beaker uncertainty is ± 5 ml (cm3).What is the absolute uncertainty?5 ml (cm3)What is the percent uncertainty?5/75.0 x 100= 6.66% 7%

Determine the uncertainties in results

Percent error = I error l x 100 accepted

When…• Percent error > Uncertainty– Systematic errors are the problem

• Uncertainty > Percent error – Random error is causing the inaccurate data

Determine the uncertainties in results

Error Propagation: If the measurement is added or subtracted, then absolute uncertainty in multiple measurements is added together.

Determine the uncertainties in results

Example:If you are trying to find the temperature of a

reaction, find the uncertainty of the initial temperature and the uncertainty of the final temperature and add the absolute uncertainty values together.

Determine the uncertainties in results

Example: Find the change in temperatureInitial Temp: 22.1 ± 0 .1oC Final Temp: 43.0 ± 0.1oCChange in temp: 43.1-22.1 = 20.9Uncertainty: 0.1 + 0.1 = 0.2

Final Answer: Change in Temp is 20.9 ± 0.2 oC

Determine the uncertainties in results

Error Propagation: If the measurement requires multiplying or dividing: percent uncertainty in multiple measurements is added together.

Determine the uncertainties in results

Example:If you are trying to find the density of an object,

find the uncertainty of the mass, the uncertainty of the volume, you add the percent uncertainty for each to get the uncertainty of the density.

Determine the uncertainties in results

Example: Find the Density given:Mass: 25.45 ± 0.01 g and Volume: 10.3 ± 0.05

mLDensity: 25.45/10.3 = 2.47 g/mL% uncertainty Mass: (0.01/25.45) x 100 = 0.04%% uncertainty Volume: (0.05/10.3) x 100 = .5%0.04 + .5 = .54%Final Answer: Density is 2.47 ± .54%

Determine the uncertainty in results

Uncertainty in Results (Error Propagation) 1. Calculate the uncertainty

a. From the smallest division (on a graduated cylinder or glassware)

b. From the last significant figure in a measurement (a balance or digital thermometer)

c. From data provided by the manufacturer (printed on the apparatus)

2. Calculate the percent error3. Comment on the error

a. Is the uncertainty greater or less than the % error?b. Is the error random or systematic? Explain.