Download Significant Figures…………

Experimental Error
Experimental Error
Significant Figures
“The Rules for Zeros”
“The last significant digit in a measured quantity
always has associated uncertainty”
•Zeros are significant when they occur in the
middle of a number
KNOWN
Example:
1.2034 x 105
e.g. 9.63 mL
•Zeros are significant when they occur at the end
of a number on the right-hand side of a decimal
point
* Interpolation = Estimation of readings
to the nearest tenth of distance
between scale readings
5
Example:
1.2340 x 105
NOTE: Assumes that zero is accurate estimate.
Experimental Error
Experimental Error
Arithmetic and Significant Figures
Significant Figures
1. Addition/Subtraction
“Some numbers are exact - with an infinite
number of unwritten significant digits”
Express numbers with the same exponent
Significant figures are limited to the leastcertain number
Example:
5 people
1.234 x 10-5
+ 6.78 x 10-9
= 5.0 people
= 5.00 people
1.234
x 10-5
+ 0.000678 x 10-5
1.234678 x 10-5
= 5.000 people
Significant
Significant Figures…………
Number of all certain digits plus the first uncertain digit
31
30.5 mL
30.67 mL
30.6 mL
30.68 mL
30.7 mL
30.69 mL
3 sig figs
31
Significant Figures…………
Rules for determining the number of sig figs
1) Disregard all initial zeros
30
30
Answer: 1.235 x 10-5
4 sig figs
0.03068 L
2) Disregard all final (terminal) zeros unless they
follow a decimal point
3) All remaining digits including zeros between nonzero digits are significant
Tip: Express data in scientific notation to avoid confusion
in determining whether terminal zeros are significant
2.0 L or 2000 mL best expressed as 2.0 X 103 mL
2 sig figs ? sig figs
2 sig figs
1
Significant Figures…………
Numerical Computation Conventions
When adding and subtracting, express the numbers to
the same power of ten.
3.4
2.432 X 106 = 2.432 X 106
0.020
6.512 X 104 = 0.06512 X 106
7.31
-1.227 X 105 = -0.1227 X 106
10.73
= 2.37442 X 106
Answer: 10.7
Answer: 2.374 X106
For Multiplication or Division, round the answer to contain the same
# of sig figs as the original number with the smallest # of sig figs.
Chapter 4 - Lecture # 3
Overview
(next 3 lectures)
Basic Tools (continued)
•Microsoft Excel - Analysis of Data
Statistics
•Descriptive Statistics
•Statistical Tests
•Least Squares and Calibration
Introduction to Statistics
“Statistics give us tools to accept conclusions that have a
high probability of being correct and to reject conclusions
that do not.”
1. Descriptive Statistics - Describes estimate of an
actual value, and the uncertainty associated with
estimate
2. Statistical Tests - Allow us to compare estimates and
uncertainties, and make conclusions about these
comparisons.
Arithmetic and Significant Figures
1. Addition/Subtraction
The number of significant figures in the answer may
exceed or be less than that in the original data.
If the numbers being added do not have
the same number of significant figures, we
are limited by the least-certain one.
5.345 (4 sig figs)
+ 6.728 (4 sig figs)
12.073 (5 sig figs)
7.26 X 1014
- 6.69 X 1014
0.57 X 1014
Good Laboratory Practice (GLP)
“embodies a set of principles that provides a
framework within which laboratory studies are
planned, performed, monitored, recorded, reported
and archived…GLP helps assure regulatory
authorities that the data submitted are true reflection
of the results obtained during the study and can
therefore by relied upon when
making…assessments.”
Introduction to Statistics
Statistics deals only with RANDOM ERROR,
not systematic (determinate) error
Measurements affected by random error will
approach a Gaussian Distribution as the
number of measurements increases.
2
Class Experiment………..
Gaussian Distribution
Flip a coin 10 times and tabulate the results as follows:
“Bell-Shaped Curve”
Number of Tails
|||| |
Number of
Measurements
Number of Heads
||||
14
12
10
6
4
2
0
0
1
2
3
4
5
6
7
8
9
Systematic Error
(Less Accurate)
Number of
Measurements
8
Number of
Measurements
Frequency (Number of Students)
Quantitative Analysis Class Coin Flip Experiment
More Random Error
(Less Precise)
10
Number of Heads (out of 10)
Normal Distribution
Introduction to Statistics
Number of Standard Deviations from Mean
Population vs. Sample
Frequency
A population is the whole set of points
measurable. A sample is a subset of the
population that typically gets measured. The
population mean is denoted as μ and the
standard deviation as σ. The sample mean is
denoted as x and the standard deviation as s.
…“Population” is an infinite number of the
same measurement
-4
-3
-2
-1
0
1
2
3
4
Number of SD’s
By definition, 68% of all measurements
will fall within 1 SD of the Mean
95 % of measurements fall within
2 SD’s of Mean
Number of Standard Deviations from Mean
Frequency
Frequency
Number of Standard Deviations from Mean
68 %
16 %
-4
-3
-2
-1
0
1
Number of SD’s
2
95 %
2.5 %
16 %
3
4
-4
-3
-2
-1
0
2.5 %
1
2
3
4
Number of SD’s
3
Descriptive Statistics
Descriptive Statistics
Mean (Average)
x
=
Σi xi
n
Σi xi = the sum of the measurements
n = number of measurements
10
9
8
7
6
5
4
3
2
1
Mode
(most common measurement)
“Outlier”
24.0
24.1
24.2
24.3
24.4
24.5
24.6
24.7
24.8
24.9
25.0
25.1
25.2
25.3
25.4
25.5
25.6
25.7
25.8
25.9
26.0
26.1
26.2
26.3
26.4
26.5
26.6
26.7
26.8
26.9
27.0
Number
Example: You calibrate a 25-mL Class A pipet by dispensing 25 mL of water
into a pre-weighed volumetric flask, and “weigh by difference” to measure the
mass of water and subsequently determine the actual volume of water. You
do this thirty (30) times, and obtain the data shown graphically below.
Descriptive Statistics
Example: You calibrate a 25-mL Class A pipet by dispensing 25 mL of water
into a pre-weighed volumetric flask, and “weigh by difference” to measure the
mass of water and subsequently determine the actual volume of water. You
do this thirty (30) times, and obtain the data shown graphically below.
24.7, 24.9, 25.0, 25.0, 25.1, 25.1, 25.1, 25.2, 25.2, 25.2,
25.2, 25.2, 25.3, 25.3, 25.3, 25.3, 25.3, 25.3, 25.3, 25.4,
25.4, 25.4, 25.4, 25.5, 25.5, 25.6, 25.6, 25.7, 25.8, 26.5
n = 30
x = 24.7+24.9+25.0+25.0+25.1+25.1+25.1+25.2 ...+26.5
30
=25.3266667
=25.3
4
Descriptive Statistics
10
9
8
7
6
5
4
3
2
1
Mean ( x ) = 25.3
Approximates µ
µ is the mean for
the populatoin
(“an infinite set of
data”)
Descriptive Statistics
Mean /Average (x)
Not as precise? How
do we quantify?
24.0
24.1
24.2
24.3
24.4
24.5
24.6
24.7
24.8
24.9
25.0
25.1
25.2
25.3
25.4
25.5
25.6
25.7
25.8
25.9
26.0
26.1
26.2
26.3
26.4
26.5
26.6
26.7
26.8
26.9
27.0
Number
Example: You calibrate a 25-mL Class A pipet by dispensing 25 mL of water
into a pre-weighed volumetric flask, and “weigh by difference” to measure the
mass of water and subsequently determine the actual volume of water. You
do this thirty (30) times, and obtain the data shown graphically below.
10
9
8
7
6
5
4
3
2
1
Descriptive Statistics
~ 68% of the measurements
-s +s
Approximates σ
σ is the standard
deviation for the
population (“an
infinite set of data”)
24.0
24.1
24.2
24.3
24.4
24.5
24.6
24.7
24.8
24.9
25.0
25.1
25.2
25.3
25.4
25.5
25.6
25.7
25.8
25.9
26.0
26.1
26.2
26.3
26.4
26.5
26.6
26.7
26.8
26.9
27.0
Number
Standard Deviation (s)
“measures how closely the data are clustered
about the mean”
Small s = Precise
10
9
8
7
6
5
4
3
2
1
Example: You calibrate a 25-mL Class A pipet by dispensing 25 mL of water
into a pre-weighed volumetric flask, and “weigh by difference” to measure the
mass of water and subsequently determine the actual volume of water. You
do this thirty (30) times, and obtain the data shown graphically below.
Median = middle number in a series of measurements
(ordered low to high); less prone to outliers than mean.
24.7, 24.9, 25.0, 25.0, 25.1, 25.1, 25.1, 25.2, 25.2, 25.2,
25.2, 25.2, 25.3, 25.3, 25.3, 25.3, 25.3, 25.3, 25.3, 25.4,
25.4, 25.4, 25.4, 25.5, 25.5, 25.6, 25.6, 25.7, 25.8, 26.5
24.0
24.1
24.2
24.3
24.4
24.5
24.6
24.7
24.8
24.9
25.0
25.1
25.2
25.3
25.4
25.5
25.6
25.7
25.8
25.9
26.0
26.1
26.2
26.3
26.4
26.5
26.6
26.7
26.8
26.9
27.0
Number
Example: You calibrate a 25-mL Class A pipet by dispensing 25 mL of water
into a pre-weighed volumetric flask, and “weigh by difference” to measure the
mass of water and subsequently determine the actual volume of water. You
do this thirty (30) times, and obtain the data shown graphically below.
Descriptive Statistics
= 25.3 + 25.3
2
= 25.3
Descriptive Statistics
Range = difference between the lowest and highest
Data Set 1
24.7, 24.9, 25.0, 25.0, 25.1, 25.1, 25.1, 25.2, 25.2, 25.2,
25.2, 25.2, 25.3, 25.3, 25.3, 25.3, 25.3, 25.3, 25.3, 25.4,
25.4, 25.4, 25.4, 25.5, 25.5, 25.6, 25.6, 25.7, 25.8, 26.5
Range = 26.5 - 24.7 = 1.8
Range = 25.8 - 24.7 = 1.1
Outlier?
Data Set 2
24.2, 24.3, 24.4, 24.5, 24.6, 24.7, 24.8, 24.9, 25.0, 25.1,
25.1, 25.2, 25.2, 25.3, 25.3, 25.3, 25.4, 25.4, 25.5, 25.5,
25.6, 25.7, 25.8, 25.9, 26.0, 26.1, 26.2, 26.3, 26.4, 26.5
Range = 26.5 - 24.2 = 2.3
Descriptive Statistics
Standard Deviation (s)
“measures how closely the data are clustered
about the mean”
s
=
Σi(xi - x)2
n-1
*Note: “n-1” is referred to as the “degrees of freedom”
“Degrees of Freedom”
=
The pieces of independent
information available.
…we know average (so not “available”), so only n-1 available.
5
Descriptive Statistics
Descriptive Statistics
Standard Deviation (s)
“measures how closely the data are clustered
about the mean”
Example: You calibrate a 25-mL Class A pipet by dispensing 25 mL of water
into a pre-weighed volumetric flask, and “weigh by difference” to measure the
mass of water and subsequently determine the actual volume of water. You
do this thirty (30) times, and obtain the data shown graphically below.
(30)-1
=0.32156228
=0.3
Number
s = (24.7-11.3)2+(24.9-11.3)2+(25.0-11.3)2+(25.0-11.3)2+…
10
9
8
7
6
5
4
3
2
1
The mean and
standard deviation
end at the same
decimal place!
25.3 ± 0.3 mg
Descriptive Statistics
Descriptive Statistics
s
Σi(xi - x)2
n-1
= 0.63278258
= 0.6
Variance
“square of the standard deviation”
=
25.3 ± 0.6 mg
s2
Not as
precise!
%CV =
x 100
x
s
μ
y =
-σ +σ
~ 95% of the measurements
1
e-(x-μ)
2/2σ2
σsqrt2pie
x-μ
σ
(Table 4-1: Area)
z
+2 s
y
24.0
24.1
24.2
24.3
24.4
24.5
24.6
24.7
24.8
24.9
25.0
25.1
25.2
25.3
25.4
25.5
25.6
25.7
25.8
25.9
26.0
26.1
26.2
26.3
26.4
26.5
26.6
26.7
26.8
26.9
27.0
Number
s/x
Gaussian Curve and
Probability
Standard Deviation (s)
“measures how closely the data are clustered
about the mean”
25.3 ± 0.6 mg ~95% of the data
-2 s
Σi(xi - x)2
n-1
(“Relative Standard Deviation”)
Descriptive Statistics
10
9
8
7
6
5
4
3
2
1
=
Coefficient of Variation (%CV)
68% of the data
24.0
24.1
24.2
24.3
24.4
24.5
24.6
24.7
24.8
24.9
25.0
25.1
25.2
25.3
25.4
25.5
25.6
25.7
25.8
25.9
26.0
26.1
26.2
26.3
26.4
26.5
26.6
26.7
26.8
26.9
27.0
Number
Standard Deviation (s)
“measures how closely the data are clustered
about the mean”
10
9
8
7
6
5
4
3
2
1
~68% of the data
0.3 0.3
24.0
24.1
24.2
24.3
24.4
24.5
24.6
24.7
24.8
24.9
25.0
25.1
25.2
25.3
25.4
25.5
25.6
25.7
25.8
25.9
26.0
26.1
26.2
26.3
26.4
26.5
26.6
26.7
26.8
26.9
27.0
24.7, 24.9, 25.0, 25.0, 25.1, 25.1, 25.1, 25.2, 25.2, 25.2,
25.2, 25.2, 25.3, 25.3, 25.3, 25.3, 25.3, 25.3, 25.3, 25.4,
25.4, 25.4, 25.4, 25.5, 25.5, 25.6, 25.6, 25.7, 25.8, 26.5
=
≈
x-x
s
-∞
x
+∞
6
Descriptive Statistics
z ≈ 24.7-25.3
0.3
z≈2
Area from μ to z=2
= 0.4773
-s = -0.3
*Note: Area from μ to - ∞
= 0.5000
+s = +0.3
Area from z=2 to - ∞
= 0.5000-0.4773
=0.0227
=2.27%
Probability
of ≤ 24.7?
Statistical Tests
Rejection of Measurements….
Q-Test for “Bad Data”
In the case where you suspect something went wrong,
you use the Q-test. (Minimum of 3 measurements)
Qcalculated
Q-test = |suspect value - nearest value| / total range
Calculate Qexp.
If Qexp ≥ Qcritical then reject.
=
gap/range
Range = “total spread of the data”
Gap =
d
X1 X2 X3 X4 X5
(from Table 4.1)
24.0
24.1
24.2
24.3
24.4
24.5
24.6
24.7
24.8
24.9
25.0
25.1
25.2
25.3
25.4
25.5
25.6
25.7
25.8
25.9
26.0
26.1
26.2
26.3
26.4
26.5
26.6
26.7
26.8
26.9
27.0
Number
Mean ( x ) = 25.3
10
9
8
7
6
5
4
3
2
1
“difference between questionable point
and nearest value”
X6
Qexp = d/w
If Qcalculated > Qtable then data point should be discarded.
w
Q-Test for “Bad Data”
10
9
8
7
6
5
4
3
2
1
“Bad” data?
“Outlier”
Qcalculated
=
gap/range
24.7, 24.9, 25.0, 25.0, 25.1, 25.1,
25.1, 25.2, 25.2, 25.2, 25.2, 25.2,
25.3, 25.3, 25.3, 25.3, 25.3, 25.3,
25.3, 25.4, 25.4, 25.4, 25.4, 25.5,
25.5, 25.6, 25.6, 25.7, 25.8, 26.5
gap
Qcalculated =
(26.5-25.8)
(26.5-24.7)
24.0
24.1
24.2
24.3
24.4
24.5
24.6
24.7
24.8
24.9
25.0
25.1
25.2
25.3
25.4
25.5
25.6
25.7
25.8
25.9
26.0
26.1
26.2
26.3
26.4
26.5
26.6
26.7
26.8
26.9
27.0
Number
Example: You calibrate a 25-mL Class A pipet by dispensing 25 mL of water
into a pre-weighed volumetric flask, and “weigh by difference” to measure the
mass of water and subsequently determine the actual volume of water. You
do this thirty (30) times, and obtain the data shown graphically below.
= 0.7/1.8
= 0.4
0.4 > 0.298
Discard?
n
3
4
5
6
7
8
9
10
15
20
25
30
Qtable
(95% confidence)
0.970
0.829
0.710
0.625
0.568
0.526
0.493
0.466
0.384
0.342
0.317
0.298
7
We can assign a confidence level
to our measurements…..
If we can accept a 5% error level, we can say that
these values are reported with a 95% confidence limit.
For a small number of measurements, we must
consult a t value table ()
• choose confidence level
• determine number of degrees of freedom (n-1)
• plug t value into the following equation:
Rectangular Distribution
Frequency
Frequency
Bimodal Distribution
49
50
51
52
53
54
55
56
57
58
59
60
49
Measurements
50
51
52
53
54
55
56
57
Measurements
Systematic Error
Frequency
Frequency
Skewed Distribution
46
48
50
52
Measurements
54
56
-4
-2
0
2
4
6
8
Measurements
8