Transcript Precision & accuracy_12Feb13

Measurements
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Accuracy and Precision
Significant Figures
Rounding
Combinations of like units
– Area, Volume
• Combinations of unlike units
– Density, speed
• Dimensional analysis
– Converting units, cancellation of like units
Measurements
• All science depends on measurements
– “If you can’t measure it, you can’t control it”
• Our interest is in physical measurements
– Fundamentals: Mass, Length, time, temperature
– Combinations: Area, Volume, density, speed
• Measurement requires some rules
– Deciding what’s relevant, Significant Figures
• How many decimals to believe, Rounding
– Manipulating data
• Average, range, deviation, combining measurements
Accuracy and Precision
• Accuracy is the degree of conformity of a measured or
calculated quantity to its actual (true) value.
• Precision, also called reproducibility or repeatability, is
the degree to which further measurements or
calculations show the same or similar results.
• A measurement can be accurate but not precise; precise
but not accurate; neither; or both.
• A result is valid if it is both accurate and precise
• Related terms are error (random variability) and bias
(non-random or directed effects) caused by a consistent
and possibly unrelated factor.
Accuracy
• Degree of error achieving the measurement goal
• Your personal “cubit” measurement accuracy
was somewhat limited by the tool used and
ability to read it (about 1/16 inch).
• Cubit measured in class has not changed much
since biblical times, still about 18 inches, so the
average value has remained relatively accurate
over hundreds (possibly thousands) of years.
Good accuracy
This example shows good accuracy, but low precision
Precision
• How well multiple measurements agree with one
another to provide a consistent value. (e.g. tight
grouping, low dispersion, “all together” series of
events).
• The “cubit” is not a very precise measure of
distance, since it varies between observers
using the same definition. No two people are the
same, so length data is dispersed. (e.g. poor
groupings, inconsistent results).
Target analogy
This example has high precision, but poor accuracy
Accuracy versus Precision
Big Slip Video
• Is this example precise or accurate?
• Experiment repeated by Myth Busters
– Performed experiment into a lake
• Hitting water less damaging than dirt
• Show Video ….
Ancient Dimensions … the "Cubit"
Defined as distance between elbow (upright on table) to farthest fingertip
Pyramid of Giza, height when built
Noah's Ark length, Biblical reference
Person measured
1 Penka
2 Danielle
3 Jason
4 Rey
5 Mohammed
6 Dinesh
7 James
8 Jose
9 Michael
10 Olga
Our Class
inch/cubit
18.0
15.0
18.5
18.0
20.0
19.5
18.5
19.0
20.5
17.0
280
300
Inch Deviation
from Average
0.40
3.40
0.10
0.40
1.60
1.10
0.10
0.60
2.10
1.40
Sum of Squares
N = number
N-1
Sum / (N-1)
Std Dev = sqrt
Average
Maximum
Miimum
Range
18.4
20.5
15.0
5.5
If "True Value" =
18.0
inches, historical value
"error" for class is
2.2%
(True- Measured)/true
Cubits
Cubits
Variance, or
Deviation^2
0.160
11.560
0.010
0.160
2.560
1.210
0.010
0.360
4.410
1.960
22.400
10
9
2.5
1.6
Reviewing Cubit Data
• Our class data is a useful example
• Accuracy was good, agrees with history
– Historical value 18 inches
• Precision not as good, a lot of variation
– Size varies greatly between individuals
– Average is more consistent
Statistical Terms
• Mean value
– Sum of measurements divided by their number (18.4)
• Mode
– The most common or “popular” value (18.5 twice)
• Range
– The largest versus smallest measured value (5.5)
• Deviation (from the average)
– How each measurement differs from average
• Standard Deviation
– A mathematical way to minimize influence of “flyers”
– Our result was 1.6, a lot less than range of 5.5
– “RMS” (Root Mean Square) used a lot in Engineering
Deviation & Variation
• Variance is the average of the squared
differences between data points and the mean.
Variance is tabulated in units squared.
• Standard deviation is the square root of the sum
of variances, and measures the spread of data
about the mean, with the same units.
• Said more formally, the standard deviation is the
root mean square (RMS) deviation of values
from their arithmetic mean.
Standard Deviation
Standard Deviation, why bother?
• Range a poor indicator of accuracy
– One bad measurement controls the range
• Averaging scheme redefines error
– RMS (root mean squared) is common tool
– Moves error to an average value basis
– Suppresses random error contribution
Significant Figures
• How accurate is your data?
– Output cannot be more accurate than input
• Can you believe your calculator?
– Unintended (unbelievable?) precision
• How to multiple, divide, add & subtract
– Different rules apply
• Exact values, sig fig rules do NOT apply
– A dozen is exactly 12
– 1 inch defined as exactly 2.54 cm
Significant Figures
• “Sig Figs” = need to establish values having realistic influence
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1cup of sugar to 3 of flour does not imply exact ratio of 0.3333333
Unintended accuracy termed “superfluous precision”
Need to define actual measurement precision intended
“Cup of flour” in recipe could be +/- 10% or 0.9 to 1.1 cup
• Can’t have more Sig-Figs than least accurate measurement
– Final “Sig Fig” called “Uncertainty Digit” … least accurately known
– adding .000001 gram sugar to 1.1 gram flour = 1.1 gram mixture
• Exact Values (definitions) exempt from Sig Fig restrictions
– 1 inch ≡ 2.54 cm (accuracy “infinite”, NOT limited to 3 sig figs)
– 1 dozen ≡ 12, quartet ≡ 4, 1 yard ≡ 3 feet … NO tolerances apply
Significant Figures Accounting
• Leading zeros don’t count
– Often just a scale factor (0.000001 = microgram)
• Middle zeros between numbers always count
– 1.001 measurement has 4 decades of accuracy
• Trailing zeros MIGHT count
– YES if part of measured or defined value, 1.001
– YES if placed intentionally, 7000 grains = 1 pound
– NO if to right of non-decimal point
• 1,000 has 1 sig-fig … but 1,000.0 has 5 sig-figs
– NO if only to demonstrate scale
• Carl Sagan’s “BILLIONS and BILLIONS of stars”
– Does NOT mean 1,000,000,000 + 1 = 1,000,000,001
– Need to consider context
Sig-Fig Examples
• Zeros in front
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0.093827 has 5 sig-figs
0.0008 has 1 sig-fig
0.012 has 2 sig-figs
Zeros in front of a number are NOT significant
• Zeros between
– 60.8 has 3 significant figures
– 39008 has 5 sig-figs
– Zeros between numbers always significant
• Zeros at end … potential ambiguity
– 35.00 has 4 sig-figs
– 8,000.000 has 7 sig-figs
– 1,000 could be 1 or 4 … if 4 intended, best to write 1.000E4
more Sig-Fig Accounting
• Addition & Subtraction
– Lowest Significant Figure determines outcome
– 1.01 + 1.00000001 = 1.01
• Multiplication & Division
– Lowest Significant Figure determines outcome
– 1.01 x 1.0000001 = 1.01
• Round-Off
– Calculations can yield more sig-figs than justified
– Must reduce answer to lowest sig-fig component
more Sig-Fig Accounting
• Round-Off
– Calculations can yield more sig-figs than justified
– Must reduce result to lowest sig-fig component
• Methodology
– If value beyond last sig-fig is ≥5, round UP
• For 3 sig fig 5.255123 becomes 5.26
– If value beyond last sig-fig is <5, round OFF
• For 3 sig figs, 5.254459 becomes 5.25
Sig-Fig Examples
• Zeros between
– 60.8 has 3 significant figures
– 39008 has 5 sig-figs
• Zeros in front
– 0.093827 has 5 sig-figs
– 0.0008 has 1 sig-fig
– 0.012 has 2 sig-figs
• Zeros at end
– 35.00 has 4 sig-figs
– 8,000.000 has 7 sig-figs
– 1,000 could be 1 or 4 … if 4 intended, best to write 1.000E4
Rounding Off
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A big source of error in calculations
Students often simplify too soon
Errors interact and make things worse
Round off at the end of the calculation
– Apply rules at the end
Don’t round off until the end
Example shows common error
A
B
C=A*B
Initial value
reference value
interim result
Good
20
5,280
105,600
D
E
F=C*E/D
reference value (exact)
reference value
Interim result
4.00
1.06
27,984
quart / gallon
quart / liter
feet / liter
4.00
1.00
26,000
G
H
I
J
K=F*G*H / (I*J)
reference value (exact)
reference value (exact)
reference value (exact)
reference value (exact)
final result
12
2.54
100
1,000
8.530
inches / foot
cm / inch
cm/meter
km / meter
km / liter
12
2.5
100
1,000
7.800
Rounded to 3 sig fig
Error
8.53
miles / gallon
feet / mile
feet / gallon
NOT so good
20
5,200
104,000
Moral: round at the end, not along the way !
7.80
8.55%
Non-Linear representations
• Exponential Growth (or decline)
– Changes associated with exponent of 10 value
– Example: exp of 2100x, exp of 31000x
• Moore’s Law, Chain Reaction of Uranium
• Logarithmic Scale
– Some differences too large to put on a linear scale
• Hearing, visual acuity, earthquakes, concentration of ions
– Logarithm scale “compresses” scales
• “decibel” for sound, “pH” for acid concentration
• Richter scale for earthquakes
– Richter 9 (SF 1906) is 1000x that of Richter 6 (mild shake)
Significant Figure Summary
• Leading zeros don’t count
– Often just a scale factor (0.000001 = microgram)
• Middle zeros between numbers always count
– 1.001 measurement has 4 decades of accuracy
• Trailing zeros MIGHT count
– YES if part of measured or defined value, 1.001
– YES if placed intentionally, 7000 grains = 1 pound
– NO if to right of non-decimal point
• 1,000 has 1 sig-fig … but 1,000.0 has 5 sig-figs
– NO if only to demonstrate scale
• Carl Sagan’s “BILLIONS and BILLIONS of stars”
– Does NOT mean 1,000,000,000 + 1 = 1,000,000,001
Sig-Fig Summary
• Addition & Subtraction
– Lowest Significant Figure determines outcome
– 1.01 + 1.00000001 = 1.01
• Multiplication & Division
– Lowest Significant Figure determines outcome
– 1.01 x 1.0000001 = 1.01
• Round-Off
– Calculations can yield more sig-figs than justified
– Must reduce answer to lowest sig-fig component