v. limits of measurement 1. accuracy and precision
TRANSCRIPT
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.
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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.
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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)