Download Making Measurements

Making Measurements
Precision vs Accuracy

Accuracy : A measure of how close a measurement comes to the
actual, accepted or true value of whatever is measured.

Example: Over two trials, a student measures the boiling point of
ethanol and then calculates the average.

Trial No.
°C
1
79.2
2
78.8
Average
79.0
Accepted Value = 78.4 °C
She then checks a chemistry handbook (CRC) to see how close her
measurements are to the actual value.
Precision vs Accuracy

Precision : A measure of how close a series of measurements
are to one another. This is best determined by the deviation of
the data points.

Example: Two students independently determine the average
boiling point of ethanol.

Trial No.
Student A
Student B
1
79.2
79.7
2
78.8
76.9
Average
79.0
78.3
Deviation
0.4
2.8
Accepted Value = 78.4 °C
Student A’s data is less accurate but more precise than student B’s.
Determining Error

Error : The difference between the accepted value and the
experimental value.
Formula: Error = experimental value – accepted value

Percent error : The absolute value of the error divided by the
accepted value, multiplied by 100.
Formula: % Error =

‫ ׀‬error ‫׀‬
Accepted value
X 100
Calculate the % error for the data given below.
Student A = 79.0 °C
Accepted Value = 78.4 °C
%E =
‫ ׀‬79.0-78.4 ‫׀‬
78.4
%E = 0.8 %
X 100
Scientific Notation
Scientific Notation

Scientific Notation : An expression of numbers in the form
m x 10n where m is ≥ 1 and < 10 and n is an integer..

Using scientific notation makes it easier to work with
numbers that are very large and very small.

Example #1: A single gram of hydrogen contains approximately
602 000 000 000 000 000 000 000 atoms
which can be rewritten in scientific notation as 6.02 x 1023

Example #2: The mass of an atom of gold is
0.000 000 000 000 000 000 000 327 gram
which can be rewritten in scientific notation as 3.27 x 10-22
Scientific Notation

In scientific notation, there is a coefficient and an exponent, or power.
6.02 x 1023
Coefficient
Exponent

The exponent value is determined by moving a decimal point.

For example: When changing a large number to scientific notation,
the decimal is moved left until one non-zero digit remains.
6 02 000 000 000 000 000 000 000 atoms
Scientific Notation

Remember to count the number of place values as you
move the decimal.
6 02 000 000 000 000 000 000 000
23 places

Now drop the “trailing” zeros and add an “x 10”.
atoms
Scientific Notation

Remember to count the number of place values as you
move the decimal.
6 02 x 10
atoms
23 places

Now drop the “trailing” zeros and add an “x 10”.

Place the 23 as an exponent of the “10”.

You have just converted standard notation to scientific notation
602 000 000 000 000 000 000 000
6.02 x 1023
Scientific Notation

When changing a very small number to scientific notation, the
decimal is moved to the right until it passes a non-zero number.
0 000 000 000 000 000 000 000 3 27 gram
- 22 places

Again, count the number of place values the decimal moved.
Scientific Notation

When changing a very small number to scientific notation, the
decimal is moved to the right until it passes a non-zero number.
0 000 000 000 000 000 000 000 3 27 gram
- 22 places

Again, count the number of place values the decimal moved.

Now drop the “leading” zeros and add an “x 10”.
Scientific Notation

When changing a very small number to scientific notation, the
decimal is moved to the right until it passes a non-zero number.
x 10
3 27 gram
- 22 places

Again, count the number of place values the decimal moved.

Now drop the “leading” zeros and add an “x 10”.
Scientific Notation

When changing a very small number to scientific notation, the
decimal is moved to the right until it passes a non-zero number.
3 27 x 10
gram
- 22

Again, count the number of place values the decimal moved.

Now drop the “leading” zeros and add an “x 10”.

Finally, place the -22 as an exponent of the “10”.

Again you have converted standard notation to scientific notation
0.000 000 000 000 000 000 000 327
3.27 x 10-22
Scientific Notation

Normally, scientific notation is not used for measurements that
produce an exponent of +1 or -1.
Examples : 89.1 = 8.91 x 101 (not usually done)
0.32 = 3.2 x 10-1 (not usually done)
Practice problems

Change the following number to proper scientific notation.
0.000 000 12

= 1.2 x 10-7
Change the following number to proper standard notation.
4.3 x 105
= 430 000
Recording Measurements

Assume you are using a thermometer and want to
record a temperature. To do this, you must first
determine the instrument’s precision, or I.P.
°C
25
The precision of an instrument is equal to the
smallest division on the instrument’s scale.

Pictured to the right is a thermometer scale.
What is the scale’s precision?
The I.P. is = 0.1 °C
24
Smallest
division
23
22
Recording Measurements

The next thing to do is record the temperature and unit.
How would you record the temperature shown?


°C
25
If you recorded the temperature as 24.3 °C
then you were not being as precise as you
could be with the scale given.
If you look carefully at the top of the thermometer’s
fluid you will see it rises a bit higher than 24.3 °C.
You might now record the temperature as
24.31 °C or 24.32 °C
24
23
22
Recording Measurements

The first three digits in our measurements are
known with certainty. However, the last digit was
estimated and involves some uncertainty.
°C
25
24.32 °C
Certain or Estimated
digit
known digits
24
Significant digits

All measurements consist of known digits and
one estimated digit. Together, they are called
significant digits.
23
22
Recording Measurements

Error in measurement may be represented by a
tolerance interval.

Machines used in manufacturing often set tolerance
intervals, or ranges in which product measurements
will be tolerated before they are considered flawed.

To determine the tolerance interval in a measurement,
add and subtract (±) one-half of the precision of the
measuring instrument to the measurement.

We will now express our measurement as
24.32 °C ± 0.05 °C (±T.I.)
Significant
figures
Unit
°C
25
24
23
Tolerance
interval
22
Recording Measurements

Always round the experimental measurement or
result to the same decimal place as the uncertainty.
It would be confusing (and perhaps dishonest) to
suggest that you knew the digit in the hundredths
(or thousandths) place when you admit that you’re
unsure of the tenth’s place.

For example: How would you read the
temperature shown to the right?
It should be read as…
°C
76
75
74
73
72
73.2 °C ± 0.1 °C (±T.I.)
3rd
This
digit
was rounded
down
to match the
place value of
the T.I.
71
70
Recording Measurements



Note: It is not necessary to write the tolerance
interval for each measurement of a series of
measurements made using the same instrument.
For example: If you record a series
of temperature measurements in a
data table, then you need only state
the tolerance interval once.
Accepted value =
27.74 °C
Time
°C
(min)
(T.I. = ± 0.05°C)
1
19.05
2
21.25
3
25.85
4
31.10
5
34.00
Average =
26.25
Standard deviation (± SD)
± SD =
± 5.67
Standard error (± SE)
± SE =
± 2.53
Percent error (%E)
%E=
5.4 %
Keep in mind there are many ways of
showing uncertainty in measurement.
This is why you must indicate the
source of the uncertainty, such as…
Tolerance Interval (T.I.)
Recording Measurements
Statistic
What it is
Statistical interpretation
Symbol
Average
An estimate of the
"true" value of the
measurement
The central value
X
A measure of the
"spread" in the data
You can be reasonably sure (about 70%
sure) that if you repeat the same
measurement one more time, that
next measurement will be less than
one standard deviation away from the
average.
SD
An estimate in the
uncertainty in the
average of the
measurements
You can be reasonably sure (about 70%
sure) that if you do the entire
experiment again with the same
number of repetitions, the average
value from the new experiment will
be less than one standard error away
from the average value from this
experiment.
SE
Standard
deviation
Standard
error
Significant Figures
Determining Significant Figures

Significant figures are all the digits that can be known precisely
in a measurement, plus a last estimated digit.

The rules for recognizing significant figures are as follows:
√ Zeros within a number are always significant.
Both 4308 and 40.05 contain four sig. figs.
√ Zeros that do nothing but set the decimal point
are not significant.
570 000 and 0.010 and 310 contain two sig. figs.
√ Trailing zeros that aren’t needed to hold a decimal
point are significant.
Both 4.00 and 0.0320 contain three sig. figs.
Determining Significant Figures

Here is a “trick” that can help you with significant figures.

Question: How many sig. figs. are in 0.00180 ?
Step 1: Check to see if the number has a decimal. If yes, think
“present.” If no, think “absent.”
In our example of 0.00180, a decimal is present.
Step 2: Note that Present starts with a “P” and so does Pacific.
Pacific
Ocean
(decimal
present)
Determining Significant Figures
Step 3: Now place the number inside the U.S.A. pictured below.
Step 4: Draw an arrow from the Pacific Ocean through the number
until you encounter a non-zero digit.
√ Rule: All digits to the right of the arrow tip are significant.
In our example, 0.00180 has three significant figures.
Pacific
Ocean
(decimal
present)
0.00180
Determining Significant Figures

New Question: How many sig. figs. are in 403 200 ?
Step 1: Check to see if the number has a decimal. If yes, think
“present.” If no, think “absent.”
In our example of 403 200, a decimal is absent.
Step 2: Note that Absent starts with an “A” and so does Atlantic.
Pacific
Ocean
Atlantic
Ocean
(decimal
present)
(decimal
absent)
Determining Significant Figures
Step 3: Now place the number inside the U.S.A. pictured below.
Step 4: Draw an arrow from the Atlantic Ocean through the number
until you encounter a non-zero digit.
√ Rule: All digits to the left of the arrow tip are significant.
In our example, 403 200 has four significant figures.
Pacific
Ocean
(decimal
present)
403 200
Atlantic
Ocean
(decimal
absent)
Determining Significant Figures

How many sig. figs are in each of the following measurements?
280.00
five
2.8000 x 10 2
five
4.5 x 10 2
two
450
two
0.0003
one
0.00030
two
2.00 x 10 -4
three
100.0030
seven
Determining Significant Figures

Let’s apply what you have learned.
How would you record the following measurement?
mm
149
150
200
It should have been recorded as 150.00 mm ± 0.05 mm

Now change this measurement into scientific notation.
It should have been written as 1.5000 x 10 2 ± 0.05 mm


Note that the three zeros after the numeral 5 must be retained in
order to uphold the precision of the measurement.
How many significant figures does 1.5000 x 102 contain?
It contains five.
Calculating with Significant Figures
Calculating with Significant Figures

In general, a calculated answer cannot be more precise than the
least precise measurement from which it is calculated.
This is analogous to saying that a chain cannot
be stronger than its weakest link.

Here is the rule for Adding or Subtracting significant figures
Round the answer to the same number of
decimal places (not digits) as the measurement
with the least number of decimal places.

Here is the rule for Multiplying or Dividing significant figures
Round the answer to the same number of
significant figures as the measurement with
the least number of significant figures.
Calculating with Significant Figures

Example addition problem:
Step 1: Stack the numbers and
align them by decimal location
12.52 + 349.0 + 8.24
369.76
Step 2: Add the numbers
Step 3: Locate the measurement
with the least number of digits
to the right of the decimal point
The first measurement
(349.0 meters) has the
least number of digits
(one) to the right of the
decimal point.
Step 4:The answer must be
rounded to one digit after the
decimal point.
369.8 or 3.698 x 102
Calculating with Significant Figures

Example multiplication problem:
x
Step 1: Multiply the numbers
Step 2: Locate the measurement
with the least number of
significant figures.
Step 3: The answer must be
rounded to the same number
of significant figures as the
measurement with the least
number of significant figures
0.70 m
2.10 m
1.47 m2
The first measurement
(0.70) has smallest
number of significant
figures (two).
1.5 m2
Calculating with Scientific Notation

Example addition problem in scientific notation
Evaluate
5.2 x 10-2
+ 1.82 x 10-3
Change to same
exponents
10th’s place
5.2 x 10-2
+ 0.182 x 10-2
5.382 x 10-2
5.4 x 10-2

Example subtraction problem in scientific notation
Evaluate
7.0 x 105
- 5.2 x 104
Change to same
exponents
10th’s place
7.0 x 105
- 0.52 x 105
6.48 x 105
6.5 x 105
10th’s place
Calculating with Scientific Notation

Example multiplication problem in scientific notation
Evaluate
Multiply the
coefficients
2 x 10 -3 Add the
x 3.6 x 10 4 exponents
7.2 x 10 1
7 x 10 1
Round to 1 sig. fig.

Example division problem in scientific notation
Evaluate
Divide the
coefficients
1.2 x 10 7 Subtract the
4.7 x 10 2 exponents
= 0.255319148 x 10 5
Round to
2 sig. figs.
2.6 x 10 4
Calculating with Significant Figures
Problem
5.43
0.023
Calculator
answer
236.0869565
Rounded
answer
Scientific
notation
240
(2 sf)
2.4 x 102
236 x 0.2
47.2
50
(1 sf)
5 x 101
0.300 + 9 250
9250.3
9250
(1 p)
9.25 x 103
236.04 - 9.2
226.84
226.8 (0.1 p)
2.268 x 102
6.70 x 10 4 - 3.6 x 10 -2* 66999.964
67000
(1 p)
6.7 x 104
2.04 x 10 -3 x 1.2 x 10 2
0.24
(2 sf)
2.4 x 10-1
* Change to 67000 x 100 – 0.036 x 100
0.2448