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