V. Limits of Measurement 1. Accuracy and Precision.

V. Limits of Measurement

1. Accuracy and Precision

• Accuracy - a measure of how close a measurement is to the true value of the quantity being measured.

Example: Accuracy• Who is more accurate when

measuring a book that has a true length of 17.0cm?

Susan: 17.0cm, 16.0cm, 18.0cm, 15.0cm

Amy: 15.5cm, 15.0cm, 15.2cm, 15.3cm

• Precision – a measure of how close a series of measurements are to one another. A measure of how exact a measurement is.

Example: Precision

Who is more precise when measuring the same 17.0cm book?

Susan: 17.0cm, 16.0cm, 18.0cm, 15.0cm

Amy: 15.5cm, 15.0cm, 15.2cm, 15.3cm

Example: Evaluate whether the following are precise, accurate or both.

Accurate

Not Precise

Not Accurate

Precise

Accurate

Precise

2. Significant Figures

• The significant figures in a measurement include all of the digits that are known, plus one last digit that is estimated.

8

Exact NumbersAn exact number is obtained when you count

objects or use a defined relationship.

- Counting objects are always exact2 soccer balls4 pizzas

- Exact relationships, predefined values, not measured1 foot = 12 inches1 meter = 100 cm

For instance is 1 foot = 12.000000000001 inches? No 1 ft is EXACTLY 12 inches.

9

Solution

A. Exact numbers are obtained by

2. counting

3. definition

B. Measured numbers are obtained by

1. using a measuring tool

2.1 Uncertainty in Measurement 40.16 cm

2.1. Uncertainty in Measurement

• A measurement always has some degree of uncertainty.

2.1 Uncertainty in Measurement

• Different people estimate differently.

• Record all certain numbers and one estimated number.

2.1 Measurement and Significant Figures• Every experimental

measurement has a degree of uncertainty.

• The volume, V, at right is certain in the 10’s place, 10mL<V<20mL

• The 1’s digit is also certain, 17mL<V<18mL

• A best guess is needed for the tenths place.

Chapter Two 13

14

What is the Length?

1 2 3 4 cm

• We can see the markings between 1.6-1.7cm• We can’t see the markings between the .6-.7• We must guess between .6 & .7• We record 1.67 cm as our measurement• The last digit an 7 was our guess...stop there

Learning Check

What is the length of the wooden stick?1) 4.5 cm 2) 4.54 cm 3) 4.547 cm

16

8.00 cm or 3 (2.2/8)?

Below are two measurements of the mass of the same object. The same quantity is being described at two different levels of precision or certainty.

2.2 Significant Figures • Significant figures are the meaningful figures in our

measurements and they allow us to generate meaningful conclusions

• Numbers recorded in a measurement are significant. – All the certain numbers plus first estimated number

e.g. 2.85 cm • We need to be able to combine data and still produce

meaningful information• There are rules about combining data that depend on

how many significant figures we start with………

2.3 Rules for Counting Significant Figures

1. Nonzero integers always count as significant figures.

1457 has 4 significant figures

23.3 has 3 significant figures

Rules for Counting Significant Figures

2. Zeros

a. Leading zeros - never count0.0025 2 significant figures

b. Captive zeros - always count 1.008 4 significant figures

c. Trailing zeros - count only if the number is written with a decimal point 100 1 significant figure 100. 3 significant figures 120.0 4 significant figures

Rules for Counting Significant Figures

3. Exact numbers - unlimited significant figures

• Not obtained by measurement Determined by counting:

3 apples Determined by definition:

1 in. = 2.54 cm

Practice Rule #1 Zeros

45.8736

.000239

.00023900

48000.

48000

3.982106

1.00040

6

3

5

5

2

4

6

• All digits count

• Leading 0’s don’t

• Trailing 0’s do

• 0’s count in decimal form

• 0’s don’t count w/o decimal

• All digits count

• 0’s between digits count as well as trailing in decimal form

How Many Significant Figures?

1422

65,321

1.004 x 105

200

435.662

50.041

102

102.0

1.02

0.00102

0.10200

1.02 x 104

1.020 x 104

60 minutes in an hour

500 laps in the race

• One convention about trailing zero

A bar placed over ( or under) the last significant figure; any trailing zeros following this are insignificantExample:500 has 1 s.f. 5500 has 3 s.f. 500. has 3 s.f.

- Round off 52.394 to 1,2,3,4 significant figures

2.4 Scientific notationWrite number in form:

Standard decimal notation Scientific notation

2 2×100

300 3×102

4,321.768 4.321768×103

−53,000 −5.300×104

6,720,00,000 6.72000×109

0.2 2×10−1

0.000 000 007 51 7.51×10−9

Chapter Two 28

Two examples of converting standard notation to scientific notation are shown below.

Chapter Two 29

Two examples of converting scientific notation back to standard notation are shown below.

• Scientific notation is helpful for indicating how many significant figures are present in a number that has zeros at the end but to the left of a decimal point.

• The distance from the Earth to the Sun is 150,000,000 km. Written in standard notation this number could have anywhere from 2 to 9 significant figures.

• Scientific notation can indicate how many digits are significant. Writing 150,000,000 as 1.5 x 108 indicates 2 and writing it as 1.500 x 108 indicates 4.

• Scientific notation can make doing arithmetic easier.

How many sig figs?

1 10302.00

100.00 970

0.001 0.00250

10302 1.0302x104

2.5 Rules for Multiplication and

Division

• I measure the sides of a rectangle, using a ruler to the nearest 0.1cm, as 4.5cm and 9.3cm

• What does a calculator tell me the area is?• What is the range of areas that my measurements might

indicate (consider the range of lengths that my original measurements might cover)?

Rules for Multiplication and Division

• The number of significant figures in the result is the same as in the measurement with the smallest number of significant figures.

2.6 Rules for Addition and Subtraction

• The number of significant figures in the result is the same as in the measurement with the smallest number of decimal places.

2.7 Rules for Combined Units

• Multiplication / Division– When you Multiply or Divide measurements you must carry out

the same operation with the units as you do with the numbers

50 cm x 150 cm = 7500 cm2

20 m / 5 s = 4 m/s or 4 ms-1

16m / 4m = 4

• Addition / Subtraction– When you Add or Subtract measurements they must be in the

same units and the units remain the same

50 cm + 150 cm = 200 cm

20 m/s – 15 m/s = 5 m/s

32.27 1.54 = 49.6958

3.68 .07925 = 46.4353312

1.750 .0342000 = 0.05985

3.2650106 4.858 = 1.586137 107

6.0221023 1.66110-24 = 1.000000

49.7

46.4

.05985

1.586 107

1.000

Calculate the following:

.56 + .153 = .713

82000 + 5.32 = 82005.32

10.0 - 9.8742 = .12580

10 – 9.8742 = .12580

.71

82000

.1

0

Look for the last important digit

Calculate the following:

Mixed Order of Operation

8.52 + 4.1586 18.73 + 153.2 =

(8.52 + 4.1586) (18.73 + 153.2) =

239.6

2180.

= 8.52 + 77.89 + 153.2 = 239.61 =

= 12.68 171.9 = 2179.692 =

Calculate the following. Give your answer to the correct number of significant figures and use the correct units

11.7 km x 15.02 km =

12 mm x 34 mm x 9.445 mm =

14.05 m / 7 s =

108 kg / 550 m3 =

23.2 L + 14 L =

55.3 s + 11.799 s =

16.37 cm – 4.2 cm =

350.55 km – 234.348 km =

practice1.Calculate Volume of sphere with ,55.0 mr

33

33

70.06969.0

)55.0(3

4

3

4

mm

rV

2. Perimeter of the big circle

mm

mrP

5.34557.3

)55.0(22

Try the following

7.895 + 3.4=

(8.71 x 0.0301)/0.056 = =

A= =

13m

4.91m2

IV Dimension Analysis – some simple rules

1.In : The product unit is the product of the individual unit of each of those variables. (Ditto for ratios.)

2. : Different terms can only added together in a sum if each term in the sum has the same unit type. (Ditto for subtraction.)

Example 1

- impossible: 40m + 20m/s or 12.5 s - 20m2

- Can Do: 50.0m + 20.55m=70.6mand 40m/s +11m/s =51m/s

- Can Do, but need to convert into same unit:

40m + 11cm = 40m + 11cm = 40.11m

Example 2

The above expression yields:

1.5 m 3.0 kg ?

a)4.5 m kgb)4.5 g kmc)A or Bd)Impossible to evaluate (dimensionally invalid)