Curve Fitting Errors

Question

I am trying to fit a part of a dataset to the function $$ae^{-b\sqrt{x-c}}$$. My friend was using GNUPlot to this and got reasonably good results. However when I try to this with Mathematica :

NonlinearModelFit[Tmean, a*Exp[-b*Sqrt[x - c]], {{a, 45}, {b, 2}, {c, 1.8}}, x]).

-188.006 E^{1.17551\sqrt{-<<18>> + x}}

Data= {{2.76134, 6.134}, {2.7552, 6.081}, {2.7491, 6.224}, {2.74301, 6.288}, {2.73696, 6.629}, {2.73093, 6.486}, {2.72493, 6.719}, {2.71895, 6.989}, {2.713, 7.076}, {2.70708, 7.293}, {2.70118, 7.152}, {2.69531, 7.37}, {2.68946, 7.626}, {2.68364, 7.59}, {2.67784, 7.74}, {2.67207, 7.723}, {2.66633, 7.606}, {2.66061, 7.855}, {2.65491, 7.668}, {2.64923, 8.03}, {2.64359, 8.16}, {2.63796, 7.932}, {2.63236, 8.174}, {2.62678, 8.335}, {2.62123, 8.171}, {2.6157, 8.292}, {2.61019, 8.247}, {2.60471, 8.363}, {2.59925, 8.541}, {2.59381, 8.357}, {2.5884, 8.347}, {2.583, 8.404}, {2.57763, 8.414}, {2.57229, 8.363}, {2.56696, 8.509}, {2.56166, 8.556}, {2.55638, 8.297}, {2.55112, 8.562}, {2.54588, 8.335}, {2.54066, 8.359}, {2.53546, 8.581}, {2.53029, 8.311}, {2.52514, 8.308}, {2.52, 8.276}, {2.51489, 8.337}, {2.5098, 8.075}, {2.50473, 8.098}, {2.49968, 8.119}, {2.49465, 7.891}, {2.48964, 7.682}, {2.48465, 7.519}, {2.47968, 7.593}, {2.47473, 7.482}, {2.4698, 7.567}, {2.46489, 7.053}, {2.46, 7.213}, {2.45513, 7.077}, {2.45028, 6.784}, {2.44545, 6.81}, {2.44063, 6.796}, {2.43584, 6.643}, {2.43106, 6.633}, {2.42631, 6.323}, {2.42157, 5.994}, {2.41685, 6.198}, {2.41214, 6.063}, {2.40746, 5.797}, {2.40279, 5.808}, {2.39815, 5.684}, {2.39352, 5.506}, {2.38891, 5.307}, {2.38431, 5.531}, {2.37974, 5.207}, {2.37518, 5.193}, {2.37063, 5.247}, {2.36611, 4.985}, {2.3616, 4.882}, {2.35711, 4.696}, {2.35264, 4.501}, {2.34819, 3.966}, {2.34375, 4.399}, {2.33932, 4.297}, {2.33492, 4.154}, {2.33053, 4.011}, {2.32616, 3.853}, {2.3218, 3.654}, {2.31746, 3.521}, {2.31314, 3.505}, {2.30883, 3.519}, {2.30454, 3.627}, {2.30026, 3.49}, {2.296, 3.412}, {2.29176, 3.624}, {2.28753, 3.43}, {2.28332, 3.345}, {2.27912, 3.503}, {2.27494, 3.2}, {2.27077, 3.169}, {2.26662, 3.015}, {2.26249, 2.968}, {2.25836, 2.76}, {2.25426, 2.523}, {2.25017, 2.624}, {2.24609, 2.567}, {2.24203, 2.654}, {2.23798, 2.525}, {2.23395, 2.413}, {2.22993, 2.279}, {2.22593, 2.262}, {2.22194, 2.47}, {2.21796, 2.288}, {2.214, 2.324}, {2.21006, 2.204}, {2.20612, 2.224}, {2.20221, 2.316}, {2.1983, 2.183}, {2.19441, 2.323}, {2.19053, 2.085}, {2.18667, 1.981}, {2.18282, 2.047}, {2.17898, 2.096}, {2.17516, 2.062}, {2.17135, 2.091}, {2.16756, 2.057}, {2.16377, 2.06}, {2.16, 2.048}, {2.15625, 2.231}, {2.1525, 1.956}, {2.14877, 2.066}, {2.14506, 2.082}, {2.14135, 2.019}, {2.13766, 2.184}, {2.13398, 2.043}, {2.13031, 1.94}, {2.12666, 1.985}, {2.12302, 2.172}, {2.11939, 2.175}, {2.11577, 2.161}, {2.11217, 2.184}, {2.10857, 2.082}, {2.10499, 1.994}, {2.10143, 2.081}, {2.09787, 2.066}, {2.09433, 2.239}, {2.0908, 2.167}, {2.08728, 2.192}, {2.08377, 2.207}, {2.08027, 2.3}, {2.07679, 2.342}, {2.07331, 2.346}, {2.06985, 2.37}, {2.0664, 2.391}, {2.06297, 2.436}, {2.05954, 2.321}, {2.05612, 2.433}, {2.05272, 2.528}, {2.04933, 2.714}, {2.04594, 2.569}, {2.04257, 2.682}, {2.03921, 2.621}, {2.03587, 2.899}, {2.03253, 3.089}, {2.0292, 3.136}, {2.02589, 3.259}, {2.02258, 3.158}, {2.01929, 3.147}, {2.016, 3.295}, {2.01273, 3.279}, {2.00947, 3.294}, {2.00622, 3.47}, {2.00298, 3.481}, {1.99975, 3.544}, {1.99652, 3.882}, {1.99332, 4.078}, {1.99012, 3.939}, {1.98693, 4.153}, {1.98375, 4.395}, {1.98058, 4.377}, {1.97742, 4.471}, {1.97427, 5.051}, {1.97113, 5.072}, {1.968, 5.454}, {1.96488, 5.711}, {1.96178, 6.012}, {1.95868, 6.169}, {1.95559, 6.691}, {1.95251, 7.021}, {1.94944, 7.428}, {1.94638, 7.902}, {1.94333, 8.348}, {1.94028, 9.005}, {1.93725, 9.836}, {1.93423, 10.504}, {1.93122, 11.491}, {1.92821, 12.563}, {1.92522, 13.621}, {1.92224, 14.784}, {1.91926, 16.286}, {1.91629, 17.614}, {1.91334, 19.183}, {1.91039, 20.854}}

when a should be around $50$, b around $5$-$7$ and c around $1.88$. We both have the same initial guesses. What does the << >> mean and why am I getting such bad results?

Data:

Mathematica:

EDIT: Added Data

Answer 1

Your problem is that the Square root function is returning complex numbers (because its argument is negative). You can bypass this:

nlm = NonlinearModelFit[data, a*Re[Exp[-b*Sqrt[x - c]]], {{a, 45}, {b, 2}, {c, 1.8}}, x]

which gives

Normal[nlm]

4.73332 Re[E^(0.781405 Sqrt[-2.3218 + x])]

However, your data is not well fit by this function for reasons that b.gatessucks pointed out in a comment.