From 1d658cbfff511880644fbba8e329e6acf320516c Mon Sep 17 00:00:00 2001 From: "W. Trevor King" Date: Thu, 16 May 2013 01:31:13 -0400 Subject: [PATCH] calibcant/discussion.tex: Add tab:calibcant:stability Showing that spring constant measurements are reliable over day-sized time intervals. --- src/calibcant/discussion.tex | 51 ++++++++++++++++++++++++++++++++++-- src/root.bib | 35 +++++++++++++++++++++++++ 2 files changed, 84 insertions(+), 2 deletions(-) diff --git a/src/calibcant/discussion.tex b/src/calibcant/discussion.tex index cb186dd..522e909 100644 --- 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 d93800e..fce2aff 100644 --- 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, -- 2.26.2