Chapter 2Measurement & Problem

Solving

Uncertainty

• There is a certain amount of doubt in every measurement

– It is important to know the uncertainty when measurements are recorded or read

– Why is this important in science?• This helps to communicate the amount

of accuracy and precision of the measurement

Accuracy

• The closeness between a measurement and its true value.– The true value is rarely known, so

accuracy is based on how close independent studies agree

– The closer they agree, the more confidence scientists have in the accuracy of their results

Accuracy

• Accuracy is affected by determinate errors– Determinate errors are errors due to

1) poor technique2) incorrectly calibrated instruments

Which graduated cylinder is more accurate?

• 50 mL up by 1 mL

• 25 mL up by 0.5 mL

• 10 mL up by 0.1 mL

**Has a smaller scale so measurement will be closer to true value

Precision

• The reproducibility of an experiment (how close are the repeated measurements to each

other)

• Precision is affected by indeterminate errors

– Indeterminate errors- errors due to estimating the last uncertain digit of a measurement

• Random• Cannot be eliminated

Graduated Cylinder

5.72 mL

3.0 mL 0.33 mL

Precision

What are other examples of when

precision is important to you?

Which graduated cylinder is more precise?

• 50 mL up by 1 mL

• 25 mL up by 0.5 mL

• 10 mL up by 0.1 mL

**Has a smaller scale so measurement will be more precise

Is it possible to be precisely inaccurate?

YES!!

Image from: http://celebrating200years.noaa.gov/magazine/tct/tct_side1.html

What is the volume?

• Up front, two of the graduated cylinders have water in them

• 10 students to “read” the volumes– Write down your measurements on a

scratch piece of paper:Volumes of water in

50mL grad. cylinder =10mL grad. cylinder=

• 1 student to write measurements on the board and tally them if there are multiples

Is the variation in those measurements due to

determinate or indeterminate errors?

BOTH!Determinate errors- may have had poor technique

(didn’t read from bottom of meniscus, didn’t lower eyes to the level of the meniscus)

Indeterminate errors- estimating the last digit (when the level of the meniscus is between two

distinct lines)**This illustrates that there is always

uncertainty in measurements!

Uncertainty

• Instruments are not always calibrated perfectly (remember, humans do the calibrating and many times the measuring!)

• It is very important that in science we can communicate how accurate or precise our measurements are

Communicating Uncertainty

• In science, it is understood that the last digit of a number is “uncertain” or the estimated digit

(if another person were to make the same measurement, the number may be different)

*The estimate is made between the smallest divisions on the scale of the instrument (i.e. ruler, graduated cylinder, thermometer)

**Generally, the more numbers after the decimal point, the more precise the instrument is

General Rule for Measuring Uncertainty

• Unless otherwise indicated, assume that the uncertainty in a measurement is one-half the smallest division

• Examples:

________________ _________________ _______________

Which is more precise?

1002 cm or 1002.39 cm

This number is measured to more decimal places which indicates

smaller increments of measurement

In both numbers, which numbers are uncertain?

Instrument measures every 0.1 cm, so estimated to the hundredths place

Instrument measures every 10 cm, so estimated to the ones place

Things to keep in mind…

• When using an electric device, the last digit reported will be considered the uncertain digit

• “2” and “2.0” are not the same number!– By saying “2” we are saying the number is not

1 or 3– By saying “2.0” we are saying the number is

not 1.9 or 2.1

**It is the responsibility of the experimenter to write numbers in a way to reflect the precision of the instrument being measured

Standard Deviation• Values reported as results from experiments are in the

form of the “mean plus-or-minus one standard deviation” ( X ± s.d.)

– This gives a range over which we have confidence that the true value will fall• Used in scientific journals• Requires a sample of at least three• Most calculators are able to compute the mean and

standard deviation

– The standard deviation indicates how much each value differs from the average

– The size of the deviation would indicate precision in running a sequence of experiments (the smaller the deviation, the more precise the work was)

Scientific Notation

• Scientific Notation is often used to write very large or very small numbers– The numbers are expressed as a

coefficient number multiplied by an exponent of 10

coefficient

exponent

4.32 x 105

Scientific Notation

• The coefficient is a number greater than 1 but less than 10

• The exponent is:– Positive if the number is greater than

1Ex: 1,200 = 1.2 x 103

– Negative if the number is less than 1Ex: 0.023 = 2.3 x 10-2

– Zero if the number is equal to 1Ex: 1 = 100

Addition & Subtraction with Scientific Notation

1) Exponents must be made equal before you can perform the operation

Ex: 5.4x103 + 6.0x102 = 5.4x103 + 0.60x103

2) Then add or subtract the coefficients & the exponent will stay the same

Ex: 5.4x103 + 0.60x103 = 6.0x103

Multiplication and Division with Scientific Notation

1)Multiply the coefficients and add the exponents

Ex: (3.0x104) x (5.0x102) = 15.0x106 = 1.50x107

2) Divide the coefficients and subtract the exponents – be careful when subtracting a negative exponent!!

Ex: (8.0x104) ÷ (2.0x10-2) = 4.0x106

Significant Figures

• In science, numbers are only significant if they were measured. The last number reported is the “estimated” number and therefore is the uncertain number

Significant Figures

• All non-zero digits are significant• What to do with the zeros:

– Zeros between significant digits are significant (Sandwich Rule)

Ex: 307cm (3 sig figs)– Zeros at the end of a number and to

the RIGHT of a decimal are significantEx: 5.20cm (3 sig figs)

Significant Figures

• Zeros at the end of a number and to the LEFT of an assumed decimal may or may not be significant

**The presence of a decimal indicates significance, while the absence of the decimal point makes it insignificant

Ex: 750.cm (3 sig figs) 750cm (2 sigs figs)

Significant Figures

• “Cosmetic” zeros written to the LEFT of the first non-zero digit are NOT significant

Ex: 0.146cm (3 sig figs)09 cm (1 sig fig)

• Zeros that are place holders are NOT significant

Ex: .006cm (1 sig fig)0.014cm (2 sig figs)

Significant Figures

• The difference between zeros used for accuracy and those being used as place holders can only be addressed using scientific notation– 1,000,000– 1.00x106

– 1.00000x106

All zeros are placeholders (no decimal) 1 sig figThe first two zeros are significant 3 sig figsAll the zeros are significant 6 sig figs

Sig Fig Practice

1. 23.46 mL2. 0.0036 s3. 854.236 g4. 6.02x1023 molecules5. 0.98 mol6. 0023 m7. 2.000 J8. 1.00026x10-3 cm9. 824 mg

4 sig figs

2 sig figs

6 sig figs

3 sig figs

2 sig figs

2 sig figs

4 sig figs

6 sig figs

3 sig figs

Exact Values• Some numbers are exact values which involve no

uncertainty. • For example, there is an exact number of people in the classroom.

If there are 20 people, the zero is automatically significant.• Chemistry examples:

– There are 4.184 joules in a calorie

– There are exactly 2 hydrogen atoms in a water molecule

– Stoichiometric coefficients and subscripts 6CO2

– Defined quantities • (i.e. If we know 1 liter = 1000 mL, the number 1000 actually has

an infinite number of significant digits)

Significant Figures & Rounding

• Round down if last (or leftmost) digit dropped is 4 or less

Ex: 56.3231 cm 56.32cm

• Round up if the last digit dropped is 5 or more

Ex: 4.5272 cm 4.53cm

**Make sure you only use the last digit being dropped to decide which direction to round (ignore all digits to the right of it)

Addition & Subtraction of Significant Figures

• The number of places to the right of the decimal point (degree of accuracy), determines the number of sig figs reported

15.34 cm 4.1 cm+ 23.584 cm

43.024 cm

**This number has the least number of digits to the right of the decimal – Answer should be rounded to the tenth place

15.340 4.100 + 23.584

Add the decimals as you would normally (add in placement zeros if it helps you, but do not consider them when you decide on the # of sig figs

Correct Answer: 43.0 cm

Multiplication & Division of Significant Figures

• The number or reported digits is determined by the lease accurate value (the one with the fewest significant figures)

23.5 cm x 63.215 cm x 2. cm = 2,994.46 cm

This number has the least amount of sig figs, so the answer will need to have only 1 sig fig

Correct answer: 3000 cm or 3 x 103cm

Keep in mind…

• For calculations involving multiple steps, round only the final answer

• In calculations involving both multiplication/division and addition/subtraction, do the steps in parenthesis first

Determining Sig Figs in Measurements

• The last significant figure in your measurement needs to be one that you estimate

• The digits preceding this last digit, are considered “certain” because the instrument will have specific “marks” that should be exact measurements.

• The final “estimated” digit will be located between the specific “marks” (depending on the location of the measurement, but you may also decide that the estimated digit is a zero)

How many centimeters is this pencil?

Your answer should have 3 sig figs! The ruler measures 0.1cm (1mm) so your measurement should have an estimated digit in the hundredths position

~6.12 cm

How many mL are in this graduated cylinder?

http://www.proprofs.com/quiz-school/story.php?title=science-benchmark-1-review-test

Your answer should have 4 sig figs.

The graduated cylinder measures to every 0.1 mL so your estimated digit will be in the hundreths position

~40.30 mL

How long is this pen?

http://www.concord.org/~ddamelin/chemsite/b_measurement/sig_fig.html

The tip of the pen is between 4.7 cm and 4.8 cm. You will need to estimate the digit between the two points