chapter 6 statistical analysis of experimental data · 2005-01-24 · statistical analysis of...
TRANSCRIPT
Chapter 6Statistical Analysis of Experimental Data
Measures of Central TendencyGiven a set of data: { x1, x2, x3, … xN }
►Mean (average):
►Median:Put data in ascending order, pick the ______ value. (If even-numbered, pick the _______ of the two middle values.)
=µ
Standard Deviation of Population
►Deviation: How far a point is from __
►Standard deviation of a population, σ, given by Eq. 6.5:
►Population: _______________
=σ
=id
“Normal” or Gaussian Distribution
( ) =xf
“Normal” Distribution Information:
►____ of population values are within ±__of mean value, µ
►____ of population values are within _____ of mean value, µ
►_____ of population values are within ____ of mean value, µ
Note – the “normal” distribution is NOT the only oneused in engineering (binomial, Poisson, Weibull, etc.)
Standard Deviation of Sample
►Standard deviation of n samples, S, given by Eq. 6.6
►sample is a ______ of all possible values
=id
=S
SampleSample meanmean
Exercise #1aWhat is the expected valueand the 95% confidence uncertainty interval for this data?
xTensile
Sample Strength, MN/m2
1 2.482 2.763 2.964 2.725 2.626 2.65
Exercise #1bTensile
Sample Strength, MN/m^2 di (di)^21 2.482 2.763 2.964 2.725 2.626 2.65
Avg: Sum:
Exercise #1c=XS
∈x (95% confidence interval)
∈x
≤≤ x
Rejecting Questionable Data – p.142
Modified Thomson τ technique1. Determine mean and std. dev. SX2. Find ______ deviation from mean,3. If then _____ data point4. If data point ________________ mean
and standard deviation5. ______ process with new mean and
standard deviation
x( )xxii −=δ
ixrejectreject
Table 6.8 Values of
Thompson’s τ
Problem #20.102 sec0.106 sec0.100 sec0.101 sec0.105 sec0.119 sec0.103 sec0.104 sec
Exercise #2a
Find the expected value and the standard deviation for these experimental time constant data
=x=XS Should any of these
experimental data values be eliminated?
Exercise #2bLargest deviation is
=iδ
From Table 6.8 we find for n = __, τ = _____
=⋅ XSτ
Problem #20.102 sec0.106 sec0.100 sec0.101 sec0.105 sec0.103 sec0.104 sec
Exercise #2c
Find the expected value and the standard deviation for these experimental time constant data
=x Should any of these experimental data values be eliminated?
=XS
Exercise #2bLargest deviation is
=iδ
From Table 6.8 we find for n = __, τ = _____
=⋅ XSτ
Correlation of Experimental Data
►Least-squares fit to a straight line is the most common
Fits to other curves are often more appropriate, i.e., exponential
►Some data must be transformed before fitting to straight line (section 6.6.3)
►Excel’s Trendline function can be used
Use Excel to plot X-Y dataTime Voltage
1 351.5 232 173 124 95 76 68 4
10 3
R2 – “Goodness of Fit”
( )( )
( ) ( )∑∑
∑
==
=
−−
−−=
n
1i
2
i
n
1i
2
i
n
1iii
xy
yy*xx
yyxxr 1r1 xy ≤≤−
Excel - Trendline►in Excel -
select the data seriesclick Chart\Add Trendline
►The numerical values for the fitted line can be displayed by selecting Display Equation on Chart under
Chart\Add Trendline\Options►The R2 (“goodness of fit) value can also be
displayed
Warning!
►Note that Trendline WILL fit a line of the desired type to the data - even if it is not appropriate!
►The data we plotted are NOT well represented by a straight line
RR22 should be close to 1!should be close to 1!
Plot X-Y data with uncertainties
Time Voltage Uncert.1 35 1.5
1.5 23 1.022 17 0.783 12 0.584 9 0.465 7 0.386 6 0.348 4 0.26
10 3 0.22
Add Error Bars for Uncertainty
Select Custom
Plot with Error Bars
0
5
10
15
20
25
30
35
40
0 1 2 3 4 5 6 7 8 9 10 11Time, sec
Vol
tage
What do you need to know how to do in Excel?
► Remove the default gray background from Excel graphs
► Add error bars to plots of experimental data
► Change axis label locations (should almost always be along the bottom of the graph)
► Re-label “x” and “y” values after a trendline curve fit
► Reduce number of significant digits displayed
► Remove labels from legend box (without removing the data from the graph!)
► Rescale axis to fit data (no long gaps where no data occurs)
► Create logarithmic axes with gridlines
► Re-size the fonts in a graph
► Cut and paste (as a picture) from Excel to Word