Download Systematic errors

Uncertainty and Measurements
There are errors associated with any measurement.
Random error
Systematic error
Human error
Random error – These errors can be caused by a variety
of sources:
1.
Inability to read a measurement beyond the smallest division
2.
Limited accuracy of measuring instruments
3.
Fluctuations in temperature/pressure
Errors in Measurements
Systematic errors – Usually caused by a faulty instrument
or improper instrument calibration. This results in
measurements that are always too large or always to small.
Human error – These are errors caused by carelessness or
simply making a mistake.
These errors are completely UNACCEPTABLE!!
We need to account for random error by recognizing the
limitations of our measuring instrument.
We need to spot systematic errors and attempt to correct
the problem.
Accuracy vs. Precision
Accuracy is an indication of how close our measurement is
to the known accepted value or the “right answer”.
Precision is an indication of the repeatability of a
measurement.
Precise
Not Accurate
Not Precise
Not Accurate
Very Precise
Very Accurate
Percent error is a measure of accuracy.
% error =
Accepted value – Experimental value
Accepted value
x 100
Percent Difference
We use percent difference as a measure of precision. This
gives us an indication of how close two experimental values
are to one another.
Exp value 1 – Exp value 2
% Difference =
x 100
Exp value 1 + Exp value 2
2
Significant Figures
We use a certain number of digits when we
report the results of a measurement. The
number of digits tells us the degree of
precision or how close a particular instrument
can measure.
The last digit in a number is always assumed
to be uncertain.
If we measure something and get 10.34 cm,
we are really saying that we are sure about
the 10.3 cm, but we are estimating the 4.
So our measurement is between 10.33 cm
and 10.35 cm. Or, 10.34cm (+/-) 0.01 cm
Sig Fig Rules
1. Leading zeros are never significant.
2. Zeros that occur between non- zero
digits are significant.
3. Trailing zeros are only significant
when there is a decimal point.
How many significant figures???
1,156,000
218
0.0068
20
20.
20.0
4
3
2
1
2
3
More Sig Fig Rules
When you multiply or divide, the result must
contain the same number of significant figures as the
number with the least number of significant figures
used in the calculation.
What your calculator says
(2.45)(2.5)
(2.786)
= 2.198492462
The number with the least number of sig figs is “2.5”
(2 sig figs). Therefore, answer can only have 2 sig figs
So, we round the answer to 2.2
More Sig Fig Rules
When you add or subtract the result can only contain
as many decimal places as the number with the fewest
number of decimal places used in the calculation.
2.345 + 4.67 – 1.23 + 4.5 = 10.285
What the calculator says
4.5 only has one decimal place, so the answer can
only have one decimal place.
So, we round the
answer to 10.3
Note: In this example, the answer has
3 sig figs and 4.5 only has 2 sig figs.
When doing calculations, it is always a good idea to
keep a few extra sig figs in the calculations, and round
off the final answer to the correct number of significant
figures or decimal places.
When rounding numbers, if the last number is a 4 or
below, round down. For a 5 or above, round up.
3.4545 = 3.46 (3 sig figs)
3.4435 = 3.44 (3 sig figs)
Scientific Notation
Scientific notation is a shorthand way to express very
large or very small numbers using the exponent 10 raised
to some integer power.
Remember any number raised to the zero power is
one. 100 = 1
To express a number in scientific notation, write the
number as a number between 1 and 9.999999999….
and then multiply the number by 10 raised to some
integer power. Positive exponents are for big numbers
>1 and negative exponents are for small numbers <1.
Express numbers in scientific notation using 3 significant figures
234,357,432 =
2.34 x 108
Moved 8 decimal places to the
left. So I need to multiply the new
number (2.34) by 108 so it is
equivalent to the original number
0.000000001230045600 =
Moved 9 places to the right
1.23 x 10-9
Metric Prefixes
Table 2-2 on page 35 of the text has complete table.
There are several that we use over and over and over….
Prefix
Abbreviation
Value
mega
M
106
kilo
k
103
centi
c
10-2
milli
m
10-3
micro
m
Greek letter
“mu”
10-6
Scientific Notation on the Calculator
DO NOT USE THE MULTIPLICATION KEY “x” and the
number 10 when you enter a number in scientific notation.
You will almost always make a mistake and be off by
a factor of 10.
Most calculators have an “EE” key. Some calculators
have an “EXP” key.
1. Enter the number between 1 and 9.999 (ex.. 2360000) and then push the EE key.
2. Your calculator will show 2.36E
3. Enter the exponent (ex. 6).
4. Your calculator will show 2.36E6
5. This is equivalent to: 2.36 x 106 = 2360000