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Section 2.3Using Scientific Measurements

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Accuracy & Precision

Accuracy

Refers to the closeness of measurements to the correct

or accepted value of the quantity measured

Closeness to the correct value

Example:

Data Collected: Actual length= 5.1 cm

5.2 cm

5.1 cm

5.0 cm

5.1 cm

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Precision

Refers to the closeness of a set of measurements of the

same quantity made in the same way

Closeness of values to each other

Example:

Data Collected: Actual length= 5.1 cm

4.2 cm4.1 cm

4.0 cm

4.1 cm

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Percentage Error

Percentage Error Formula:

Used to determine the accuracy of a value by comparing

it quantitatively to the correct or accepted value

% Error= ValueexperimentalValue accepted x100

Value accepted

Percentage Error Formula: Used to determine the accuracy of a value by comparing

it quantitatively to the correct or accepted value

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Percentage Error


Value accepted

Example: What is the percentage error for a mass measurement of

17.7 g, given that the correct value is 21.2 g?

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Percentage Error


Value accepted

Example: A volume is measured experimentally as 4.26 mL. What

is the percentage error, given that the correct value is

4.15 mL?

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Uncertainty in Measurements

Uncertainty always exists in any measurement

What affects the precision and/or accuracy of a

measurement? Skill of measurer (human errorcorrect it!)

Conditions of measurement

Instruments

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Uncertainty in Measurements What is the length of the nail?

Definitely between 2.8 and 2.9 cm

Value is about halfway between 2.8 and 2.9 cm

Hundredths place is somewhat uncertain

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Uncertainty in Measurements

When measuring record all certainnumbers and one uncertain (estimated)

number.

Length of Nail = 2.85 Estimated Number

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Measurement with a

Graduated Cylinder

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Measurement with a

Graduated Cylinder

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Measurement with an

Electronic Balance

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Significant Figures

Significant Figures are all of the numbers recorded in ameasurement, including all the certain numbers plus the first

estimated number

Why are significant figures important when taking data in thelaboratory?

Significant figures indicate the precision of the measured value to

someone looking at the data.

Example:Mass= 1100 grams (mass has been rounded to the nearest hundred

grams)

Mass= 1100.0 grams (mass has been rounded to the nearest tenth of a

gram)

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Rules for Determining

Significant Figures1. Zeros appearing between nonzero digits are significant.

Examples: 40.7 L 3 sig figs

87 009 km 5 sig figs

2. Zeros appearing in front of all nonzero digits are notsignificant.

Examples: 0.095 897 m 5 sig figs

0.0009 kg 1 sig fig

3. Zeros at the end of a number and to the right of a decimalpoint are significant.

Examples: 85.00 g 4 sig figs

9.000 000 000 mm 10 sig figs

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Rules for Determining

Significant Figures4. Zeros at the end of a number but to the left of a decimal point

may or may not be significant.

If a zero has not been measured or estimated but is just a

placeholder it is not significant. (no decimal point)

Examples: 2000 m 1 sig fig

29 310 cg 4 sig figs

A decimal point placed after zeros indicates that they are

significant.Examples: 2000. m 4 sig figs

29310. cg 5 sig figs

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Determining the Number of

Significant Figures5. All numbers in the coefficient of a number expressed in scientific notationare significant, including zeros.

Example: 7.3021 x 10-4 5 sig figs

6. Counting numbers and defined conversion factors within the same systemof measurement are exact and have an infinite number of significant figures.

Example: 2 students infinite sig figs

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Significant Figures

Pretend that the number you are evaluating is sitting on a map of the

United States

If a decimal point is present in the number, you are going towards thePacific Ocean. Start counting from the right and stop when you reach

the last non-zero digit.

Example: 1.20 ________ significant figures

190.113 ________ significant figures

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Significant Figures

If a decimal point is absent in the number, you are

going towards the Atlantic Ocean. Start counting from the left and

stop when you reach the last non-zero digit.

Examples: 250 ________ significant figures

601,820 ________ significant figures

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Practice 1: How many significant figuresare in each of the following numbers?

1. 92

2. 78.04

3. 405.34

4. 0.23

5. 23.40

6. 15.40

7. 1.2 x 103

8. 210

9. 0.00120

10. 801.5

11. 0.0478

12. 230

13. 230.

14. 54.00

15. 0.00610

16. 0.0102

17. 1,000

18. 9.010 x 10-6

19. 101.0100

20. 2,370.0

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Practice 1: How many significant figuresare in each of the following numbers?

21. Why are significant figures important when taking data in

the laboratory?

22. Why are significant figures NOT important when solving

problems in math class?

23. Using two different instruments, a student measured thelength of their foot to be 27 cm and 27.00 cm. Explain the

difference between these two measurements.

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Practice 2: Rounding Using

Significant Figures

Value # of significantfigures

Rounded

Value

24 4.31589 3

25 83,692.1 2

26 0.00574800 3

27 2,591.7742 5

28 0.0219983 4

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Significant Figures


Rounded

Value

29 0.000123 2

30 23.842 4

31 7,563,874.5748 9

32 32.847 3

33 0.291 1

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Significant Figures


Rounded

Value

34 0.00473 2

35 382,739.47362 6

36 83.75 3

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Rounding using Significant

Figures

Decide how many significant figures are needed

Round to that many digits, counting from the left (start counting

from the first nonzero digit)

Examine the number to the right of the last significant figure-If the digit is less than 5Leave the last significant figure

alone

-If the digit is 5 or greaterRound the last significant figure

up by 1

Change the remaining digits to zeros if the number is greater than 1

or- eliminate the digits if the number is less than 1

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Significant Figures in

Calculations

A calculated answer cannot be more

precise than the measuring tool so a

calculated answer must match the leastprecise measurement.

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Significant Figures in Addition

and Subtraction In addition and subtraction problems,

the answer must have the same

number of figures to the right of the

decimal point as there are in the

measurement having the fewest

figures to the right of the decimalpoint.

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Significant Figures in Addition

and SubtractionUse the following problem to answer the questions below.

12.52 g + 349.0 g + 8.24 g

What is the rule for rounding the answer to a calculation involvingaddition and subtraction of measurements?

Identify the place value (tens, ones, tenths, hundredths) of the lastsignificant digit in each measurement.

12.52 g ________ 349.0 g ________ 8.24 g ________

To what place value should the answer to this calculation be rounded?

Calculated answer without rounding:

Final answer with the appropriate number of significant figures:

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Practice 3: Complete the followingcalculations and round to the appropriate number of

significant figures.37. 12.52 g + 349.0 g + 8.24 g

38. 1.327 mg + 9.45 mg + 103.38 mg

39. 56.1 cm - 2.001 cm3.11 cm

40. 101.004 g + 45.0 g75.34 g

41. 8 g + 2.981 g + 8.217 g

42. 114.21 g + 3041.2 g + 0.042 g + 349.5 g

43. 78.43 g + 21.019 g + 83 g

44. 90.023 cm5.90 cm

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Multiplication and Division

In multiplication and division problems,

the answer cannot have more significant

figures than the measurement in theproblem with the fewest significant

figures.

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Multiplication and DivisionUse the following problem to answer the questions below.

8.913 m x 20.005 m

What is the rule for rounding the answer to a calculation involvingmultiplication and division of measurements?

8.913 m ____ sig figs 20.005 ____ sig figs

How many significant figures should the answer to this calculation

have?

Calculated answer without rounding

Final answer with the appropriate number of significant figures:

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45. 3.11 m x 56.1 m x 2.001 m

46. 904 L 83.41 L

47. 9.345 dg 7.2 dg 320 dg

48. 410 mm x 178.8 mm x 321 mm

49. 56.3 g x 1.7346 g 100.2 g

50. 8.3 hL x 2.27 hL

51. 2.56 cm x 4.652 cm x 8.70 cm

52. 0.81 mg 450 mg

Practice 3:Complete the followingcalculations and round to the appropriate number

of significant figures.

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Scientific Notation

Used to express very large or very small numbers

Written as a coefficient between 1 and 10 (not including10) multiplied by 10 raised to an exponent.

General Format: M x 10n

M= coefficient 1< M

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Scientific Notation

When converting between real numbers and scientific notation:

Positive Exponent = Large Real #

(greater than 10)

Negative Exponent = Small Real #

(smaller than 1)

*If the exponent is 0 dont move the decimal

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Real Number to Scientific

NotationWhen converting from a real number to scientific notation

a) move the decimal so that one number is to the left

b) count the number of times the decimal place was

moved to get the value of the exponent

c) if the real number is large than exponent is +

if the real number is small than exponent is -

Example 1:

93,000,000

Example 2:

0.000167

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Practice: Convert the following real

numbers to scientific notation.

1) length of a football field, 91.4 meters

2) diameter of a carbon atom, 0.000 000 000 154 meter

3) radius of Earth, 6 378 888 meters

4) the diameter of a human hair, 0.000 008 meter

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Scientific Notation to Real

NumbersWhen converting from scientific notation to a real number

a) if the exponent is + make the number large

if the exponent ismake the number smaller

b) number of times the decimal is moved = exponentc) add zeros in the empty spaces

Example 1:

2.11 x 10-9

Example 2:

7.3418 x 102

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Convert the following from scientific

notation to real numbers.

1) average distance between the center of the sun and the center of

Earth, 1.496 x 1011meters

2) mass of a fly, 3.27 x 10-1gram

3) the number of atoms of hydrogen in a gram, 6.02 x 1023atoms

4) the number of stars in a galaxy, 1.1 x 1010stars

dd b

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Addition & Subtraction Using

Scientific Notation These operations can be performed only if the values have the

same exponent (n factor)

Make sure the answer is in correct scientific notation format

Example:

4.2 X 104 kg + 7.9 x 103kg

l l S f

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Multiplication Using Scientific

Notation M factors are multiplied and the exponents are added

algebraically

Example: 5.23 x 106m x 7.1 x 10-2m

Di i i U i S i ifi

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Division Using Scientific

Notation M factors are divided and the exponent of the denominators is

subtracted from that of the numerator

Example:

5.44 x 107 g

8.1 x 104 mol

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2.3 Sig Figs