calibcant/discussion.tex: Add tab:calibcant:stability

Showing that spring constant measurements are reliable over day-sized
time intervals.

diff --git a/src/calibcant/discussion.tex b/src/calibcant/discussion.tex
index cb186dd0772ac2ac532c7ef25378f6eb1b0eb253..522e9094ace518e610cd4bead57d3d7bea949674 100644 (file)
--- a/src/calibcant/discussion.tex
+++ b/src/calibcant/discussion.tex
@@ -83,8 +83,10 @@ thermal expansion as the room warms up).  However TODO
 For example, on a recent calibration run\footnote{2013-02-07T08-20-46}
 I measured $\sigma_p=35.68\pm0.87\U{V/$\mu$m}$,
 $T=298.151\pm0.033\U{K}$, and $\avg{V_p^2}=96.90\pm0.99\U{mV$^2$}$,
-which gives $\kappa=54.1\pm2.7\U{mN/m}$.  The uncertainty
-contributions from each term are
+which gives $\kappa=54.1\pm2.7\U{mN/m}$.  These numbers are very
+similar to those obtained with a different cantilever from the same
+batch measured a month later (\cref{tab:calibcant:stability}).  The
+uncertainty contributions from each term are
 \begin{align}
   4\p({\frac{\sigma_{\sigma_p}}{\sigma_p}})^2 &= 2.38\E{-3}\U{N$^2$/m$^2$} \\
   \p({\frac{\sigma_{T}}{T}})^2 &= 1.29\E{-8}\U{N$^2$/m$^2$} \\
@@ -117,6 +119,51 @@ the calculated $\kappa$.
   \end{center}
 \end{figure}
 
+\begin{table}
+  \begin{center}
+    \begin{tabular}{c c r l r l}
+      \toprule
+      \multicolumn{2}{c}{Timestamp:} &
+      \multicolumn{2}{c}{2013-03-03T16-37-12} &
+      \multicolumn{2}{c}{2013-03-04T12-21-54} \\
+      \midrule
+      Quantity & Units & Mean & Std.~Dev. & Mean & Std.~Dev. \\
+      \midrule
+      $\sigma_p$    & \bareU{V/$\mu$m} & 46.22   & 0.76  & 41.30   & 0.21 \\
+      $T$           & \bareU{K}        & 296.302 & 0.021 & 294.272 & 0.022 \\
+      $\avg{V_p^2}$ & \bareU{mV$^2$}   & 108.3   & 1.1   & 105.5   & 2.16 \\
+      $\kappa$      & \bareU{mN/m}     & 67.3    & 2.5   & 65.6    & 1.5 \\
+      \bottomrule
+    \end{tabular}
+    \caption{Measured spring constant calibration parameters (mean and
+      standard deviation) for a single cantilever on two consecutive
+      days.  The measured parameters have changes slightly because the
+      laser alignment and buffer temperature drift over time, but the
+      measured $\kappa$ are not significantly different ($p=0.9$, as
+      measured with a two-tailed Welch's
+      $t$-test\citep{welch38,welch47}).\label{tab:calibcant:stability}}
+    % Using Welch's t test
+    %     http://en.wikipedia.org/wiki/Welch%27s_t_test
+    %   from math import sqrt  # using Python in the following
+    %   x1 = 67.3
+    %   v1 = 2.5**2
+    %   n1 = 10    # sortof :p
+    %   x2 = 65.6
+    %   v2 = 1.5**2
+    %   n2 = 10
+    %   t = (x1 - x2) / sqrt(v1/n1 + v2/n2)
+    %     = 1.8
+    % Degrees of freedom with the Welch-Satterthwaithe equation
+    %   df = (v1/n1 + v2/n2)**2 / ( (v1/n1)**2/(n1-1) + (v2/n2)**2/(n2-1) )
+    % Test null hypothesis that means are equal (two-tailed t)
+    %     http://en.wikipedia.org/wiki/Two-tailed_test
+    %   from scipy.stats.mstats import betai
+    %   p = betai(0.5*df, 0.5, float(df) / (df + t*t))
+    %     = 0.09
+    % With arrays, we could have used scipy.stats.mstats.ttest_ind().
+  \end{center}
+\end{table}
+
 \subsection{Archiving experimental data}
 \label{sec:calibcant:discussion:data}
 

diff --git a/src/root.bib b/src/root.bib
index d93800eb2c93bfd8ff7cb28a583655f4c876425f..fce2aff89c47f1fe426fd67a3e59eb4a2d67d4b1 100644 (file)
--- a/src/root.bib
+++ b/src/root.bib
@@ -104,6 +104,7 @@
 @string{Biochem = "Biochemistry"}
 @string{BBABE = "Biochimica et Biophysica Acta (BBA) - Bioenergetics"}
 @string{BIOINFO = "Bioinformatics (Oxford, England)"}
+@string{Biomet = "Biometrika"}
 @string{BPJ = "Biophysical Journal"}
 %string{BPJ = "Biophys. J."}
 @string{BIOSENSE = "Biosensors and Bioelectronics"}
@@ -1060,6 +1061,7 @@
 @string{YWei = "Wei, Yen"}
 @string{ALWeisenhorn = "Weisenhorn, A.~L."}
 @string{JWeissenbach = "Weissenbach, J."}
+@string{BLWelch = "Welch, Bernard Lewis"}
 @string{GWen = "Wen, G."}
 @string{MWen = "Wen, M."}
 @string{JWetter = "Wetter, J."}
@@ -3579,6 +3581,39 @@
     jstor_issuetitle = ""
 }
 
+@article{ welch38,
+  author = BLWelch,
+  title = {The significance of the difference between two means when
+    the population variances are unequal},
+  year = 1938,
+  month = feb,
+  journal = Biomet,
+  volume = 29,
+  number = "3-4",
+  pages = {350--362},
+  keywords = "Population",
+  issn = "0006-3444",
+  url = "http://www.jstor.org/stable/2332010",
+  language = "eng",
+}
+
+@article{ welch47,
+  author = BLWelch,
+  title = {The generalization of {Student's} problems when several
+    different population variances are involved},
+  year = 1947,
+  month = jan,
+  journal = Biomet,
+  volume = 34,
+  number = "1-2",
+  pages = {28--35},
+  keywords = "Population",
+  issn = "0006-3444",
+  url = "http://www.ncbi.nlm.nih.gov/pubmed/20287819",
+  jstor_url = "http://www.jstor.org/stable/2332510",
+  language = "eng",
+}
+
 @article { granzier97,
     author = HLGranzier #" and "# MSKellermayer #" and "# MHelmes #" and "#
         KTrombitas,