11.1 & 11.2 Uncertainty & Error

*Quantities that are counted are exact; there is no uncertainty about them…they have UNLIMITED significant digits. Two examples are found in these statements:

i) There are 22 people in the room.

ii) Ten dimes have the same value as one dollar.

*Measured quantities are much different because they are never exact. Suppose you are finding the mass of a friend in kilograms. You might try a bathroom scale and get 56 kg, then a scale in a physician’s office and get 55.8 kg, and finally a sensitive electronic scale in a scientific laboratory and get 55.778 kg. Not one of the measurements is exact; even the most precise measurement, the last one, has an uncertainty.

*(↑ decimal places = ↑ precision)

*In any measurement, the digits that are reliably known are called significant digits. These include the digits known for certain and the single last digit that is ESTIMATED.

1. All NON-ZERO numbers are significant. (e.g., 1,292 mm has four significant digits.)

2. “Captive” zeros (i.e. zeroes placed between other digits) ARE significant. (e.g. 400,006 cm has 6 sig. digs).

3. Leading zeros are never significant. (e.g. 0.0000089 kg has 2 significant digits.)

5. Counted quantities are exact and infinite number of significant figures. (i.e. there is NO uncertainty about them.)

*Determine the number of significant digits in the following numbers:

Round the following to the indicated number of significant digits.

*When performing calculations that involve a series of mathematical steps, don’t round off numbers until you are ready to state the final answer…use all the decimal places reported by your calculator.

a) 2150 (2 s.d.) -___________ c) 0.0346 (2 s.d.) - _________ e) 1.994 (3 s.d.) - _________

b) 0.0256 (2 s.d.) - ________ d) 0.0050129 (3 s.d.) - _______ f) 2149.99 (3 s.d.) - ________

a) ADDING or SUBTRACTING:

RULE: Round off the final answer to the same number of _____________ places as the least precise number.

e.g. Add the following measurements: 123.0 cm + 12.40 cm + 5.380 cm

Solution:

123.0 cm

012.40 cm

005.380 cm

140.780 cm

*Thus, 140.780 cm should be rounded to _____________ cm since 123.0 cm has only one decimal place.

*Perform the following operations expressing the answer in the correct number of significant digits.

a) 1.35 x 2.267 = g) 0.2129 + 0.002 + 0.03 = m) 150 / 4 =

b) 1 035 / 42 = h) 101.4 + 25 + 201 = n) 505 – 450.25 =

c) 12.01 + 35.2 + 6 = i) 1.0 + 2.04 + 5.03 = o) 1.252 x 0.115 x 0.012 =

d) 55.46 – 28.9 = j) 2.5 x 1.1111 = p) 0.15 + 1.15 + 2.051 =

e) 0.021 x 3.2 x 100.1 = k) 8.314 x 2.5x10^-2 = q) 41.11 + 20.5 + 18.333 =

f) 1.278 x 10³ x 1.4267 x 10² = l) 35.45 x 2.25 = *r) 3.76x10⁵ – 276 =

Rule 1: When converting from a LARGER UNIT TO SMALLER UNIT -->________________________

Examples:

Metre to decimeter --> multiply by______ 2.13 m =_______________dm

Meter to centimeter --> multiply by______ 2.13 m = _______________cm

Meter to millimeter --> multiply by ______ 2.13 m = _______________mm

Decimeter to centimeter --> multiply by _____

Decimeter to millimeter --> multiply by ______

Rule 2: When converting from a SMALLER UNIT TO A LARGER UNIT --> _____________________

Examples:

*Practice --> Complete the following conversions:

Examples: Converting From One Unit to Another

c) 5.0 x 10⁴ GigaWatts (GW) = ? Watts

5.0 x 10⁴ GW = (5.0 x10⁴ GW) x

a) 20 milliseconds (ms) to seconds

b) 3.28 g to megagrams

c) 105 MHz to KHz (*1 MHz = ______ KHz)

d) 1.3 megametres (Mm) to micrometres (μm) (*2 steps)

e) 14.6 m/min to km/h

a) ACCURACY:

• a measure of how close a result is to the theoretical / accepted / documented value for the quantity being determined

Example: Determination of the ideal gas constant (Accepted value = 8.314 J mol^-1 K^-1)

Which of the following experimental values is more accurate?

Experimental value #1 = 8.317 J mol^-1 K^-1

Experimental value #2 = 8.103 J mol^-1 K^-1

The accuracy of a measurement is usually measured as a percent error from the theoretical value using the following expression:

*Calculate the percent error for the two above values quoted for the ideal gas constant.

b) PRECISION:

à the exactness of a measurement (more decimal places = ↑ precision)

• how close a series of measurements are to one another (close = precise ; not close = imprecise)

• usually quoted as a value

• ↓ _______________ error = ↑ precision

Example:

Consider the following ideal gas constant measurements:

Group 1 8.34 +/- 0.03 J mol^-1 K^-1

Group 2 8.513 +/- 0.006 J mol^-1 K^-1

Which of the measurements is:

a) more accurate? Explain.

b) more precise? Explain.

c) Calculate the relative(percent) error associated with each measurement.

a) SYSTEMATIC ERRORS/UNCERTANTIES (affect accuracy)

Affect ALL measurements in the SAME way [i.e. measurements that are either consistently too high or too low…due to systematic error(s)]

• REPEATED measurements WILL NOT reveal this type of uncertainty regardless of the number of trials performed.

• Usually difficult to detect unless the expected/actual value is known.

• Can sometimes be remedied/ameliorated/lessened.

Due to:

a) Miscalibrations / “faulty” or damaged equipment

Examples:

i) A stopwatch running 2 seconds too slow.

*Repeated measurements will not reveal this uncertainty.

*Remedy = check clock against a more reliable one (If doubt the reliability of a measuring device try to check it against a device that is more reliable.)

ii) A miscalibrated pH meter will always display a reading that is either higher or lower than the actual value.

iii) A thermometer reporting a temperature that is 2 degrees higher than the actual value.

iv) A stretched ruler.

OR

b) Poor experimental technique (“human error”)

Examples:

i) Reading a meniscus incorrectly / parallax error?

ii) Reading scale incorrectly

iii) Unaccounted heat loss in a calorimetry experiment

*A note on “human error”:

Please don't refer to "human error." Examples of so-called human error include misreading a ruler, adding the wrong reagent to a reaction mixture, mis-timing the reaction, miscalculations, or any kind of mistake. Scientists would never report the results of an experiment affected by human error -instead, they repeat the experiment more carefully!

Points will be deducted from your lab report if you discuss "human error" instead of "experimental error."

b) RANDOM ERRORS/UNCERTANTIES (affect precision)

Actual value may be higher OR lower than the value you record….in this manner, they are “random”...just as likely to overestimate as to underestimate a measurement.

• REPEATED measurements WILL reveal this type of uncertainty.

• They are PREDICTABLE and UNAVOIDABLE.

• Random errors can generally not be ameliorated.

  • Arise mostly from the INADEQUACIES or LIMITATIONS inherent to all measuring instruments.

  • The degree of random error can be QUANTIFIED.

  • The random error is equivalent to the uncertainty in measurement. This is usually given by the manufacturer of the equipment and expressed as +/ - a certain value.

• *If this information is not available, a good guideline is to estimate the uncertainty at HALF of the SMALLEST division on a scale for an ANALOGUE device and the SMALLEST division for a DIGITAL device.

• Note when the uncertainty is recorded, it should be to the same number of decimal places as the measured value. For example a balance reading to 53.457g +/- 0.001 (↑ decimal places = ↑ sig. digs. = ↑ precision)

Example 1:

à Imagine trying to determine the period of a pendulum.

Best strategy = Make repeated measurements; calculate average (…the best average of the period is the average value)

*the mean is the best estimate of a measurement based on a set of measured values

*standard deviation = ?

e.g. Four measurements(in seconds) = 2.3, 2.4, 2.5, 2.4 ; average = 2.4

Range = 2.3 to 2.5

Example 2: multiple students take temperature of room (discuss both system and random error)

• Sources of random errors cannot always be identified. Possible sources:

a) observational e.g. reading burette, judging a colour change

b) environmental e.g. convection currents

*standard deviation and precision

When the final uncertainty arising from random errors is calculated, this can then be compared with the experimental error as described above. If the experimental error is larger than the total uncertainty, then random error alone cannot explain the discrepancy and systematic errors must be involved.

Almost all measurements are subject to both systematic and random uncertainties.

- the study of uncertainty in measurement

- all measurements (no matter how carefully made) have an uncertainty to them, which is referred to as an error

- uncertainty (error), is not due to mistakes and hence cannot be avoided. Therefore, it is important to minimize the uncertainties or errors in measurements

- uncertainty is usually due to the instrument/tool which is used to make a measurement

b)

With the ruler used in a), the measurement of the length of the pencil would be 5.4 +/- 0.2 cm

The uncertainty (error) is 0.2 cm, hence this measurement indicates that the pencil can have a length from 5.2 cm to 5.6 cm (5.2 cm £ length £ 5.6 cm). The value of 0.2 cm is sometimes called the absolute uncertainty, or absolute error.

With the ruler used in b), the measurement of the length of the pencil would be 5.4 +/- 0.1 cm

The absolute error is 0.1 cm, hence this measurement indicates that the pencil can have a length from 5.3 cm to 5.5 cm (5.3 cm £ length £ 5.5 cm). This measurement has less error (is more certain) than the previous measurement since the ruler has finer increments.

The relative error is a comparison between the absolute error of the measurement and the measurement. The relative error is often represented as a percent. For case b, the relative error would be

relative error = absolute error/measurement = 0.1/5.4 = 0.019 or 1.9%

time (seconds): 0.64 0.69 0.54 0.44 0.75 0.90 0.43

Averaging the results would yield 0.63 seconds. The highest result was 0.90 seconds, while the lowest result yielded 0.43 seconds. So you could report your result as

0.63 +/- 0.30 seconds OR 0.63 +/- 0.20 seconds

For measurements that involve a stopwatch, the general rule of thumb is to use an error that is between 0.20 to 0.30 seconds. This error is due to human reaction time.

For both of the examples, the uncertainty in the measurement needs to be justified. In other words, you must indicate “why you chose the error you chose”.

- for laboratory experiments, error analysis is used to compare your experimental results to theoretical results to show that your answer “agrees” with theory

- if your experimental answer does not agree with the theory, then one must redesign the experiment or account for these differences (perhaps the current theory is incorrect and needs to be altered)

- used in all branches of science to validate results

Example 3: The dimensions of a rectangular shaped lawn are measured. The length of the lawn is measured to be 10.2 m +/- 0.1 m. The width of the lawn is measured to be 5.6 +/- 0.1 m.

Example 4:

The volume of water that is measured is 50 ml +/- 2 ml. The mass of the water is 50.2 +/- 0.1 grams

a) Determine the range (possible values) for the measurement of the volume of water.

b) Determine the range for the measurement of the mass of the water.

c) Calculate the relative error for the volume of water.

d) Calculate the relative error for the mass of water.

e) Determine the range of values for density (highest and lowest possible value for density).

f) Calculate the absolute error in density.

g) Calculate the relative error in density.

h) If you wanted to decrease the absolute/relative error in density, and had to either purchase a new scale to improve the precision on the measurement of mass or a new volumetric flask to improve the precision on the measurement of volume, which one would you choose and why (explain).

Example 5:

A Hotwheels car was traveling at constant velocity.

The distance it covered was 30.0 +/- 0.1 cm in 10.05 +/- 0.20 seconds.

a) Determine the range (possible values) for the measurement of distance.

b) Calculate the relative error for the measurement of distance.

c) Determine the range for the measurement of time.

d) Calculate the relative error for the measurement of time.

e) Determine the range of values for the calculation of speed based on the high and low values for time and distance.

f) Calculate the absolute error in speed.

g) Calculate the relative error in speed.

h) Explain how you could decrease the absolute error in speed?

**The uncertainty of a calculation is always greater than the individual uncertainties.

RULE 1: When ADDING or SUBTRACTING measurements, ADD THE ABSOLUTE UNCERTAINTIES. (Justin, Al foil, density of coke etc. activity)

RULE 2: When MULTIPLYING or DIVIDING measurements, ADD THE RELATIVE UNCERTAINTIES.

Determine the quantity of heat required to raise the temperature of 75.00 g of water from 25.1°C to 85.5°C. The specific heat capacity of water is 4.18 Jg^-1K^-1. The uncertainties are ± 0.01 g and ±0.1°C.

DATA:

Mass of water = 75.00 g ± 0.01 g

T₁ = 25.1°C ± 0.1°C ; T₂ = 85.5°C ± 0.1°C

STEP 1: Calculate ΔT. (Remember: __________ the _________________uncertainties.)

ΔT = T₂ - T₁ = (85.5°C ± 0.1°C) - (25.1°C ± 0.1°C) =

STEP 2:

a) Calculate the relative(%) uncertainties for both mass and ΔT.

b) Calculate the sum of the relative uncertainties.

STEP 3: Calculate q.

STEP 4: a) Report q with its relative uncertainty.

b) Convert relative uncertainty to absolute uncertainty.

q = mc ΔT

=