Download Measurement of length - Southern Adventist University

Quantifying
measurement error
Blake Laing
Southern Adventist University
I
uncertainty
Measurements have no meaning without a quantified experimental
error/uncertainty
Uncertainty
𝜌𝑎𝑙𝑙𝑜𝑦 = 13.8
𝑔
± 0.1 3
𝑐𝑚
𝜌𝑔𝑜𝑙𝑑 = 15.5 ± 0.1 𝑔/𝑐𝑚3
George: 𝜌 = 15 ± 1.5 𝑔/𝑐𝑚3
What can he conclude?
Martha: 𝜌 = 13.9 ± 0.2 𝑔/𝑐𝑚3
What can she conclude?
Uncertainty due to measurement error
A personal error is a mistake.
No need to quantify, but we should be able to
recognize mistaken data
Two other measurement errors which are not
accidental
Random error
Causes repeated measurements to be different
Causes wide “margin of error”
Systematic error
Repeated measurements are consistent
All measurements are shifted in a predictable way
Can be recognized and corrected by “shifting back”
Accuracy is not precision
Precision
“how close to each other”
Accuracy
“how close to expected”
Random error
Different error every time
Limits precision
Systematic error
Same error every time
Limits accuracy
Random error
Quantified by statistics
Statistics in real life
If weekly average > 120 mg/dL, must
take insulin
Same breakfast: toast with almond
butter
Same breakfast: toast with almond
butter
Morning blood glucose
mass concentration
glucose
day
(mg/dL)
1
110
2
119
Different results
3
7
13
123
129
90
Summary
Max: 129 mg/dL
Min: 90 mg/dL
Mean: 109.5 mg/dL
Random or systematic error?
That “distance” is called the standard deviation 𝜎
68% within 𝑥 − 𝜎, 𝑥 + 𝜎 or 𝑥 ± 𝜎
A 68% of past measurements were within 𝑥 ± 𝜎
There was a 68% probability for each measurement
to be within 𝑥 ± 𝜎.
68% confidence interval
I can say with 68% confidence that the next
measurement will be within 𝑥 ± 𝜎.
The precision of each measurement is
quantified by 𝜎
Systematic error: comparison to known
68% probability that absolute error < 𝜎
3
Frequency
Gaussian distribution, or “bell-curve”
68% within some “distance” of mean
4
2
0
85
1
90
0
2
2
1
95 100 105 110 115 120 125
Blood glucose (mg/dL) (bin maximum)
1
130
0
135
Random error/precision in one
measurement is quantified by 𝜎
68% confidence interval
I can say with 68% confidence that the next
measurement will be within 𝑥 ± 𝜎.
means that 68% of repeated measurements will
be within one standard deviation of the average
95% confidence interval
95% of previous measurements within
𝑥 ± 2𝜎.
I can say with 95% confidence that the next
measurement will be within 𝑥 ± 2𝜎.
means that 95% of repeated measurements will
be within two standard deviation of the average
Good way to state precision of instrument
Precision of the mean quantified by α
Let’s take 100 measurements!
Will standard deviation decrease?
Shouldn’t we know mean value more precisely?
Precision of the mean, or “error of the mean”
is quantified by the standard error.
𝜎
← 𝑠𝑡𝑎𝑦𝑠 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒
𝛼=
𝑁 ← 𝑔𝑒𝑡𝑠 𝑙𝑎𝑟𝑔𝑒𝑟
68% probability that the mean of a many more
measurements would be within 𝑥 ± 𝛼
If there were no systematic error…
the mean of many more measurements would be
equal to the true value
There is a 68% probability that the true value is
within 𝑥 ± 𝛼
More common: 95% confidence int. 𝑥 ± 𝟐𝛼
Measure with sanity
Blood glucose concentration
glucose
(mg/dL)
day
1
110
Maximum:
129
2
119
Minimum:
90
3
129
Mean:
4
109
5
6
68% CI for the next measurement
109.5 ± 10 𝑚𝑔/𝑑𝐿=110 ± 10 𝑚𝑔/𝑑𝐿
mg/dL
95% CI for the next measurement
mg/dL
110 ± 20 𝑚𝑔/𝑑𝐿
109.5
mg/dL
σ:
10
mg/dL
123
N:
16
Days
106
α:
4
mg/dL
Measure with sanity
Blood glucose concentration
glucose
(mg/dL)
day
1
110
Maximum:
129
2
119
Minimum:
90
3
129
Mean:
109.5
68% CI for the next measurement
109.5 ± 10 𝑚𝑔/𝑑𝐿=110 ± 10 𝑚𝑔/𝑑𝐿
mg/dL
95% CI for the next measurement
mg/dL
110 ± 20 𝑚𝑔/𝑑𝐿
mg/dL 68% CI for mean value
4
109
σ:
10
mg/dL
5
123
N:
16
Days
6
106
α:
4
110 ± 4 𝑚𝑔/𝑑𝐿
95% CI for mean value
110 ± 8 𝑚𝑔/𝑑𝐿
mg/dL
Systematic error
Comparison to expectation
Systematic error: compare to “known”
Suppose that medical laboratory glucometer measures
123.2 ± .2 𝑚𝑔/𝑑𝐿 (68% CI)
Compare home device to this
112 ± 10 𝑚𝑔/𝑑𝐿 (68% CI)
Absolute error:
𝐸−𝑇
= 112 − 123.2 mg/dL
=11.2 mg/dL
=10 mg/dL
Compared to what?
COMPARE ABS. ERR. TO EXPECTATION
𝐸−𝑇
𝑃𝐸 =
× 100%
𝑇
112 − 123.2
𝑃𝐸 =
× 100%
123.2
11.2
𝑃𝐸 =
× 100% = 9. 09%
123.2
COMPARE TO RANDOM ERROR IN HOME
DEVICE
Is the absolute error large
compared to the standard error?
Then the mean for the home
device has a significant
systematic error.
How many standard errors?
𝐴𝑏𝑠. 𝐸𝑟𝑟.
=1
𝛼
May not need to be calibrated
Quantifying measurement error
Problem
Source of
error
Measure
Relative measure
Poor
accuracy
Systematic
error
Absolute error:
Percent error
Poor
precision
Random error One measurement: std.
dev. σ
x − x𝑡ℎ𝑒𝑜𝑟𝑦
Mean value: std. err. 𝛼
x − x𝑡ℎ𝑒𝑜𝑟𝑦
× 100%
x𝑡ℎ𝑒𝑜𝑟𝑦
Percentage std. err.
𝛼
× 100%
x
Notes
Experimental notes
Advice from previous students
“Take the time to get well acquainted with standard deviation and standard error on
your first few labs... you'll be seeing them all year!”
“Learn how to quantify measurements in the beginning - believe me.
I didn't fully learn how to use the tools of the trade till the beginning of the second
semester, and it would have paid to learn it first.”
“Know the significant figures for sure: locking in the understanding at the start of the
semester saves you A LOT of points.”
Notes from the reader
Need precise, quantitative answers to questions
Less wordy “fluff”, more equations/numbers.
In every questions it is implied to use or refer to the appropriate “tool for the job”,
such as percent error.
Need careful articulation of words to be able to have a carefully-articulated
understanding.
Common mistakes on significant figures
use calculated standard error to determine correct sig figs on the mean
When calculating percent error, watch for the loss of sig figs when subtracting
Because I always back up
my argument with an
incisive quantitative
analysis.
Quantifying
measurement error
necessary to form quantitative conclusions
See Dr. Laing bleed for
science
Was that glucometer really so bad?
Class data
Expected value: 140 mg/dL
168
Frequency
Mean value: about 267±1 mg/dL
88
66
68% CI
21
0
0
0
0
3
2
0
0
2
157 175 192 209 227 244 261 279 296 313 331 348 365
Glucose concentration (bin maximum) (mg/dL)
A faulty assumption is a systematic error
Two hours after breakfast
concentration
(mg/dL)
Trial
1
104
N=
32
2
99
Max=
110
mg/dL
3
106
Min=
83
mg/dL
4
102
5
99
Mean=
6
94
σ=
6
mg/dL
7
94
α=
1
mg/dL
94.41mg/dL
Aqueous glucose vs whole blood
blood has a pH of about 7.4 (basic)
Distilled water has a pH < 7 (acidic)
Different density
Standard deviation about half of aqueous
glucose solution
What is the 95% CI for each
measurement?
What is the 95% CI for mean?
Reader notes
It appears that a number of people don’t have a solid grasp on what the 68% confidence intervals xav
± σn m or xav ± αn mean.
CI for each measurement: 𝑥 ± 𝜎 = (𝑥 − 𝜎, 𝑥 + 𝜎) is a range of possible values of the
measurement
About 68% of the measurements were within this range.
Implies that each measurement had a 68% probability of being within that range
Implies that it is exceedingly unlikely to be due to random error if one additional measurement is 10 𝜎 away
CI for mean value 𝑥 ± 𝛼 = (𝑥 − 𝛼, 𝑥 + 𝛼)
Implies that if there is no systematic error, there is a 68% probability that the true value is within this range
Less wordy, more equations/numbers. In every questions it is implied to use or refer to the
appropriate “tool for the job”, such as percent error.
Statements like “Systematic error is 180” are concerning.
Need careful articulation of words to be able to have a carefully-articulated understanding.
Feel free to use pencil on everything but raw data
Question 1
Does the standard deviation get much smaller as more measurements are taken?
How about the standard error? Demonstrate by making a table of the standard
deviation and standard error for 5, 25, and 50 data points using your data, and for all
points of the class data. Would σ or α be more appropriate to describe the precision
of an instrument?
Number
Standard deviation σ
Standard error α
5
10ish
4ish
25
10ish
2ish
50
10ish
2ish
300
10ish
1ish or less
Notes
Post-analysis