Transcript Significant Figures

ACCURACY & PRECISION
IN MEASUREMENT
GO EAGLES
FIGHT !!!!! WIN!!!
ACCURACY & PRECISION
•
ACCURACY:
– HOW CLOSE YOU ARE TO THE
ACTUAL VALUE
– DEPENDS ON THE PERSON
MEASURING
– CALCULATED BY THE
FORMULA:
% ERROR = (YV – AV) X 100 ÷ AV
WHERE: YV IS YOUR MEASURED
VALUE & AV IS THE ACCEPTED
VALUE
•
PRECISION:
– HOW FINELY TUNED
YOUR
MEASUREMENTS ARE
OR HOW CLOSE THEY
CAN BE TO EACH
OTHER
– DEPENDS ON THE
MEASURING TOOL
– DETERMINED BY THE
NUMBER OF
SIGNIFICANT DIGITS
• ACCURACY & PRECISION MAY BE DEMONSTRATED BY
SHOOTING AT A TARGET.
•
ACCURACY IS REPRESENTED BY HITTING THE BULLS
EYE (THE ACCEPTED VALUE)
•
PRECISION IS REPRESENTED BY A TIGHT GROUPING OF
SHOTS (THEY ARE FINELY TUNED)
Accuracy & Precision
Precision without
Accuracy
Accuracy without
Precision
No Precision &
No Accuracy
SIGNIFICANT DIGITS
A MEASUREMENT FOR PRECISION
SIGNIFICANT DIGITS & PRECISION
• THE PRECISION OF A MEASUREMENT IS THE
SMALLEST POSSIBLE UNIT THAT COULD BE
MEASURED.
• SIGNIFICANT DIGITS (SD) ARE THE NUMBERS
THAT RESULT FROM A MEASUREMENT.
• WHEN A MEASUREMENT IS CONVERTED WE NEED
TO MAKE SURE WE KNOW WHICH DIGITS ARE
SIGNIFICANT AND KEEP THEM IN OUR
CONVERSION
• ALL DIGITS THAT ARE MEASURED ARE
SIGNIFICANT
SIGNIFICANT DIGITS & PRECISION
• HOW MANY DIGITS
ARE THERE IN THE
MEASUREMENT?
• ALL OF THESE DIGITS
ARE SIGNIFICANT
• THERE ARE 3 SD
WHAT IS THE LENGTH OF
THE BAR?
0
CM
1
2
LENGTH OF BAR
= 3.23 CM
3
4
SIGNIFICANT DIGITS & PRECISION
• IF WE CONVERTED TO THAT MEASUREMENT OF
3.23 CM TO MM WHAT WOULD WE GET?
• RIGHT! 32 300 MM
• HOW MANY DIGITS IN OUR CONVERTED NUMBER?
• ARE THEY ALL SIGNIFICANT DIGITS (MEASURED)?
•
• WHICH ONES WERE MEASURED AND WHICH ONES
WERE ADDED BECAUSE WE CONVERTED?
• IF WE KNOW THE SIGNIFICANT DIGITS WE CAN
KNOW THE PRECISION OF OUR ORIGINAL
MEASUREMENT
• WHAT IF WE DIDN’T KNOW THE ORIGINAL
MEASUREMENT – SUCH AS 0.005670 HM. HOW WOULD
WE KNOW THE PRECISION OF OUR MEASUREMENT.
• THE RULES SHOWING HOW TO DETERMINE THE NUMBER
OF SIGNIFICANT DIGITS IS SHOWN IN YOUR LAB MANUAL
ON P. 19. THOUGH YOU CAN HANDLE THEM, THEY ARE
SOMEWHAT COMPLEX.
Significant Figures
Physical Science
What is a significant figure?
• There are 2 kinds of
numbers:
–Exact: the amount of
money in your account.
Known with certainty.
WHAT IS A SIGNIFICANT FIGURE?
– APPROXIMATE:
– WEIGHT, HEIGHT—ANYTHING MEASURED. NO
MEASUREMENT IS PERFECT.
WHEN TO USE SIGNIFICANT FIGURES
• WHEN A MEASUREMENT IS RECORDED ONLY
THOSE DIGITS THAT ARE DEPENDABLE ARE
WRITTEN DOWN.
– IF YOU MEASURED THE WIDTH OF A PAPER
WITH YOUR RULER YOU MIGHT RECORD
21.7CM.
TO A MATHEMATICIAN 21.70, OR 21.700 IS THE
SAME, BUT, TO A SCIENTIST 21.7CM AND
21.70CM IS NOT THE SAME
• GRADE SCHOOL PLACEHOLDERS AND NUMBER LINE
• 21.700CM TO A SCIENTIST MEANS THE MEASUREMENT IS
ACCURATE TO WITHIN ONE THOUSANDTH OF A CM.
• BUT, TO A SCIENTIST 21.7CM AND 21.70CM IS NOT THE
SAME
• IF YOU USED AN ORDINARY RULER, THE SMALLEST
MARKING IS THE MM, SO YOUR MEASUREMENT HAS TO BE
RECORDED AS 21.7CM.
HOW DO I KNOW HOW MANY SIG FIGS?
• RULE: ALL DIGITS ARE SIGNIFICANT STARTING WITH
THE FIRST NON-ZERO DIGIT ON THE LEFT.
• EXCEPTION TO RULE: IN WHOLE NUMBERS THAT END IN
ZERO, THE ZEROS AT THE END ARE NOT SIGNIFICANT.
GO EAGLES
• 2ND EXCEPTION TO RULE: IF ZEROS ARE SANDWICHED
BETWEEN NON-ZERO DIGITS, THE ZEROS BECOME
SIGNIFICANT.
• 3RD EXCEPTION TO RULE: IF ZEROS ARE AT THE END OF
A NUMBER THAT HAS A DECIMAL, THE ZEROS ARE
SIGNIFICANT.
HOW MANY SIG FIGS?
•7
• 40
• 0.5
• 0.00003
5
• 7 x 10
• 7,000,000
•1
•1
•1
•1
•1
•1
• 3RD EXCEPTION TO RULE: THESE ZEROS ARE SHOWING
HOW ACCURATE THE MEASUREMENT OR CALCULATION
ARE.
•
•
•
•
•
•
1.2
2100
56.76
4.00
0.0792
7,083,000,000
•
•
•
•
•
•
2
2
4
3
3
4
•
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•
•
•
•
3401
2100
2100.0
5.00
0.00412
8,000,050,000
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•
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•
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4
2
5
3
3
6
RULES FOR SIGNIFICANT
FIGURES
RULE #1
• ALL NON ZERO DIGITS ARE ALWAYS SIGNIFICANT
• HOW MANY SIGNIFICANT DIGITS ARE IN THE
FOLLOWING NUMBERS?
•274
•3 SIGNIFICANT FIGURES
•25.632
•5 SIGNIFICANT DIGITS
•8.987
•4 SIGNIFICANT FIGURES
RULE #2
• ALL ZEROS BETWEEN SIGNIFICANT DIGITS ARE
ALWAYS SIGNIFICANT
• HOW MANY SIGNIFICANT DIGITS ARE IN THE
FOLLOWING NUMBERS?
504
3 SIGNIFICANT FIGURES
60002
5 SIGNIFICANT DIGITS
9.077
4 SIGNIFICANT FIGURES
RULE #3
• ALL FINAL ZEROS TO THE RIGHT OF THE DECIMAL
ARE SIGNIFICANT
• HOW MANY SIGNIFICANT DIGITS ARE IN THE
FOLLOWING NUMBERS?
32.0
3 Significant Figures
19.000
5 Significant Digits
105.0020
7 Significant Figures
RULE #4
• ALL ZEROS THAT ACT AS PLACE HOLDERS ARE
NOT SIGNIFICANT
• ANOTHER WAY TO SAY THIS IS: ZEROS ARE ONLY
SIGNIFICANT IF THEY ARE BETWEEN SIGNIFICANT
DIGITS OR ARE THE VERY FINAL THING AT THE
END OF A DECIMAL
FOR EXAMPLE
HOW MANY SIGNIFICANT DIGITS ARE IN
THE FOLLOWING NUMBERS?
0.0002
6.02 x 1023
100.000
150000
800
1 Significant Digit
3 Significant Digits
6 Significant Digits
2 Significant Digits
1 Significant Digit
RULE #5
• ALL COUNTING NUMBERS AND CONSTANTS HAVE
AN INFINITE NUMBER OF SIGNIFICANT DIGITS
• FOR EXAMPLE:
1 HOUR = 60 MINUTES
12 INCHES = 1 FOOT
24 HOURS = 1 DAY
HOW MANY SIGNIFICANT DIGITS ARE
IN THE FOLLOWING NUMBERS?
0.0073
100.020
2500
7.90 x 10-3
670.0
0.00001
18.84
2 SIGNIFICANT DIGITS
6 SIGNIFICANT DIGITS
2 SIGNIFICANT DIGITS
3 SIGNIFICANT DIGITS
4 SIGNIFICANT DIGITS
1 SIGNIFICANT DIGIT
4 SIGNIFICANT DIGITS
Rules Rounding Significant
Digits
Rule #1
• If the digit to the immediate right of the last
significant digit is less that 5, do not round up
the last significant digit.
• For example, let’s say you have the number
43.82 and you want 3 significant digits
• The last number that you want is the 8 –
43.82
• The number to the right of the 8 is a 2
• Therefore, you would not round up & the
number would be 43.8
Rounding Rule #2
• If the digit to the immediate right of the last
significant digit is greater that a 5, you round
up the last significant figure
• Let’s say you have the number 234.87 and
you want 4 significant digits
• 234.87 – The last number you want is the 8
and the number to the right is a 7
• Therefore, you would round up & get 234.9
Rounding Rule #3
• If the number to the immediate right of the
last significant is a 5, and that 5 is followed by
a non zero digit, round up
• 78.657 (you want 3 significant digits)
• The number you want is the 6
• The 6 is followed by a 5 and the 5 is followed
by a non zero number
• Therefore, you round up
• 78.7
Rounding Rule #4
• If the number to the immediate right of the
last significant is a 5, and that 5 is followed by
a zero, you look at the last significant digit
and make it even.
• 2.5350 (want 3 significant digits)
• The number to the right of the digit you want
is a 5 followed by a 0
• Therefore you want the final digit to be even
• 2.54
Say you have this number
• 2.5250
(want 3 significant digits)
• The number to the right of the digit you
want is a 5 followed by a 0
• Therefore you want the final digit to be
even and it already is
• 2.52
Let’s try these examples…
200.99
(want 3 SF)
201
18.22
(want 2 SF)
18
135.50
(want 3 SF)
136
0.00299
(want 1 SF)
0.003
98.59
(want 2 SF)
99
Scientific Notation
• Scientific notation is used to express
very large or very small numbers
• I consists of a number between 1 & 10
followed by x 10 to an exponent
• The exponent can be determined by the
number of decimal places you have to
move to get only 1 number in front of
the decimal
Large Numbers
• If the number you start with is greater than 1,
the exponent will be positive
• Write the number 39923 in scientific notation
• First move the decimal until 1 number is in
front – 3.9923
• Now at x 10 – 3.9923 x 10
• Now count the number of decimal places that
you moved (4)
• Since the number you started with was
greater than 1, the exponent will be positive
• 3.9923 x 10 4
Small Numbers
• If the number you start with is less than 1, the
exponent will be negative
• Write the number 0.0052 in scientific notation
• First move the decimal until 1 number is in
front – 5.2
• Now at x 10 – 5.2 x 10
• Now count the number of decimal places that
you moved (3)
• Since the number you started with was less
than 1, the exponent will be negative
• 5.2 x 10 -3
Scientific Notation Examples
Place the following numbers in scientific notation:
99.343
9.9343 x 101
4000.1
4.0001 x 103
0.000375
3.75 x 10-4
0.0234
2.34 x 10-2
94577.1
9.45771 x 104
Going from Scientific Notation
to Ordinary Notation
• You start with the number and move the
decimal the same number of spaces as
the exponent.
• If the exponent is positive, the number
will be greater than 1
• If the exponent is negative, the number
will be less than 1
Going to Ordinary Notation
Examples
Place the following numbers in ordinary notation:
3 x 106
6.26x 109
5 x 10-4
8.45 x 10-7
2.25 x 103
3000000
6260000000
0.0005
0.000000845
2250
Accuracy - Calculating % Error
How Close Are You to the Accepted
Value (Bull’s Eye)
Accuracy - Calculating % Error
• If a student measured the room width at
8.46 m and the accepted value was 9.45 m
what was their accuracy?
• Using the formula:
% error = (YV – AV) x 100 ÷ AV
– Where YV is the student’s measured value &
AV is the accepted value
Accuracy - Calculating % Error
• Since YV = 8.46 m, AV = 9.45 m
• % Error = (8.46 m – 9.45 m) x 100 ÷ 9.45
m
•
=
-0.99 m
x 100 ÷ 9.45
m
•
=
-99 m
÷ 9.45
m
•
=
-10.5 %
• Note that the meter unit cancels during the division
& the unit is %. The (-) shows that YV was low
Acceptable error is +/- 5%
Values from –5% up to 5% are acceptable
Values less than –5% or greater than 5% must be remeasured
remeasure -5%
5% remeasure