A.0 SF’s-Uncertainty-Accuracy-Precision

Objectives:• Convert standard notation to scientific notation

• Convert scientific notation to standard notation

• Perform calculations with numbers in Sci-Not

• Determine the #SF’s in a measurement

• Round a calculated answer to the correct #SF’s

• Round a calculated answer to the correct decimal place

• Calculate % error

• Honors-Calculate relative precision

• Describe the difference between precision and accuracy

• Know how accuracy is quantified

• Know how precision is quantified

Scientific Notation

Scientific notation is a format for writing very large and very small numbers.

Numbers written in scientific notation have two parts:• The decimal part

• The exponential part

Numbers not in scientific notation are in standard notation. Examples:

43, 000,000,000

0.00567

6.78 x 10 -8

Scientific NotationA number is expressed in scientific notation when it is in the form:

a x 10 n

where a is between 1 and 10 and n is an integer. Both a and n can

be negative or positive

Sci-Not: 6.3 x 10 6

7.930 x 10 -4

- 5 x 10 24

Not in Sci-Not: - 63 x 10 9

0.62 x 10 - 8

When changing scientific notation to

standard notation, the exponent tells

how to move the decimal:

With a positive exponent (big number), move the decimal to

the right (you’re multiplying by 1000):

4.08 x 103 = 4080

When changing scientific notation to

standard notation, the exponent tells

how to move the decimal:

With a negative exponent (small number), move the decimal to

the left (you’re multiplying by 0.0001):

8.93 x 10-3 = 0.00893

When changing standard notation to scientific notation follow these steps:Dry erase board example:68,940,0000.000569

1) Move the decimal point in the number to obtain a number between 1 and 10 (the decimal part).

2) Write the decimal part multiplied by 10 raised to the number of places you moved the decimal.

3) The exponent is positive if you moved the decimal point to the left (it is a big number) and negative if you moved the decimal point to the right (it is a small number)

Standard notation to Sci-Not

835000 = ?

8.35 x 105 Positive exponent means big number

move decimal to the right.

0.000893 = ?

8.93 x 10-4 Negative exponent means small number

move decimal to the left.

Scientific Notation Practice

Change to Sci-Not:

1) 68700

2) 0.0043

Change to Standard Notation:

3) 5.78 x 10-5

4) 1.6 x 10 6

Answers

1)6.87 x 104

2)4.3 x 10-3

3)0.0000578

4)1 600 000

Uncertainty and Significant FiguresSF’s-Why Are They Important?

•All measuring devices have limitations and therefore measurements always involve some uncertainty.

•Significant figures are used to report all precisely known numbers + one estimated digit

•The # of SF’s indicates the precision of the instrument.

Measurement

The number of significant figures in a measurement.

Uncertainty•What is the diameter of a quarter ?

•The answer depends on the precision of the ruler used.

Uncertainty•The top ruler is more precise (and more expensive).

•In the measurement 2.33 cm, the last digit is estimated as a “3” by using the calibration marks. The other two digits are certain.

•In the measurement 2.3 cm, the last digit is estimated as a “3” by using the calibration marks. The other digit is certain.

Measurement and Uncertainty•When measuring an object, the last digit should always be estimated (between two calibration marks).

•Consider the line below. What is the length in cm ?

•When the measurement falls on a calibration mark, the last digit is always zero.

•Length = 1.70 cm

Measurement and Uncertainty

•Consider the line below. What is the length in cm ?

Length = 4.00 cm

Measurement and Uncertainty

What is the difference in the following lengths ?They represent the length of the same line.

4.23 cm4.2 cm4 cm

Measurement and Uncertainty

•What is the difference in the following lengths ?

•They represent the length of the same line.

•4.23 cm--scientist used a fairly precise ruler

•4.2 cm--scientist used a less precise ruler-he must have run out of $$

•4 cm--they cheaped-out ! This ruler would have no calibration marks !

0 cm 10 cm

Why do we care about Significant Figures (SF’s) ?

There are 2 kinds of numbers:

Exact: the amount of money in your account,

Metric unit equalities. These are known with

certainty.

Approximate: weight, height—anything

MEASURED, has a limit to the certainty,

depending on the instrument used.

Why do we care about Significant Figures (SF’s) ?

3.50 inch to a scientist means the

measurement is precise to within one

hundredth of an inch.

Every measurement is a reflection of

the precision of the measuring

instrument.

Why do we care about Significant Figures (SF’s) ?

To a mathematician 3.5 inches, or

3.50 inches is the same.

But, to a scientist 3.5 inches and

3.50 inches is NOT the same

Pacific-Atlantic Rule

P A

If a decimal point is present, start on the Pacific (P) side

and begin counting at the first non-zero digit all the way to

the end.

If a decimal is absent, start on the Atlantic (A) side and

begin counting at the first non-zero digit all the way to the

end.

P A

Decimal

Is present Is absent

0.0050074

5 sig figs

If a

6003895400

8 sig figs

then then

Pacific Rule Examples:

123.003 = decimal present, start on “P” side,

begin counting # SF’s = 6Examples (Do with a study buddy):

0.00024 How many SF’s ?

0.453 How many SF’s ?

40.9 How many SF’s ?

43.00 How many SF’s ?

1.010 How many SF’s ?

1.50 How many SF’s ?

2

3

3

4

4

3

Atlantic Rule

Examples:

204,000 = decimal is absent, start on “A” side, begin counting at first non-zero digit until you hit the end of the number. # SF’s = 3

Examples (CO):

7003 How many SF’s ?

300 How many SF’s ?

27,300 How many SF’s ?

56 How many SF’s?

4

1

3

2

Drill and Practice

Underline the significant digits in each of the following numbers.

0.00506970

500.6790

250005

4970350000

5678493

6 sig figs

7 sig figs

6 sig figs

6 sig figs

7 sig figs

Scientific Notation

All digits in the decimal part of in a

number written in scientific notation

are significant.

Examples:

5.6 x 10 4 Has 2 SF’s

8.90 x 10 3 Has 3 SF’s

Use the Pacific-Atlantic Rules to determine the # of SF’s

a) 18.3 b) 1.83 x 102 c)0.00183

d) 183.0 e) 50 f) 505

g) 0.0050

h) 5000 i) 200.00 j) 0.00220

k) 22020 l) 22.20

How many Sig-Figs ?

a) 3 b) 3 c) 3

d) 4 e) 1 f) 3

g) 2

h) 1 i) 5 j) 3

k) 4 l) 4

Significant Numbers in Calculations

A calculated answer cannot be more precise than the measuring tool.

A calculated answer must match the least precise measurement.

There are two different rules for rounding and reporting a calculated answer

--Multiplying or dividing

--Adding or subtracting

Sig. Fig. Math Rules

• Multiplication / Division:

Your answer can’t have more sig. figs. than the number in the problem with the least amt. of sig. figs.

Example = 60.56227892 m x 35.25 m

Calculator says – 2134.890832 m2 (too many SF’s !)

Don’t forget units!

How many SF’s should be in final answer?

4 SF’s

Answer - 2135 m2

Multiplication & Division

The final answer has the same number of sig-figs as the number with the least number of sig-figs.

123m x 5.35m = 658.05m2 = 658m2 (3 SF’s)

16cm2 x 2cm = 32cm3 = 30cm3 (1 SF)

16 x 2.0g = 32g = 32g (2 SF’s)

Multiplication & Division

The final answer has the same number of sig-figs as the number with the least number of sig-figs.

823dm3 ÷ 4.0dm = 205.75 dm2 = 210 dm2

(2 SF’s)

84.7 g ÷ 20 mL = 4.235 g/mL = 4 g/mL (1 SF)

Sometimes the answer must be written in scientific notation to express the correct number of SF’s

25 m x 4.0 m = 100 m2

The answer should have 2 SF’s. If left as 100 m2 , it would only have 1 SF.

So change it to Sci-Not: 1.0 x 10 2 m2 now it has 2 SF’s

Be Careful with “Domino Rounding”

A. Round the following to 4 SF’s:

37.4959

B. Round the following to 3 SF’s:

68.4499

Answer: 37.50

Answer: 68.4

Express Result in Correct # of SF’sPay attention to units!

5.35 x 20.2 x 5.0 = _______

2.00 x 500 = _______

2.50 cm x 55.5 cm = ________

5.0 m x 4.000 x 102 m = _________

Multiplication Examples

5.35 x 20.2 x 5.0 = 540.35 = 540

2.00 x 500 = 1000

2.50 cm x 55.5 cm = 138.75cm 2

= 139cm 2

5.0 m x 4.000 x 102m = 2000m 2 = 2.0 x 10 3 m 2

( 2 SF’s)

Express Result in Correct # of SF’s

0.030 ÷ 2.00 = __________

9101 ÷ 8.8 = __________

(3.94 x 10 7 m2 ) ÷ (8.4 x 10 -25 m) = _____________

Express Result in Correct # of SF’s0.030 ÷ 2.00 = 0.015

9101 ÷ 8.8 = 1034 = 1.0 x 10 3 ** must be is Sci-Not !

(3.94 x 10 7 m2 ) ÷ (8.4 x 10 -25 m) =

4.7 x 10 31 m

Addition and SubtractionWhen adding or subtracting, look at the decimal places. Find the measurement that is least precise (# places past the decimal). Round to the least precise place.

Ex. a) 13.64 + 0.075 + 67 b) 267.8 – 9.36

13.64

0.075

67.

80.71581

267.8

9.36

258.44258.4

–

+

+

Round to the

ones place.

Round to the

tenths place.

Examples

4.021 =

4.0283.1425 =

83.14

i)

83.25

0.1075–

4.02

0.001+

ii)

1.8183 =

1.82

0.2983

1.52+

iii)

Round to the

Hundredths place.

Round to the

Hundredths place.

Round to the

Hundredths place.

Examples

42, 000 53.9

+ 698 + 60

42, 698 = 113.9 =

43, 000 110

Round to the

Thousands place.

Round to the

Tens place.

Learning Check-Study Buddy

In each calculation, round the answer to the correct number of significant figures.

A. 235.05 + 19.6 + 2.1 =

1) 256.75 2) 256.8 3) 257

B. 58.925 - 18.2 =

1) 40.725 2) 40.73 3) 40.7

Solution

A. 235.05 + 19.6 + 2.1 =

2) 256.8

B. 58.925 - 18.2 =

3) 40.7

Sig. Figs.-Mixed Operations

• Significant Figures in Mixed operations (Use PEMDAS)

(1.7 x 106 ÷ 2.63 x 105) + 7.33 = ???

Step 1: Divide the numbers in

the parenthesis. How many sig

figs?

Step 2: Add the numbers. How

many decimal places to keep?

(6.463878327…) + 7.33

6.463878327…

+ 7.33

13.7938

Step 3: Round answer to the

appropriate decimal place. 13.8 or 1.38 x 101

Precision and Accuracy

Precision

--Refers to how close the measurements in a series are to

each other.

--Can indicate how well your tools are working, not what

the tools are measuring.

--Can indicatethe technique of the scientist-poor

technique leads to poor precision and vice versa.

Accuracy

--Refers to how close each measurement is to the

accepted, true, or literature value.

--Checks how “right” or “correct” your answer is.

--Only need one measurement and the “true” value.

Precision, Accuracy, and Error

Precision refers to how close the measurements in a series

are to each other.

Accuracy refers to how close each measurement is to the

actual value.

Systematic error produces values that are either all higher

or all lower than the actual value.

This error is part of the experimental system.

Random error produces values that are both higher and

lower than the actual value.

Figure 1.9

precise and accurate

precise but not accurate

Precision and accuracy in a laboratory calibration.

systematic error

random error

Precision and accuracy in the laboratory.Figure 1.9continued

Quantifying Accuracy

Percent error, sometimes referred to as percentage error, is an

expression of the difference between a measured value and the

known or accepted value. It is often used in science to report the

difference between experimental values and accepted (true)

values. The smaller the value, the better the accuracy. Units are

%, usually rounded to the hundredths place.

The formula for calculating percent error is:

Absolute value bars

Calculating Percent Error

The literature value for the atomic mass of an isotope of nickel is 57.9

g/mol. If a laboratory experimenter determined the mass to be 56.5

g/mol, what is the percent error (rounded to the hundredths place)?

See board.

Answer: 2.42 %

Quantifying Precision

Take multiple measurements of the same thing under

the same conditions. Describe the “spread” of the

data relative to the average. The smaller the value,

the better the relative precision. Units are %. Usually

rounded to the hundredths place.

Calculate the Relative Precision:

relative precision = |(largest value – smallest value)| 100%

average value

Honors

Only

Quantifying Precision

relative precision = |(largest value – smallest value)| 100%

average value

Honors

Only

Calculate the relative precision for the data given

below. Round your answer to the hundredths place

and include the appropriate units. (Answer: 1.78 %)

Trial Volume (mL)

1 28.22 27.93 28.44 28.1

Average = 28.15

Relative Precision =

(28.4 - 27.9) / 28.15 * 100 =

1.77620 = 1.78 %

Do #1 on p. 20 now (Do #2 for homework)

Why Perform Multiple Trials?

--Can show reliability and reproducibility, this can add

validity to the conclusions you draw

--Multiple trials can give you information about only

precision.

Poor technique can contribute to poor precision.

Performing many trials will NOT ensure good accuracy.

In fact, nothing can be determined about accuracy with

just data from trials. You need a true (literature) value in

order to determine accuracy,