Transcript Error, Accuracy, Precision, and Standard Deviation Notes

Error, Accuracy,
Precision, and
Standard Deviation
Notes
Errors
Two types of errors: random and systematic
• Random errors: uncontrollable events, like
air currents, temperature variations, and
electrical variations.
The error can be minimized by taking a large
number of measurements (100 or more).
Random errors are recognized by the fact that
the values are BOTH above and below the
true value
• Systematic errors: controllable events.
Find the error, fix the error and repeat the
experiment.
Types of systematic errors:
Equipment : the equipment is worn, out of
calibration or broken, fix the equipment and
repeat the measurements.
Technique: some part of the procedure is
incorrect. Examples: not looking at eye level
at a graduated cylinder or balance, cars are not
released at the same time.
Bias: eliminating a number because you do not
like it. Unless a known error has occurred (can
eliminate then) you cannot throw out a value
because it is different.
• Systematic errors are recognized because all
of the values are EITHER above or below the
true value.
• All of the numbers we measure need to be
evaluated and accuracy, precision and
standard deviation are the tools we use to do
this.
Accuracy
• How close a value is to the true or accepted
value (an average can be compared to the
accepted value)
• Only one measurement is necessary for
calculating an accuracy but many numbers is
preferred and the accuracy of the average is
then taken.
Accuracy:
% error = True value – experimental value
True value
* The experimental value can be the average
Desired value is zero.
x100
Precision
• How close a set of values are to each other.
• Requires at least 2 values; more are better.
% difference = High – Low x 100
Average
* Desired value is zero.
Acceptable ranges are arbitrary but for Physics
we will use
0-1% Excellent
1-7% Good
7-15% Fair
15 and up Redo (Unacceptable)
• What is the precision is high, how can it be
fixed?
• Look to see if there is an outlier in the set
and statistically try to eliminate it.
Standard Deviation (S)
(sample standard deviation)
• Population Standard Deviation
(σx on the calculator)
The standard deviation of the entire population
of data
• Sample Standard Deviation
(Sx on the calculator)
The standard deviation of a small sample of the
whole population – this is all that we are able to
collect.
√ Σ(x-ave)2
n-1
√ - the square root of the entire thing
Σ – sum of
x – a value
ave – average of all values
n – the number of values
√ Σ(x-ave)2
n-1
Take the value, subtract the average and square
this number. (Do this for all values.)
Add all of these together.
Subtract one from number of values.
Divide your sum by this difference.
Take the square root of the whole thing.
Example
Values
2.54
2.55
2.56
2.57
2.58
Ave = 12.80/5
= 2.560
Example
Values
value – average
2.54
-.02
2.55
-.01
2.56
0
2.57
.01
2.58
.02
Ave = 12.80/5
= 2.560
difference2
0.0004
0.0001
0
0.0001
0.0004
Example
Values
value – average
difference squared
2.54
-.02
0.0004
2.55
-.01
0.0001
2.56
0
0
2.57
.01
0.0001
2.58
.02
0.0004
12.80/5
0.0010
Average = 2.560
n-1 = 5-1 = 4
0.0010/4 = 0.00025
√ 0.00025 = 0.016
Standard deviation values are hard to interpret
(2.560 + 0.016)
Hard to say from the numbers whether they
are good or not.
Therefore, we use Relative Standard Deviation.
• Relative Standard Deviation
= s/average x 100
0.016 x100 = 0.63%
2.560
Easier to interpret: 2.560 + 0.63%
very close
If you have a value that does not fit the set,
you must statistically show if it is an
outlier.
Two methods to do so are:
1. 2 standard deviations
2. q test
2 Standard Deviations
If have a set of values, is 2.79 an outlier?
2.54
2.55
2.56
2.79
2.57
2.58
2.54
2.55
2.56
2.79
2.57
2.58
Average = 2.598
Sx = 0.09
2.598 + .18 = 2.78
2.598 -.18 = 2.42
Range of values 2.42 to 2.78
The value 2.79 would be an outlier because it is
beyond 2 standard deviations from the average.
Q test
Questionable value – closest value numerically
Range of all values
= q value
Compare the results to the Q values, if your
questionable value is larger than the 95%
confidence Q value, then it is an outlier.
2.79 – 2.58 = 0.84
2.79 – 2.54
Number of values
3
4
5
6
7
8
9
10
95% confidence Q value
0.943
0.754
0.640
0.564
0.510
0.469
0.438
0.412
Using the Calculator for Standard
Deviation
Plug the values into the calculator
Hit STAT button
Select 1: Edit
Enter the list of data
Hit STAT button
Select CALC menu
Select 1: 1-Var Stats
Hit Enter
Avg 2.56 (¯x)
Sum 12.8 (Σx)
Sx = 0.0158 = 0.016 (sf of standard deviation values is first nonzero digit unless it is a one then keep 2 digits)