1_accuracy_precision_measurement

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14 Νοε 2013 (πριν από 3 χρόνια και 10 μήνες)

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Scientific Measurement


Investigation: Det’n of Thickness of Al foil


Counted versus Measured Quantities


Significant Digits and Measurement



-

Rules for Significant Figures


Accuracy and Precision


Errors in Measurement


Scientific Notation


Rules for Calculations Involving Measured
Quantities


Investigation: Determination of the Density
of Water


Investigation: Determination of
Thickness of Aluminum foil

Take note of what gauge of foil you’re given:
regular, heavy duty, etc.

What equipment/info. do you need to find
the thickness of the Al foil?


square of foil


mm ruler


electronic balance


access to PT at front of room

Brainstorm . . . and get busy!

Write your answer on the board.


Counted Quantities vs

Measured Quantities

How many measuring spoons are
illustrated?

Four. Are you sure?

Yes.


How many people are

in this room?

Do you have 100% confidence in this value?

Yes.


Counted quantities have
no uncertainty

associated with them.

If there are 15 people in a room, this may
also be expressed as

15.00000000000000000000000000

etc


There is an
infinite number of significant
figures
associated with a

COUNTED QUANTITY.

Measured Quantities

Is it possible to use the illustrated devices to
measure with absolute certainty?

No.

All
measurements


have some

uncertainty.

Significant Figures and
Measurement

When measuring, ALWAYS ESTIMATE
BETWEEN THE TWO SMALLEST SCALE
DIVISIONS to obtain the estimated
(uncertain) digit.

eg. When using a mm ruler, estimate the

tenth of a mm.






When using an
ungraduated

metre

stick,
estimate to the

tenths of a
metre
.

When measuring temperature on a
thermometer calibrated every degree, we
estimate

the tenths of a degree
.

How long is this line?



____________




the length of the line is


2.0
8

cm.

Remember

the “8” is

estimated

and is therefore

uncertain
.

How long is this line?


_____________________





The line is

4.0
0

cm long.

Read the temperature indicated on the
thermometer to the correct number of sig
figs.










Ans: 32.
6
o
C


What about a digital readout?

With a digital readout,
assume

that the

final

or rightmost

digit is estimated.


Significant Digits


All
measured quantities

have some
degree of uncertainty.


[cf. counted quantities have no uncert.]


The significant digits in a measurement
include all the digits that can be known
precisely plus a

final digit that must be

estimated.


Every measured quantity MUST have
ONE estimated, or uncertain, digit.



Rules for Significant Digits

1. All non
-
zero digits in a recorded
measurement are significant.


The measurements 24.7 m, 0.743
m, and 714 m all express
measures of length to

three

significant digits.

2. Zeros appearing between non
-
zero digits are significant.


The measurements 7003 m, 40.79
m, and 1.503 m all have

four

significant digits.


3.
Zeros

appearing
in front

of all nonzero
digits are
not significant
. They are acting as
place holders. The measurements 0.071 m,
0.42 m, and 0.000 099 m all have

two

significant digits.
Sci. not’n is your friend.

ie. 0.071 m =





7.1 x 10
-
2

m


0.000 099 m =





9.9 x 10
-
5

m


Check this . . .







orig. volume of coffee =






2.
0

x 10
2

mL

“new” vol. of coffee


=






2.
0

x 10
2

mL

here’s why:

vol. before addition

=


2
0
0


mL






+ 0.05

mL

“new” volume


=


2
0
0.05

mL


Taking into account
confidence limit
,

“new” volume = 2.
0

x 10
2

mL


SAME

Accuracy


How close a measurement is to the
accepted value.





e.g. the proximity of a dart to the “bullseye”


Precision

1. A precise instrument measures to more
decimal places. e.g. a mm ruler is more
precise than a cm ruler.

2. A precise set of data points are close to


one another.

eg. On a dartboard high precision has all
darts


close together, but not necessarily

at the bullseye.


Errors in Measurement


Systematic Error: The same error occurs
over and over.

eg. Taking a set of masses with an
improperly calibrated balance.



Random Error:

Errors that cannot be accounted for.

For which type of error can we most easily
correct?

systematic . . . why?


Because you make the same error each
time.


A Possible Ambiguity



The distance from Earth to moon is
382 000 km. How precise is this
distance?

We don’t know.

How can we eliminate this
ambiguity?

Use sci not’n: eg 3.8
2

x 10
5

km

Scientific Notation

Some examples:




6807 =





6.807 x 10
3


0.000 000 000 813 =







8.13 x 10
-
10

To make it easy


Always put values in scientific notation.

ie. 0.0067
0

cm



put in sci. not’n









6.7
0
x10
-
3

cm, or








6.7
0
E
-
3 cm





remember that the right
-
most zero is
estimated
, and is therefore a sig. fig.



Calculations Involving
Measured Quantities

Overlying Principle
:

A calculated value can have no more
precision than the data used to calculate it.

You can’t create precision.

You can’t create a greater confidence level
simply by doing a calculation.

“You can’t make a silk purse out of a sow’s
ear.”













Calculations Involving
Measured Quantities

1.
Multiplication and Division of Measured
Quantities

The measurement with the FEWEST
number of significant figures that goes into
a multiplication/division determines the
number of sig figs in the final answer.

Fewest # s.f. “in” = # sf in answer.

Don’t forget to round the answer.


examples

a)

Find the area of a room whose
dimensions are 23 m x 7.582 m.


A

= (23 m)
*(7.582 m)


= 174.386 m
2

says your calculator, or

A

= 1.7 x 10
2

m
2

to two sig figs.



b) Osmium, a metal, is the most dense
element.


Calculate the density of Os if a 12.3 cm
3

piece of Os has a mass of 277.98 g.

ρ

= mass/volume


= 277.9
8

g/12.
3

cm
3


= 22.
6

g/cm
3
to three sig figs.

c) A vehicle travels 278 miles on 11.70
gallons of gasoline. What is the average
fuel consumption in miles per gallon of the
vehicle, to the correct number of sf.?

2.
Addition and Subtraction of Measured
Quantities

Remember that the final answer can have
only ONE estimated digit. e.g. Add these:


2.4
5

m


4.
5


m

+
8.695


m


15.
645

m .

We must round this off to






one estimated digit, or





15.
6

m

eg. d) Here are the masses of several
plums.

56.4g, 65.5g, 62.34g, 102g, 77.8g.

Calculate the average mass of a plum.



56.
4




65.
5




62.3
4




10
2



+ 77.8


36
4.04




. . . con’t


36
4

g/5 plums =

72.8 g/plum says your calculator.

To the correct number of sf, the average
mass of a plum is

7
3

g.

The least certainty was in the “units” digit of
the 10
2

g plum.


But what about the
5

plums? This is a

counted quantity and therefore has

infinite sf.

f) Find the sum of 3.18 and 0.01315 to the
correct number of sf.

Sample Problem

Using a mm ruler, determine the thickness
of one sheet in your Chemistry textbook.

Solution:



680 pages;




thickness = 2.23 cm;

How many sig figs in each of these?





counted;


3 sf


measured



680 pages/2 pages
∙sheet
-
1

= 340 sheets




2.2
3

cm/340 sheet






= 6.5
5
882353 x 10
-
3
cm/sheet





= 6.5
6

x 10
-
3

cm/sheet, or





= 6.5
6

x 10
-
2

mm/sheet

Compare to thickness of Al foil

Al foil



= 1.
5

x 10
-
2

mm;

textbook sheet


= 6.5
6

x 10
-
2

mm


Why is the textbook sheet’s thickness more

precisely known?

mass of Al foil


2 sf

number of sheets


∞ sf (counted quantity)

Rounding Off

When doing a series of calculations
involving measured quantities, round off
only the final answer.


Allow your calculator to carry a few more
sig. digs throughout the (intermediate)
calculations to

avoid compounding errors due to rounding
.

Rounding “fives”


Look at these numerals:



1 2 3 4
5

6 7 8 9


What will happen if we always round a
5

“up” in a large set of data?


Average will be skewed slightly upward.


How can we overcome this problem?


We need to round a
5

up half the time, and
down the other half.


But how can we easily do this?

Rounding 5s (con’t)


If the digit before the
5

to be rounded is
odd, round “up”. e.g. 6.
7
5 g











6.8 g


If the digit before the
5

to be rounded is
even, do not round “up”. e.g. 3.
4
5 m











3.4 m


If there are digits after the
5

that needs to
be rounded, always round up. This makes
sense, since we’re rounding something
that’s greater than
5
.

e.g. 4.6851 g (to 3 s.f.)









4.69 g


Investigation: Determination of the
Density of Water

For each pair of students: Use the
equipment provided to determine the
density of tap water.


Write your raw data and your calculated
density of water on the board.

Report your answer to correct number of

significant figures.